# Slope Intercept Form With Undefined Slope Seven Precautions You Must Take Before Attending Slope Intercept Form With Undefined Slope

Day By Day Addendum for MATH 206

Spring 2009

Day 1

Activity: Go over syllabus. Booty roll. Functions activities.

Goals: Review advance objectives: aggregate data, abridge information, and achieve inferences.

I acquire disconnected this advance into three “units”. Assemblage 1 (Days 1 through 14) is about basal functions. Unit 2 (Days 15 through 32) is about derivatives and their uses. Assemblage 3 (Days 33 through 56) is about affiliation and multi-variable functions.

I acquire to be acknowledged in this course, you allegation READ the text carefully, alive abounding convenance problems. Our activities in chic will sometimes be altered to the arrangement you convenance and/or about-face in for the arrangement allocation of your grade; instead they will be for compassionate of the basal principles. For example, on Day 16 you will draw sample graphs and derivatives, then try to reconstruct the aboriginal graph. This is article you would never do in practice, but which I anticipate will authenticate several acquaint for us. In these notes, I will try to point out to you aback we’re doing article to accretion understanding, and aback we’re accomplishing article to accretion skills.

Each semester, I am aghast with the baby cardinal of acceptance who arise to me for advice alfresco of class. I doubtable some of you are ashamed to seek help, or you may feel I will anticipate beneath of you for not “getting it” on your own. Personally, I anticipate that if you are disturbing and cannot achieve faculty of what we are doing, and don’t seek help, you are cheating yourself out of your own education. I am actuality to help you apprentice mathematics. Please ask questions aback you acquire them; there is no such affair as a brainless question. About added acceptance acquire the aforementioned questions but are additionally too shy to ask them in class. If you are still too shy to ask questions in class, arise to my arrangement hours or achieve an appointment.

I acquire you get out of article what you put into it. Complete rarely will accession abort a chic by accessory every day, accomplishing all the assignments, and alive abounding convenance problems; about bodies abort by not applying themselves abundant – either through missing classes, or by not allocating enough time for the material. Obviously I cannot acquaint you how abundant time to absorb anniversary anniversary on this class; you allegation all acquisition the adapted antithesis for you and your life’s priorities. One aftermost allotment of advice: don’t procrastinate. I believe mathematics is abstruse best by circadian exposure. Cramming for exams may get you a casual grade, but you are abandoned cheating yourself out of compassionate and learning.

Today we will activate by discussing functions. Absolutely simply, a action is a rule. From an input value, a action gives the achievement value. The set of accessible inputs is alleged the domain, and the set of achievement ethics is alleged the range. The ascribe bulk is sometimes alleged the complete value, and the achievement bulk the abased value. One of the arch goals of mathematics is to archetypal complete apple phenomena with functions. Accordingly it is important for us to be accustomed with their uses and roles.

Throughout the course, we will try to attending at functions from four altered viewpoints. Abstracts will be presented to us in collapsed form, graphical form, algebraic form, or verbally. It will be up to us to actuate the best adapted adjustment of anecdotic the function. A common delusion that I achievement to allay is that equations are alike with functions. Equations are abandoned one adjustment of anecdotic functions. Our altercation makes an honest accomplishment to affectation functions for us in all four forms.

Today and tomorrow I would like to assay functions graphically, verbally, and algebraically. We will activate with a altercation of a academic flight amid two cities. Afresh I will acquire you assignment in groups.

In these notes, I will put the circadian assignment in gray background.

Activity: Graphical Description.

The bulk of a car goes bottomward as the car gets older, so we can anticipate of the bulk of a car, V, in bags of dollars, as a action of the age of a car, a, in years. We acquire V = f(a).

1) Draw a accessible blueprint of V adjoin a. You don’t allegation scales on the axes, but characterization anniversary arbor as V or a.

2) What does the annual f(5) = 6 acquaint you about the bulk of the car? Be abiding to use units for 5 and for 6. Characterization this as a point on your graph, and mark the 5 and the 6 on the adapted axes.

3) Put a vertical ambush of 15 on your blueprint of the function. Explain the acceptation of this vertical ambush in agreement of the bulk of the car.

4) Put a accumbent ambush of 10 on your blueprint of the function. Explain the acceptation of this accumbent ambush in agreement of the value of the car.

Goals: (In these notes, I will abridge anniversary day’s action with a annual of goals for the day.)

Introduce the course, and the abstraction of a function.

Skills: (In these notes, anniversary day I will assay abilities I acquire you should acquire afterwards alive the day’s activity, annual the adapted sections of the text, and practicing contest in the text.

· Use the “Guess and Check” adjustment of botheration solving. This abode is the aspect of the authentic method. There is annihilation bad about academic in adjustment to learn. The bigger guessers, of course, tend to get quicker results, but if you acquire adapted accoutrement to appraise your guesses, afresh alike poor guesses can be aesthetic adequately. By the way, your calculator in this chic will about use this assumption and assay adjustment to break equations. It’s aloof that your calculator works a bit faster than you can. Another accompanying abstraction is appliance assay numbers to alpha a process. That is, conceivably authoritative up a sample situation will advice you see what is action on. I animate you to use this admission often; it is the best basal lesson my adviser able me in alum school. He acclimated to say, “Start with a simple example.” That about meant bold some specific ethics for some variables, and alive from there to acquire the problem.

· Physical modeling. Abounding times actuality ashore on a complete apple botheration can be alleviated by clay the bearings with concrete items or by other simulations. Of advance many situations are infeasible; you can’t fly airplanes to simulate scheduling airline routes, but you can use adapted diagrams or tokens representing airplanes. Sometimes actually physically apery article will get you over a brainy block.

Reading: (The annual mentioned in these addendum refers to what annual you should do for the abutting day’s material.)

Section 1.1. Bring your calculator to chic every day. It will be an invaluable tool.

Day 2

Activity: Today we abide to assay functions graphically, verbally, and algebraically. I will acquire you assignment in groups again. For anniversary activity, aback you arise up with a solution, put it on the lath and we will assay addendum before chic ends.

Activity 1: Algebraic and Exact Description.

From a 24 inch length of string, anatomy two geometric shapes, a amphitheater and a square. Your ambition is to actualize the smallest complete breadth amid by both shapes. To activate this activity, I advance aggravating some specific values. For example, what if the cord is cut in half? How abundant will the square contain, and how abundant will the amphitheater contain? You will allegation to apperceive the formulas for areas of circles and squares. Afresh try a different breakdown, like ¼ of the cord for the aboveboard and ¾ for the circle. My acceptance is that if you can bulk out the areas for specific numbers, you can bulk it out for almost values, like x and 24 – x.

Activity 2: Graphical and Algebraic Description.

With our calculators, we acquire the accoutrement accessible to assay limits. Specifically, we can hone our intuition about this important affair in calculus.

1) Annual for n = 1, 10, 100, 1000, etc. What seems to be accident to the values? Are the abutting a limit? Can you explain (prove) it?

2) Annual for n = 1, 10, 100, 1000, etc. What seems to be accident to the values? Are the abutting a limit? This complete we see actuality is a complete important complete in calculus and mathematics. We will appointment it and abstraction it in added detail later.

3) Accede this series of terms: 1, 1/2, 1/3, 1/4, etc. Add alternating agreement to get a new alternation of apportioned sums. That is, acquisition 1, 1 1/2, 1 1/2 1/3, 1 1/2 1/3 1/4, etc. What seems to be accident to this sum?

4) Now try this series of agreement and echo the sums you did in 3). 1/2, 1/4, 1/8, 1/16, 1/32, etc. (These are successively abate admiral of two.) What seems to be accident to this sum?

5) Appliance , acquisition the banned as you admission x = 2 from the adapted and left. (Approaching from the adapted agency appliance ethics aloof greater than 2 and abutting from the larboard agency appliance ethics aloof abate than 2.) Additionally acquisition the bulk adapted at x = 2.

6) Echo 5) appliance .

In these notes, I will put sections of computer commands in boxes, like this one. I’m absolutely acquisitive that you already are absolutely accustomed with this machine, accepting already taken MBA I. In these notes, I accredit to the calculator as the TI-83, but the aforementioned commands administer to the TI-84.

Y = is activate on the top row of buttons, on the left. You admission equations into whichever Y-variable you appetite to use. Be authentic to enter what you want, that is, pay absorption to parentheses, typos, etc! Anniversary Y-variable whose = assurance is accent will be graphed aback the GRAPH button is pressed. In addition, if any plots at the top of the affectation are highlighted, those too will be plotted, whether you advised them to be or not!

GRAPH is activate on the top row of buttons, on the right. This button toggles amid the abstracts / afterwards admission screens and the graphing window. To leave the graphing window, columnist any key, or columnist QUIT, (found by acute 2nd MODE).

WINDOW is activate on the top row of buttons, added from the left. This opens the windows ambience screen, which tells you the ambit and characteristics of the accepted graphing window. We will mostly change abandoned 4 items: Xmin, Xmax, Ymin, and Ymax. If you like, you may boggle with the added settings.

TRACE is on the top row of buttons, added from the right. This key puts a cursor on the graphing window on one of your y-variables / functions. You may advance adapted and larboard arrow to move alongside on the alleged curve, or up and bottomward arrow to baddest added curves (if you have entered added than one y-variable.) Be careful: TRACE is dependent on the accepted window settings. If you allegation complete values, afterwards acute TRACE, blazon the x-value you allegation evaluated. TRACE will annual the anatomic bulk exactly.

ZoomFit (Zoom 0) Abounding times, you do not apperceive which is the best examination window. If you aboriginal specify the horizontal endpoints in the WINDOW settings awning (Xmax and Xmin), afresh you can columnist ZoomFit (under ZOOM menu, annual 0) to acquire the calculator acquisition the adapted Ymin and Ymax values. This action is absolutely handy; I use it a lot myself.

ZStandard (Zoom 6) If you are in adulation with the numbers amid –10 and 10, you should use ZStandard in the ZOOM menu. Otherwise, you may acquisition this key useless!

Goals: Appreciate the dynamics of collaboration. Acquire the altered botheration analytic strategies. Assay some basal limits.

Skills:

· Collaboration. One of the bigger problems I see assay afterwards assay with algebraic acceptance is their abhorrence to allocution about their algebraic frustration. Allocution about things with anniversary other! If you are too abashed to allocution to me, (or if you acquire added affidavit for not absent to babble with me) at atomic allocution to your peers. Sometimes simply adage article out loud will accessible up doors you ability not acquire otherwise opened, or an abrupt acknowledgment may affect accession else’s imagination. Of advance this doesn’t beggarly that one actuality in a accumulation of botheration solvers should do all the work; but alike if only one accumulation affiliate ‘gets’ a solution, the administration is benign to all concerned. The bedfellow gets to absolutely apprentice the abstraction as he/she is adapted to explain it; the others get to see a band-aid they missed. Ideally, anybody should be able to explain a accumulation solution; until you can explain the solution, you haven’t quite accepted the method.

· Evaluate banned numerically and graphically. By appliance numbers afterpiece and closer to the bulk in question, whether it is apprenticed or infinite, your calculator or computer can advice you to appraise limits. There is a caution, however: you allegation still use your analytic abilities to abstain actuality fooled. You may have empiric this in Exercise 2 of Action 2. Some banned are accessible to appraise (simply bung in and evaluate) while others are added complicated (the apportioned sums we saw in Contest 3 and 4 of Action 2 are about absolutely difficult to appraise or to alike adjudge if they converge.) One of the “big ideas” in calculus is differentiation, and we allegation to be adequate with banned to acquire derivatives. Accession of the “big ideas” in calculus is integration, and we allegation to acquire banned such as apportioned sums to acquire integrals.

· Recognize the harmonic series. Alike admitting agreement in a alternation may be accepting smaller and smaller, their apportioned sums may not assemble to a apprenticed number. The arrangement in Exercise 3 of Activity 2 aloft is alleged the harmonic alternation and demonstrates this appearing paradox. Abounding apportioned sums will converge, though, as you saw in Exercise 4 of Action 2, which is an archetype of a geometric series.

· Understand the analogue of the cardinal e (approximately 2.7182818). Exercise 2 of Action 3 is the analogue of the cardinal e, which we will use afresh and afresh in calculus. Remember, though, e is aloof a number, annihilation more. Don’t be abashed of it!

Reading: Breadth 1.2.

Day 3

Activity: Using the Olympic data, fit a corruption band to adumbrate the 2004 and 2008 race results. Interpreting Ante of Change.

Unit 1 is about architecture up a library of functions. To be an effective algebraic modeler, we allegation acquire a alive ability of basal functions. These accommodate beeline functions, exponential functions, polynomials, and combinations of these. The simplest and best acclimated is the beeline action (it is the abject for the acquired we will adept in Chapter 2). You should already apperceive a lot about beeline functions. Aloof to achieve abiding we all acquire the aforementioned background, today we will assay linear functions in detail.

To begin, I will annual the advantageous forms for beeline equations.

1) Slope/Intercept form: . In this form, m is the abruptness and b is the y-intercept.

2) Point/Slope form: . In this form, m is the slope, and (x1, y1) is an ordered brace on the line.

3) Two Point form: . In this form, (x1, y1) and (x2, y2) are two sets of ordered pairs on the line.

I will use do the Celsius/Fahrenheit about-face in chic to authenticate appliance these forms.

The arch abode for summarizing a beeline accord accustomed abstracts credibility on a besprinkle artifice is Atomic Squares Beeline Regression. This abode is additionally accepted as Least Squares Regression, Best Fit Regression, Beeline Regression, etc. The important point is that we are action to alarm the relationship with a beeline line, so if the besprinkle artifice shows some added shape, this abode will be inappropriate. Your tasks are to 1) arise up with a line, either by duke or with technology, that “goes through” the abstracts in some adapted way, 2) to be able to use this archetypal to alarm the accord verbally, and 3) to adumbrate numerically y-values for authentic x-values of interest.

Activity: Graphical description: Appliance beeline regression.

Begin by authoritative a besprinkle artifice of the chase times. (Use STAT PLOT. See calculator commands below. Put the years in L1 and the times in L2.) If you appetite a asperous assumption for the slope of the best applicable band through the data, you can affix two credibility spaced far afar (I will appearance you the accommodation in class, appliance the two-point form.)

Next, use the TI-83’s corruption appearance to annual the best fit. The command is STAT CALC LinReg(ax b), bold the two lists are in L1 and L2. (L1 will be the accumbent variable, years in this case.) (For corruption it is basal that you get the adjustment of the variables correct; the abstraction actuality is that you are admiration the vertical capricious from the accepted accumbent variable.)

Interpret what your two corruption coefficients mean. Make abiding you acquire units absorbed to your numbers to advice with the meanings.

Have the calculator blazon this blueprint into your Y = card (using VARS Statistics EQ RegEQ or use the commands below), and TRACE on the band to adumbrate the abutting results. Specifically, see what your archetypal says the 2004 and 2008 times should acquire been. Afresh we will attending them up and assay how predictive our archetypal is. (You can additionally use the abode in the calculator commands breadth below.)

Men’s and Women’s 100-meter birr acceptable Olympic times:

1896

Thomas Burke, United States

12 sec

1900

Francis W. Jarvis, United States

11.0 sec

1904

Archie Hahn, United States

11.0 sec

1908

Reginald Walker, South Africa

10.8 sec

1912

Ralph Craig, United States

10.8 sec

1920

Charles Paddock, United States

10.8 sec

1924

Harold Abrahams, Great Britain

10.6 sec

1928

Percy Williams, Canada

10.8 sec

Elizabeth Robinson, United States

12.2 sec

1932

Eddie Tolan, United States

10.3 sec

Stella Walsh, Poland (a)

11.9 sec

1936

Jesse Owens, United States

10.3 sec

Helen Stephens, United States

11.5 sec

1948

Harrison Dillard, United States

10.3 sec

Francina Blankers-Koen, Netherlands

11.9 sec

1952

Lindy Remigino, United States

10.4 sec

Marjorie, Jackson, Australia

11.5 sec

1956

Bobby Morrow, United States

10.5 sec

Betty Cuthbert, Australia

11.5 sec

1960

Armin Hary, Germany

10.2 sec

Wilma Rudolph, United States

11.0 sec

1964

Bob Hayes, United States

10.0 sec

Wyomia Tyus, United States

11.4 sec

1968

Jim Hines, United States

9.95 sec

Wyomia Tyus, United States

11.0 sec

1972

Valery Borzov, USSR

10.14 sec

Renate Stecher, E. Germany

11.07 sec

1976

Hasely Crawford, Trinidad

10.06 sec

Annegret Richter, W. Germany

11.08 sec

1980

Allen Wells, Britain

10.25 sec

Lyudmila Kondratyeva, USSR

11.6 sec

1984

Carl Lewis, United States

9.99 sec

Evelyn Ashford, United States

10.97 sec

1988

Carl Lewis, United States

9.92 sec

Florence Griffith-Joyner, United States

10.54 sec

1992

Linford Christie, Great Britain

9.96 sec

Gail Devers, United States

10.82 sec

1996

Donovan Bailey, Canada

9.84 sec

Gail Devers, United States

10.94 sec

2000

Maurice Greene, United States

9.87 sec

Marion Jones, United States

10.75 sec

2004

??

??

2008

??

??

(a) A 1980 dissection bent that Walsh was a man.

Men’s and Women’s 200-meter birr acceptable Olympic times:

1900

Walter Tewksbury, United States

22.2 sec

1904

Archie Hahn, United States

21.6 sec

1908

Robert Kerr, Canada

22.6 sec

1912

Ralph Craig, United States

21.7 sec

1920

Allan Woodring, United States

22 sec

1924

Jackson Sholz, United States

21.6 sec

1928

Percy Williams, Canada

21.8 sec

1932

Eddie Tolan, United States

21.2 sec

1936

Jesse Owens, United States

20.7 sec

1948

Mel Patton, United States

21.1 sec

Francina Blankers-Koen, Netherlands

24.4 sec

1952

Andrew Stanfield, United States

20.7 sec

Marjorie, Jackson, Australia

23.7 sec

1956

Bobby Morrow, United States

20.6 sec

Betty Cuthbert, Australia

23.4 sec

1960

Livio Berruti, Italy

20.5 sec

Wilma Rudolph, United States

24.0 sec

1964

Henry Carr, United States

20.3 sec

Edith McGuire, United States

23.0 sec

1968

Tommy Smith, United States

19.83 sec

Irena Szewinska, Poland

22.5 sec

1972

Valeri Borzov, USSR

20.00 sec

Renate Stecher, E. Germany

22.40 sec

1976

Donald Quarrie, Jamaica

20.23 sec

Barbel Eckert, E. Germany

22.37 sec

1980

Pietro Mennea, Italy

20.19 sec

Barbel Wockel, E. Germany

22.03 sec

1984

Carl Lewis, United States

19.80 sec

Valerie Brisco-Hooks, United States

21.81 sec

1988

Joe DeLoach, United States

19.75 sec

Florence Griffith-Joyner, United States

21.34 sec

1992

Mike Marsh, United States

20.01 sec

Gwen Torrance, United States

21.81 sec

1996

Michael Johnson, United States

19.32 sec

Marie-Jose Perec, France

22.12 sec

2000

Konstantinos Kenteris, Greece

20.09 sec

Marion Jones, United States

21.84 sec

2004

??

??

2008

??

??

STAT EDIT To admission a annual of numbers into your calculator, instead of an equation, use the STAT menu. EDIT is the affectation that allows you to admission lists of numbers. You may acquire up to 3 lists displayed in the EDIT window. It is acceptable to use the congenital lists L1 to L6, but absolutely any alleged lists may be used. You may appetite to accredit to the calculator chiral if you are absorbed in allotment and extenuative your lists. (It ability save you accepting to consistently re-enter data.)

STAT PLOT 1 On Use this awning to baptize the plot settings. You can acquire up to three plots on the awning at once.

ZOOMStat (Zoom 9) To appearance a besprinkle artifice of two lists, ZoomStat will actualize an adapted examination window. To use the TI-83 to finer appearance besprinkle plots, I acclaim turning off or de-selecting all Y-variables afore acute ZoomStat. There will be times however, aback you will appetite to acquire both a besprinkle artifice and an blueprint on the aforementioned viewing window, so it is not adapted to consistently de-select all functions.

STAT CALC ???Reg Afterwards two lists of numbers acquire been entered, we can fit ambit or curves to the abstracts with the ???Reg commands. The TI-83 will fit 10 kinds of equations; the best accepted one is LinReg. Afore you use any of the applicable routines, achieve the following: Columnist CATALOG (found by acute 2nd 0), the letter D, bottomward arrow eight times (to point to DiagnosticOn), and columnist ENTER twice.

If you appetite to abundance your adapted blueprint in the Y= annual anon from the corruption command, do this: columnist STAT CALC ???Reg, afresh announce the lists (variables) you appetite to use, afar by commas, then columnist VARS, adapted arrow, 1, and acquire the adapted Y-variable. Your adapted blueprint afresh appears in your annual of Y-variables. An archetype command is: LinReg(ax b) L1, L2, Y1. This will use L1 as the x-values, L2 as the y-values, and Y1 as the blueprint to abundance the fitted blueprint in. Be acquainted admitting that this command will overwrite annihilation you already had stored in Y1. Achieve abiding important actuality in Y1 is adored abroad afore you achieve this command.

Goals: Practice appliance corruption with the TI-83. We appetite the corruption equation, the corruption band superimposed on the plot, and we appetite to be able to use the band to adumbrate new values.

Skills:

· Fit a band to data. This may be as simple as ‘eyeballing’ a straight band to a besprinkle plot. However, to be added precise, we will use atomic squares, STAT CALC LinReg(ax b) on the TI-83, to annual the coefficients, and VARS Statistics EQ RegEQ to blazon the blueprint in the Y= menu. You should additionally be able to annual a band assimilate a besprinkle artifice (by hand) by alive the regression coefficients.

· Interpret corruption coefficients. Usually, we appetite to abandoned adapt slope, and slope is best accepted by analytic the units involved, such as inches per year or afar per gallon, etc. Because abruptness can be anticipation of as “rise” over “run”, we are attractive for the arrangement of the units complex in our two variables. Added precisely, the abruptness tells us the change in the acknowledgment variable for a assemblage change in the allegorical variable. We don’t about bother interpreting the intercept, as aught is about alfresco of the ambit of experimentation.

· Estimate/predict new observations appliance the corruption line. Already we acquire calculated a corruption equation, we can use it to adumbrate new responses. The easiest way to use the TI-83 for this is to TRACE on the corruption line. You may allegation to use up and bottomward arrows to toggle aback and alternating from the artifice to the line. You may additionally aloof use the equation itself by abacus the new x-value by the abruptness and abacus the intercept. (This is absolutely what TRACE is doing.) Note: aback appliance TRACE, and the x-value you appetite is currently alfresco the window settings (lower than Xmin or aloft Xmax) you allegation displace the window to accommodate your x-value first.

Reading: Breadth 1.4.

Day 4

Activity: Continue alive with beeline functions, declared graphical and with a table of data.

Yesterday, we acclimated the TI-83 to fit a beeline band to the Olympic chase data. Interpreting the abruptness was the best important allotment of anecdotic that relationship. Today we will work added with beeline functions, this time accustomed verbally. The added action today will be a examination of Chapter 2, as the bulk of change of a action is an important abstraction in calculus. To acquisition the bulk of change of a action (or the boilerplate of the function) over an interval, we use the two-point form.

Activity 1: Algebraic description: Appliance exact description.

Taxicab rates. Accustomed the afterward advice on the ancillary of a cab, advance an blueprint that will let you annual the book for any ambit x. Report your acknowledgment on the board.

Info on the ancillary of a cab: $2.50 FOR THE FIRST 1/9 MILE, PLUS 25 CENTS FOR EACH ADDITIONAL 1/9 MILE OR FRACTION OF A MILE.

Activity 2: Collapsed description: Appliance boilerplate ante of change.

A half-marathon runner annal the afterward times during a race. Acquisition the boilerplate dispatch of the agent from the alpha through mile 6. From afresh end of six miles through the end of the race3. For the able race. For aloof the last 3.1 miles. Report your answers on the board.

Mile

Time on Clock

Mile

Time on Clock

1

7:36

7

55:07

2

15:29

8

1:02:50

3

23:25

9

1:10:29

4

31:23

10

1:18:20

5

39:20

11

1:26:08

6

47:18

13.1

1:42:58

Goals: Understand the abruptness of the band is the Bulk of Change of the function.

Skills:

· Convert a exact description into an equation. You should be able to admit the annual of abruptness / ambush or a description of several credibility on a band from a exact description of a beeline function. By acquainted which advice is present, you afresh should be able to acquire the able anatomy for the beeline equation.

· Be able to annual boilerplate ante of change from collapsed data. Given a table of values, you should be able to annual assorted ante of change. The important abstraction is that the boilerplate bulk of change is artlessly the abruptness from a beeline equation.

Reading: Breadth 1.5.

Day 5

Activity: Economics Examples.

Several important business/economic applications use beeline functions. Today we will attending at profit, bordering costs, depreciation, and supply/demand curves. All of these accommodation can be modeled with non-linear functions, so we will appointment them later. For now, however, we will use abandoned the beeline functions.

Profit: In business settings, accumulation is affected by subtracting costs from revenue.

Marginal Costs: The abstraction of bordering costs, revenues, etc. is a notion about the abutting item’s cost, revenue, etc. Acquainted the difference amid a bordering bulk and an boilerplate bulk is analytic to appliance derivatives appropriately afterwards (Chapter 2).

Depreciation: Items lose bulk over time, and we archetypal this with different functions. With linear depreciation, we basically use a two-point form.

Supply/Demand Curves: Economists conjecture that markets can be modeled with accumulation and appeal curves, breadth the accumulation ambit applies to producers of a article and the appeal ambit applies to the consumers.

Today we will attending at examples of anniversary of the aloft accommodation from the contest on pages 30 to 32. For anniversary exercise, put your band-aid on the board.

Revenue, Cost, Profit appliance beeline functions. Marginal Cost/Revenue. Botheration 9, folio 30.

Linear Depreciation. Botheration 16, folio 30.

Supply/Demand using curves. Botheration 20, folio 31.

Goals: Recognize the appliance of beeline functions to bread-and-butter examples.

Skills:

· Understand accumulation functions. Accumulation is authentic as the aberration amid Revenue and Cost. We about byword these functions in agreement of abundance produced, q. Acquirement as a action of abundance is usually linear. Bulk as a function of abundance is usually not linear, but today we will acquire it is to achieve some calculations. Bordering bulk (revenue, profit) is the bulk (revenue, profit) of the abutting item produced. Bordering ethics are about different, based on accepted assembly levels. We will assay bordering ethics added in Chapter 2 on derivatives.

· Understand beeline depreciation. In general, abrasion is the crumbling bulk of an annual over time. The simplest anatomy of abrasion is beeline depreciation. The accepted adjustment of free a beeline blueprint for linear abrasion is to use the two-point form.

· Understand accumulation and appeal curves. Bread-and-butter admission suggests that prices and quantities produced or adapted are related. The appeal ambit suggests that as bulk increases, beneath bodies will buy an item. The accumulation ambit suggests that as bulk increases, added items will be produced. These two curves can be modeled with beeline functions, and bread-and-butter admission says they intersect at equilibrium. Later, we will assay non-linear accumulation and appeal curves (Section 6.2, Day 42).

Reading: Sections 1.5 and 1.6.

Day 6

Activity: Supply and Appeal appliance beeline functions. Quiz 1 today.

One absorbing modification we can achieve to the accumulation and appeal ambience is abacus assorted kinds of taxes. The basal catechism is how taxation affects market equilibrium. For the problems today, accede assorted credibility of view. For example, aback we allegation the ambassador the tax on an item, as opposed to charging the consumer, the ambassador behaves as if the artefact sells for less than the bulk the customer pays. Accordingly we alter p with p – t, breadth t is the bulk burdened per item. With the new equation, we now acquire a new equilibrium, and new complete profits, which we can now assay to the values afore the tax.

Supply/Demand using lines. Aftereffect of taxes. Problems 35 to 37, folio 32.

Goals: Continue alive with accumulation and appeal theory.

Skills:

· Understand accumulation and appeal curves. Bread-and-butter admission suggests that prices and quantities produced or adapted are related. The appeal ambit suggests that as bulk increases, beneath bodies will buy an item. The accumulation ambit suggests that as bulk increases, added items will be produced. These two curves can be modeled with beeline functions, and bread-and-butter admission says they intersect at equilibrium. Later, we will assay non-linear accumulation and appeal curves (Section 6.2, Day 42).

Reading: Breadth 1.5.

Day 7

Activity: Exponential Functions.

In beeline functions, as the x-value increases one unit, the y-value increases m units, breadth m is the abruptness of the line. This is accretion growth. Accession blazon of advance is multiplicative. In this affectionate of growth, aback the x-value increases one unit, the y-value increases by a agency of b. That is, instead of abacus a anchored value, we accumulate by a anchored value. This affectionate of advance is alleged exponential growth.

Famous examples of exponential advance are populations. I will attending at the US population. In Presentation 1, you will baddest an abandoned accompaniment and archetypal its growth, conceivably calmly with exponential curves. (Some populations do not abound exponentially; you will acquire to assay the advance ante to see.)

Today we will use the calculator to fit exponential curves to advance functions, like the US citizenry over time.

Activity: Clay Citizenry Growth.

The citizenry for the US is on folio 213. Appliance ratios, acquisition periods of time aback the US citizenry grew about exponentially. For your applicant eras, fit an exponential archetypal appliance regression.

STAT CALC ExpReg This corruption functions fits exponential curves. Again, the x-variable comes first, afresh the y-variable. The third parameter, if used, is the Y-variable breadth the blueprint will be stored. Example: ExpReg(ax b) L1, L2, Y1 uses data from lists L1 and L2 and stores the blueprint in Y1.

Goals: Explore exponential growth.

Skills:

· Know the anatomy of the exponential functions. Exponential equations acquire two parameters, a y-intercept, and a base. The abject is the multiplicative advance factor. The accepted blueprint is . You should be accustomed with the appearance of the exponential graphs, as able-bodied as the breadth and range.

· Know the multiplicative attributes of exponential functions. In adverse to linear functions growing at a abiding bulk over time, exponential functions abound at an accretion rate. The arrangement of alternating y-values for equally spaced x-values is a constant. This actuality is abnormally advantageous for blockage whether tabled ethics abound exponentially, but abandoned if the table has appropriately spaced ethics of the complete variable.

Reading: Sections 1.6.

Day 8

Activity: Explore the changed of exponential growth, the logarithm. Arrangement 1 due today.

To use an exponential advance function, we alpha with a accepted x-value, such as a time. The exponential blueprint then gives us the acme of the function, or the y-value. In abounding situations, however, we appetite to assignment in the other direction. That is, we apperceive the height of the action (the y-value), but appetite the time aback that happens (or the x-value). This changed is alleged a logarithmic function. I acquire activate that abounding acceptance are rather abashed by logarithms. I will try to allay this confusion by emphasizing the actuality that exponentials and logarithms accord together, much like squares and aboveboard roots do, or multiplication and assay do. There are rules we allegation apprentice to do algebra with exponential functions however; for archetype aback we break for time in an exponential growth model.

Today we will assay e, and the logarithm rules.

Activity 1: Advertent e.

As we saw on Day 1, the cardinal e is a complete of the abacus as n gets large. However, you allegation to be authentic not to let your calculator fool you. For example, try ethics of n from 1010 to 1014. With such ample ethics for n, your calculator’s absorption capabilities are exceeded. In your groups, try to arise up with an annual of what the calculator is accepting agitation with.

Activity 2: Rules.

Explore , , and , appliance assay values. For example, verify that Now attending at , , and . I will “prove” anniversary of the after-effects appliance algebra. Convenance the rules appliance 1 – 16 on folio 43.

Goals: Understand the logarithmic functions.

Skills:

· Understand the accord amid exponential and logarithmic functions. Logarithmic functions are inverses to exponential functions. This agency that we about-face the x and y ethics and their associated facts. For example, the ambit of the exponential functions is abandoned complete numbers; accordingly the breadth of the logarithmic functions is additionally abandoned positive numbers.

· Understand the analogue of the cardinal e (approximately 2.7182818). Exercise 2 of Action 2 from Day 2 is the analogue of the cardinal e, which we will use afresh and afresh in calculus. Remember, though, e is aloof a number, annihilation more. The accent of e will become added bright aback we assay derivative formulas in Chapter 3.

· Know the exponential and logarithmic backdrop and be able to use them to solve equations. To break equations for variables that arise in exponents, we allegation logarithmic functions. Therefore, you allegation apperceive the properties. In particular, you allegation be adequate appliance and . The second acreage is how we “rescue” a capricious from the exponent.

Reading: Breadth 1.7.

Day 9

Activity: Advance and Decay.

Doubling time in an exponential function is the breadth of time it takes the y-value to double. To acquisition it algebraically, acquire that a action has angled amid times x1 and x2. So, (because it is an exponential function) and y2 = 2 y1 because it has doubled. Putting these two expressions calm gives . If we now break for the change in time, x2 – x1, we will acquire activate the dispatch time.

Examples of exponential functions that are absolutely advantageous in business are the Present Bulk and Abutting Bulk formulas on folio 49. You may acquire had some acquaintance with these functions in the accounts breadth of MATH 204. We will assay them briefly as examples of exponential advance or decay.

There are several hallmarks of growth functions, and you should be able to acquaint advance from adulteration aloof by attractive at the formula. If the abject of an exponential action is greater than 1, we acquire a advance function, and vice versa. The catchy allotment of checking this affection out is the case breadth we acquire abrogating exponents. For example, . So at aboriginal we ability anticipate this is advance action because 2 > 1, but afterwards the algebra we see the abrogating backer shows this is a adulteration function, because .5 < 1.

Today we will practice appliance exponential and logarithmic functions. Afresh we will assay creating new functions from beeline and exponential functions. In authentic we will assay composed functions, which are analytic to compassionate the alternation aphorism of Chapter 3.

Activity 1: Dispatch Times. Tripling Times. Etc.

Using a graph, explore the accord amid dispatch time and the abject b. Choose a bulk for b; by academic and checking, actuate an breach breadth the y-value has doubled. Annual the dispatch time by abacus the two x-values. Echo this abacus with a altered breach breadth the y-value has doubled. You should apprehension an absorbing fact.

Repeat now for tripling time. Also, try a altered value for the base. Achieve a conjecture about the furnishings of the abject and the multiplier (doubling, tripling, etc) on the times. Can you abutment your abstracts appliance algebra?

Activity 2: Comparing admixture absorption rates.

A banal has current bulk $150 per allotment and is accepted to admission in bulk by 8% anniversary year. In anniversary case below, acquisition a blueprint for the bulk of the banal t years from now and annual the bulk of the banal in 10 years:

Interpret the 8% return as an anniversary (not continuous) rate.

Interpret the 8% return as a connected anniversary rate.

Now blueprint both functions on the aforementioned axes. What is the aftereffect of connected against anniversary compounding? Write a one-paragraph summary.

Activity 3: Appliance Present Bulk and Abutting Bulk formulas.

Work on botheration 31 page 51. Hints: Treat anniversary year as a abstracted investment. For example, the benefit is put into one annual at the bank. Afresh afterwards anniversary year, that year’s bacon is put into a abstracted account, etc. What is altered about these accounts is the breadth of time they exist. Afresh add all the annual balances calm to get the complete amount.

Goals: Become accustomed with manipulating exponential functions.

Skills:

· Know facts about Dispatch Times. The best important actuality about dispatch time is that for any exponential function, it is the aforementioned value. That is, if an exponential action doubles from time t = 3 to t = 13, it will additionally double amid t = 20 to t = 30. From our algebraic assignment on Action 1, the dispatch time is .

· Be able to use Present Bulk and Abutting Bulk formulas in applied settings. The Present Bulk and Abutting Bulk formulas are examples of exponential functions. You should apperceive facts about these formulas. For example, is an exponential action in the capricious t. The abject is (1 r), which is greater than one, so it is a advance function. P is the y-intercept.

Reading: Section 1.8.

Day 10

Activity: Transformations. Quiz 2 today.

Our affair today is transformations, or creating new functions from old. In particular, we will assay shifts, stretches/compressions, and compositions. Aback a connected is added to the y-value, we acquire a vertical shift. Aback a connected is added to the x-value, in parentheses, we acquire a accumbent shift. Aback the y-value is assorted by a constant, we acquire a vertical stretch/compression. Aback the x-value is assorted by a constant, we acquire a accumbent stretch/compression.

Composed functions are complete important to acquire for actuality able to use the alternation aphorism afterwards in Chapter 3. Often, aback we acquire a action inside parentheses, we acquire a composed function. The important accomplishment with these composed functions is anecdotic the “inner” and “outer” functions. See chic addendum for examples.

Activity: Appliance the “Rule of Four” with assorted composed functions.

We will use all four approaches (verbal, graphical, algebraic, and tabular) to become accustomed with composed functions and transformations. Verbal: botheration 35 folio 56. Graphical: problems 32-34 folio 56. Algebraic: botheration 42 folio 73. Tabular: botheration 31 folio 56.

Goals: Become accustomed with transformations, abnormally composed functions.

Skills:

· Recognize the basal functions in complicated functions, abnormally the accouterment and stretches. Abacus and abacus by constants creates accouterment and stretches. You should be able to assay the basal action actuality manipulated, and additionally the accouterment and stretches demography place.

· Be able to decompose functions into the consecutive steps. To use the alternation aphorism to booty derivatives, in Chapter 3, we allegation to be able to admit the apparatus in composed functions. The “inner” action usually is central parentheses, and the “outer” action is the action that after-effects if you alter the expression central the parentheses with x.

Reading: Breadth 1.9.

Day 11

Activity: Ability functions and Polynomials.

Power functions acquire the anatomy . Agenda the credible affinity to exponentials. It is up to you to bethink which is which. My claimed admonition is that x2 is a polynomial. You should be able to accord with apportioned and abrogating exponents. Apportioned exponents are radicals like aboveboard base (an backer of 0.5 or ½) while abrogating exponents are reciprocals ().

Polynomials are several ability functions (with complete accumulation exponents) added together. The bulk of the polynomial is the able ability of x. Annth bulk polynomial can acquire up to n – 1 arbor points. However, there are about fewer, such as with x3, which has none, but is a 3rd degree polynomial.

Today we will play about with polynomials, a able chic of functions. They can booty on a array of shapes, but we should acquire their behavior afore clearing on them as final models to our data.

Activity 1: Exploring polynomial arbor points.

Using balloon and error, actualize a cubic that has 1) aught arbor credibility 2) one arbor point, and 3) two arbor points. Now try the same affair for a quartic (4th bulk polynomial), with up to three arbor points. In anniversary case, assay the endpoint behavior by comparing the cubic or quartic to x3 or x4 with ample x-values.

After we abstraction Chapter 3, we will be able to bigger authorize aback a polynomial has 0, 1, 2, etc. arbor points.

Activity 2: Acquainted ability functions against exponentials.

Values of three functions are accustomed beneath (the numbers acquire been angled off to two decimal places). Two are ability functions and one is an exponential. Allocate them and acquisition abeyant equations. You may acquisition the corruption functions abnormally accessible here. But you can additionally use algebra as a band-aid method.

x

f(x)

x

g(x)

x

h(x)

8.4

5.93

5

3.12

.6

3.24

9

7.29

5.5

3.74

1.0

9.01

9.6

8.85

6.0

4.49

1.4

17.66

10.2

10.61

6.5

5.39

1.8

29.19

10.8

12.60

7.0

6.47

2.2

43.61

11.4

14.82

7.5

7.76

2.6

60.91

Goals: Acquire the appearance of polynomials and ability functions.

Skills:

· Know about ability functions and their attributes. Ability functions acquire a cardinal of appearance you should be acquainted of. Alike powered functions are non-negative and symmetric about x = 0. Odd powered functions are symmetric about the origin. The college the power, the quicker the action goes to infinity. Apportioned admiral are abandoned authentic for complete x-values. Abrogating admiral acquire a vertical asymptote at x = 0.

· Know the basal facts about polynomials. Polynomials are sums of power functions with complete accumulation exponents. The bulk is the bigger ability of x. Annth bulk polynomial can acquire up to n – 1 arbor points. Endpoint behavior is bent by the appellation with the bigger power.

Reading: Breadth 1.9.

Day 12

Activity: Polynomial endpoint behavior. Arrangement 2 due today.

We should additionally acquire the asymptotic behavior of polynomials. As x gets large, abandoned the appellation with the bigger exponent matters. To see this, alpha with a polynomial that has turns and gradually admission the x-value until the blueprint looks like abandoned the leading term. (See Action 1 from Day 11.)

Activity: Assay the asymptotic ascendancy of exponentials to polynomials.

No bulk the degree, no bulk the abject of a advance model, an exponential action will be larger than a ability action for ample abundant ethics of x. First attending at botheration 28 folio 96. Then change the abject to 1.5 and the ability to 10. Zoom out abundantly to verify that , for ample abundant x. (If you are having agitation award a window that verifies this, attending at the acknowledgment below, in the reading.)

When you accomplishment the activity, you may use any added time to assignment on your presentations for abutting time.

Goals: Understand the allegory of exponential and polynomial growth.

Skills:

· Know the asymptotic ascendancy of exponentials over polynomials. Slowly growing exponentials may be bedeviled by polynomials for baby x-values. However, for ample abundant x-values, exponentials (growth models) will always beat polynomials. We alarm this “endpoint behavior” and it is important in allegory functions qualitatively.

Reading: Chapter 1. (Activity window: x: 100 to 130 y: 0 to 2e21.)

Day 13

Activity: Presentation 1.

Pick one of the 50 states. (The abstracts is at the end of these notes.) Fit a archetypal to its citizenry growth. You acquire two goals: alarm the growth, and adumbrate the 2010 census. Assay linear, exponential, and polynomial models. Your presentation should argue us that you acquire alleged the best appropriate anecdotic archetypal and that your appraisal for 2010 is believable.

Reading: Chapter 1.

Day 14

Activity: Assay 1.

This aboriginal assay will awning the elementary functions of Chapter 1. Some of the questions may be assorted best or T/F. Others will crave you to appearance your formed out solution.

Reading: Section 2.1.

Day 15

Activity: Instantaneous Change.

Today we activate Chapter 2, the derivative. The acquired at a point is the abruptness of a band that is “parallel” to the ambit at that spot. We will use a array of techniques to almost this slope, depending on the array of advice accessible to us. With equations, we can use added and added complete “two point” estimates, or slopes of secant lines; afterwards Chapter 3, we will use formulas instead. If we acquire tabled data, we will not acquire complete estimates, as we can abandoned “zoom in” as abundant as the table allows. If we acquire graphs, we will acquire to assumption appliance a straightedge. In any case, we’re gluttonous the abruptness of the line, and accordingly the units are a ratio, like afar per gallon, or anxiety per second, depending on the units acclimated for the two variables.

I acquire two activities today to assay direct change, or derivative. Both chronicle to the actuality that if we zoom in abutting abundant on any continuously differentiable (or calmly curving) function, the function will resemble a beeline line. This abnormality is alleged belted linearity.

Activity 1: Exploring Belted Linearity. Appliance Tangent on the TI-83.

Graph the action on the standard window. Zoom in on what you think is the curviest spot. Accumulate zooming in, say 8 times. Appliance two points on the “line”, appraisal the blueprint of the band this zoomed-in action is abutting to. Blueprint your applicant in the aforementioned window.

Now, at your alleged x-value, use the Departure action to get an blueprint of the line. Assay to your appraisal from the “two point” adjustment above. Agenda the Departure function letters the complete blueprint of the departure line; about we are abandoned interested in the slope.

Activity 2: Ciphering the acquired at a point appliance secant lines.

The acquired at a point can be approximated with an appropriately alleged secant line, that is a band amid two acceptable credibility on the curve. The afterward exercise should advice you see what the calculator is accomplishing aback it calculates Tangent.

Fill in the table, appliance x = 7, and f(x) = sin(x). Assay your answers with the others in your group. You may be accepting different answers. If so, explain whose ethics are “correct”. Note that the two y-values basal the numerator of the secant abruptness are either or .

h

f(x – h)

f(x h)

Secant abruptness

.1

.01

.001

Now blueprint f(x) = sin(x), authoritative abiding that your window includes the point breadth x = 7. Use the DRAW-Tangent affection and draw a departure band on your window. Now, use the dy/dx key on the CALC menu. How do these two techniques assay numerically? Graphically? Is one bigger over the other?

Compare the analogue of the derivative (page 135) with your calculations aback bushing out the table. Observe how the calculator computes acquired ethics with dy/dx. However, sometimes we cannot use our calculators (perhaps a connected in the blueprint has an alien or variable value) and we allegation use our algebra skills. Specifically, apprehension how our authors do algebraic derivatives on folio 137. Don’t fear, though, you won’t be able to use this adjustment for all problems, so we will allegation added accoutrement (theorems) to advice us, and aback we absolutely calculate derivatives, we will use rules, not this definition.

Zoom In (ZOOM 2) allows us to achieve the window “closer” by a agency of four. To use it, columnist ZOOM 2, afresh move the cursor to breadth you appetite the new window to be centered, afresh afresh columnist ENTER.

The CALC menu (found by acute 2nd TRACE) is best advantageous for calculus. The functions in this card will acquiesce us to acquisition minimum and best values, find roots of equations, and achieve the adverse and affiliation activities of calculus. We will assay the syntax of these commands as we use them. Today we use dy/dx, which gives the abruptness of the departure band at that point.

The DRAW card (found by acute 2nd PRGM) will acquiesce you to draw assorted ambit and shapes on your window. In particular, we will appetite to draw departure ambit to curves. These departure ambit are beeline ambit that aloof blow a ambit at a point, and are in some faculty alongside to the ambit at that point. DRAW – Departure can be acclimated in two ways: from an complete graph, or from the abacus screen. To use it for an complete blueprint in the graphing window, achieve abiding you acquire the point of absorption on-screen. Afresh columnist DRAW – Tangent. (Select the ambit you appetite appliance up or bottomward arrow, if you acquire added than one ambit graphed.) Acquire the x-value you appetite by appliance adapted or larboard arrow or by accounting the x-value of interest. Finally columnist ENTER. The command syntax from the abacus awning is: DRAW – Tangent(Y#, x), breadth Y# is the ambit of absorption (such as Y1, or Y2, etc.) and x is the point at which you appetite to acquire the tangent band drawn.

Goals: Understand that best functions we attending at are “locally linear”. Acquire slopes of secant ambit as approximations for the abruptness of the departure line.

Skills:

· Understand the analogue of acquired as the abruptness of the departure line. The departure band aloof touches a ambit at the point of interest, and is in a loose faculty alongside to the line. The abruptness of this band is the acquired at that point. Because it is the abruptness of a straight line, we apperceive abundant about its features: it is a bulk of change (rise over run), it is important to apperceive the sign, etc.

· Evaluate derivatives numerically. If your calculator can aftermath afterwards ethics for a action (whether from a blueprint or aloof from some calculation), and the ascribe ethics can be arbitrarily abutting calm (that is what h abutting aught means), afresh you can annual a acquired numerically. You allegation annual the slopes of some secant lines, and should appraise several such slopes, authoritative abiding the complete in actuality does exist. You allegation additionally apprehend you may acquire the abandoned estimated the bulk of the derivative, and the exact value may abandoned be abutting to the value you have. For added exact values, either use the algebraic approach, or attending advanced to the theorems we will appointment in Chapter 3.

· Evaluate derivatives graphically. If you can byword a action in the anatomy of an equation, afresh your graphing calculator can advice you annual a acquired at specific ascribe values. The TI-83 can draw departure ambit at assorted places on a curve, and can calculate derivatives numerically as well, displayed on the graphing window.

· Understand the analogue of the derivative. You should be adequate with the bend of a complete of slopes of secant line. You should additionally be adequate with the equations and . (Note that the added blueprint is absolutely the complete of slopes of secant line. See folio 135.) This aftermost announcement differs hardly from Action 2 today; I abandoned anticipate it makes added faculty to centermost the secant band on the x-value instead of benign the adapted side. It should achieve no aberration in the limit, but about we can only achieve h so baby appliance our TI-83.

· Know several methods of ciphering the acquired at a point. If we acquire a formula, we can use successively narrower intervals and use the “two point” anatomy for a band to appraisal a abruptness at a point. Afterwards Chapter 3, we may be able to use a blueprint approach. If we have collapsed data, we can abandoned appraisal almost the abruptness of the departure line, using secant lines. If we acquire a graph, we can appraisal slopes appliance a straightedge.

Reading: Breadth 2.2.

Day 16

Activity: Interpreting the acquired function.

The acquired is a abruptness of a function at a authentic point. If we appraise the acquired at abounding such x-values, and blueprint the result, we acquire the acquired function. This is a graph, aloof like the original function, but with altered interpretations, as the y-values are now the slopes at anniversary x-value, instead of the aboriginal functional values. Today we will activate by ciphering the acquired action from collapsed data. Afresh we will appraisal anatomic ethics by alive the acquired at a point.

Comment on notations: There are two main notations mathematicians acquire acclimated to baptize derivatives. I will use them interchangeably, afterwards thinking, as it is added attributes to me. These notations are:

1) Prime notation. .

2) Leibniz notation. This characters reminds us that acquired is a arrangement of differences, a slope. Either we use Dy/Dx or dy/dx. One advantage of Leibniz characters is that we get to see the complete variables involved. Abounding times with the “f-prime” characters we aloof say “f-prime”. This isn’t complete illuminative. What are the variables! Unfortunately, though, Leibniz characters doesn’t acquiesce us to specify which x-value we’re talking about. In fact, to baptize which x-value we’re appliance becomes absolutely cumbersome. Folio 113 shows you the messiness.

After the activities, we will attending at a accessible action on the calculator that will almost the acquired at all x-values in the graphing window. See calculator commands below.

Activity 1: Ciphering the acquired appliance collapsed data.

Using the following half-marathon times, acquisition the estimated acquired function. Agenda it will be difficult to estimate the abruptness at the alpha and end. You don’t acquire the affluence of credibility afore and after. Discuss with your accumulation associates what is reasonable.

Mile

Time on Clock

Mile

Time on Clock

1

7:36

7

55:07

2

15:29

8

1:02:50

3

23:25

9

1:10:29

4

31:23

10

1:18:20

5

39:20

11

1:26:08

6

47:18

13.1

1:42:58

Activity 2: Ciphering appliance belted linearity.

Work on botheration 2, genitalia d and e, on folio 133. These genitalia are about admiration new ethics appliance belted breadth (or in this case extrapolating as 7 anxiety is aloft the accessible data).

Goals: Seeing the acquired as a function. Ciphering appliance belted linearity.

Skills:

· Evaluate derivatives from collapsed data. Aback advice is accessible in collapsed form, we cannot “zoom in” to get a complete of secant slopes. We acquire abandoned a few choices to appraisal the acquired at anniversary x-value. Generally, the best advantage is to boilerplate the secant abruptness afore the point with the secant abruptness afterwards the point. This is algebraically agnate to award the secant slope for the two credibility afore and after.

· Understand belted breadth and how to use it appraisal new values. If we are abutting abundant to a point breadth we apperceive the departure slope, we can action the departure band a abbreviate way and use it to appraisal the bulk of the action at that new point. Caution: if the band is complete “curvy” at this spot, our departure band will ailing represent the function, so it is important to abandoned use this adjustment complete abutting to the accepted acquired value.

Reading: Breadth 2.3.

Day 17

Activity: Abstraction the derivative function.

Today we will try an exercise using graphs.

After the activity, we will attending at a accessible action on the calculator that will almost the acquired at all x-values in the graphing window. See calculator commands below.

Activity: Ciphering the acquired appliance a graph, and advice back.

Each of you will sketch an almost action on a allotment of paper, labeling it “Original Curve” and putting your name on it. You will afresh canyon your blueprint to accession else; they will blueprint the acquired function on a abstracted breadth of paper, labeled with “Derivative Ambit for <insert name here>”. The actuality drawing the acquired will acquire to anxiously appraisal the slopes, so a calibration is needed. I will appearance you in class the adjustment I use to appraisal these slopes. It involves agreement a beeline bend departure to the curve, and award the dispatch over run for that angle. This is afresh for a cardinal of x-values.

After abstraction the derivative, the added actuality will canyon the acquired blueprint to a third person (keep the aboriginal abreast to assay with later); the third actuality will attempt to alter the aboriginal blueprint based abandoned on the advice from the acquired graph. Caution: this aftermost allotment is tricky, as the starting breadth is not unique. You allegation to arbitrarily aces a y-intercept to get started. From there, the acquired blueprint shows you how abrupt the blueprint needs to be at that point, so draw a little band articulation with that slope. Move over slightly, and echo the process.

I will appearance you an archetype in chic afore you attack this activity. If anybody has done the estimates correctly, the blueprint the third actuality draws should bout the “Original Curve” graph. If there are discrepancies, the two sketchers should boldness them. It ability be that the actuality cartoon the acquired fabricated poor estimates, or it may be that the third actuality didn’t construe the advice well.

nDeriv( (MATH 8) will aftermath an appraisal for the acquired at a point. The syntax is nDeriv( expression, variable, value). announcement is the blueprint for the function. I will about use Y#, accepting already stored the action in a Y variable. capricious is about x, but you have some adaptability actuality in case you appetite accession letter to be the variable. bulk is whatever cardinal you’re absorbed in. Aback appliance nDeriv( to blueprint the complete derivative action in the graphing window, use x actuality instead. Example: nDeriv( Y1, x, x).

Goals: Apprehend that the acquired can be viewed as a function.

Skills:

· Interpret the acquired verbally. For problems with a real-world setting, you should be able to use the bulk of the acquired at a point in an English sentence. For example, you may say, “At a assembly affiliated of 1,000 car seats, we can apprehend profits to rise $10 for every added car bench produced.” If you are accepting agitation with this exact description of the derivative, one affair that may advice is to pay abutting absorption to the units involved, for instance dollars, or cardinal of car seats produced. The examples in Breadth 2.3 should help you acquire this exact delivery and interpretations of the derivative.

· Know how to use the TI-83 to aftermath a blueprint of the estimated acquired of a formula. The command nDeriv( will appraisal the acquired numerically with a baby secant line. If we use this in the Y= window, we can blueprint the complete acquired action on the graphing window. The syntax for this is nDeriv( Y#, x, x).

Reading: Breadth 2.4.

Day 18

Activity: Introduction to the Second Derivative.

After advertent that the acquired is a action aloof like the aboriginal curve, there is no acumen we cannot booty the acquired of the derivative. This is alleged the added derivative, and about reflects advantageous advice in complete apple problems. It is the change in the change of a function. The second acquired additionally can be anticipation of as the curvature of a function. You acquire apparently apparent this idea already in agreement of concavity. In particular, if the added derivative is positive, we say we acquire complete concavity, and the added way about for abrogating values.

When we attending at the advice from the aboriginal and added derivatives, there are four capital situations of interest. The aboriginal acquired can be either complete or abrogating (or zero, but we will abode that bearings later) and the added acquired can be either complete or negative.

1) Complete aboriginal derivative, complete added derivative: an accretion ambit that is accepting steeper.

2) Complete first, abrogating second: an accretion ambit that is leveling off, abutting a peak.

3) Abrogating first, complete second: a abbreviating ambit that is leveling off, abutting a minimum.

4) Abrogating first, abrogating second: a abbreviating ambit that is falling faster.

The Leibniz characters gets a little messy for added derivatives. The best way to byword them is to use the prime notation, abacus accession prime for the added derivative. We usually say “f bifold prime”, apropos to . Agenda the messy Leibniz characters on folio 119.

One important appliance of the acquired is the abstraction of bordering analysis. In fact, the appellation bordering is alike with derivative. If either the bulk action or the acquirement action is a beeline line, afresh the bordering bulk or acquirement is artlessly the abruptness of that line. We will attending at this affair afresh in Breadth 4.4 (Day 28), afterwards we assay the adjustment formulas to differentiation.

Activity 1: Comparing a action to its aboriginal and added derivatives.

Enter , forth with its aboriginal and added derivatives, in the Y= window. (See calculator commands below.) Baddest abandoned the added acquired and use the window -2 < x < 2 and -2 < y < 2. Achieve statements about the original action accustomed what you see about the added derivative. Echo appliance aloof the aboriginal derivative. Afore graphing the aboriginal function, achieve a annual that satisfies your statements. Afresh assay and see how abutting you were. If you are off in any of your statements, anxiously appraise breadth you went wrong.

Activity 2: Interpreting derivatives in a complete apple setting.

Problem 30 page 132. Genitalia c and d are especially important; you allegation be able to convert the algebraic advice into complete apple uses. In this case, the ambience of crumbling graduation ante is complete important to academy officials.

There isn’t a abstracted command on the TI-83 for the added derivative; it is simply the acquired of the aboriginal derivative. The easiest way to get the calculator to appraisal the added derivative action is to use these two Y= functions. Put your blueprint in Y1. In Y2, put nDeriv( Y1, x, x). In Y3, put nDeriv( Y2, x, x).

Goals: Investigate the backdrop of the added derivative.

Skills:

· Be able to blueprint the added acquired on the TI-83. Appliance nDeriv( will aftermath a afterwards acquired of a formula. If we echo the command on the new formula, we will almost the added derivative. I acclaim befitting these two commands in Y2 and Y3 for the blow of the semester. Put the blueprint you appetite to assay in Y1. Use Y4 to Y0 for any added functions you appetite to graph.

· Understand what the added acquired says about the concavity of a function. The added acquired measures the concavity of a function. Aback it is positive, we apperceive the original action is bowl-shaped (concave up); aback it is negative, the original action is angled (concave down). Aback the added acquired is zero, it is neither bowl-shaped nor humped; rather it is complete about beeline at that point. Beforehand we talked about local linearity; aback the added acquired is zero, we ability anticipate of that point of the ambit actuality alike more locally linear!

· Be able to catechumen added acquired facts into accustomed English. Because the added acquired is a change in the aboriginal derivative, aback we catechumen to an English description, we acquire to allocution about the bulk of change in the bulk of change. For example, the dispatch of the car is increasing. Sometimes we acquire adapted words for these derivatives. With the motion of an object, the aboriginal acquired is speed and the added acquired is acceleration.

Reading: Sections 2.4.

Day 19

Activity: Economic Examples. Quiz 3 today.

Description.

Activity: Bordering bulk and revenue.

Problem 10 page 129. Appraisal ethics for the bordering bulk and acquirement at both 50 and 90. Use these abstracts in your answers.

Goals: Goal.

Skills:

· Realize that bordering costs/revenues/etc are artlessly derivatives. Marginal costs, revenues, profit, etc are important annual in economics. Because the bordering bulk is the cost of the abutting item, we are just talking about the abruptness of the departure line, which is the derivative. Similarly for revenue, the derivative is the bordering revenue. We will assay these annual added in Breadth 4.4.

Reading: Breadth 3.1.

Day 20

Activity: Appliance Polynomial derivative formulas. Arrangement 3 due today.

A calculator approximation for the acquired action is convenient, but there will be times aback we would rather acquire an exact formula. Fortunately, there are theorems (shortcuts) we can use. We won’t prove abounding of these results, but we will use them to aftermath formulas. Chapter 3, therefore, is abandoned anxious with the algebraic point of view. Aback we acquire collapsed data, graphs, or exact descriptions, we cannot use these theorems.

Several of the theorems administer to any function. Others are specific to authentic forms. The general rules are the accretion connected rule, the multiplicative connected rule, the addition/subtraction rule, the artefact rule, the caliber rule, and the chain rule. The specific functions are the ability rule, the exponential rule, and the logarithmic rule.

Additive connected rule: For this rule, we can achieve a quick altercation to see the answer. What happens to the abruptness of a curve aback we add a connected to it? Abacus the aforementioned connected to every bulk artlessly lifts or lowers the complete ambit that much, but doesn’t change the appearance at all. Thus, the accretion connected aphorism is that there is no change to the derivative. .

Multiplicative connected rule: It is a little harder to verbally prove this rule, but we can see for beeline lines that abacus by a connected increases the abruptness by that constant. With algebra, and the analogue of acquired on folio 135, we can ascertain that the acquired of a multiplied action is assorted by the aforementioned amount. .

Addition/subtraction rule: Again, appliance algebra is the easiest way to prove this rule, but we will acquire the aftereffect on faith. (If you would like to see the algebra, see me afterwards class.) Basically, the acquired of a sum is the sum of the derivatives. .

Power rule: To prove the ability rule, we allegation the binomial theorem, and lots of algebra. Again, we will acquire this aftereffect on faith. . Aback we amalgamate this aphorism with the multiplicative connected rule, we get the best accepted aphorism we’ll use: . We allegation to use this aphorism for reciprocals and radicals, as they can be accounting as exponents. This agency you will acquire to admit that aboveboard roots, and reciprocals, are ability functions. We will do some examples in class.

Activity: Try some basal expressions.

For anniversary of the afterward functions, artifice the action in Y1, its nDeriv( in Y2, and your applicant acknowledgment in Y4. Appliance trace, assay to see if your acknowledgment is right. (Compare Y2 to Y4.)

1)

2)

3)

4)

Goals: Learn and use the basal rules for adverse shortcuts.

Skills:

· Know the Aphorism for Sums. .

· Know the Aphorism for Powers. . Agenda that n can be any number, including fractions and negatives.

· Realize that your nDeriv( action will verify that you acquire a absolute derivative. By graphing the afterwards acquired on your calculator (nDeriv), forth with what you anticipate the acknowledgment is, you can verify if your acknowledgment is correct. You can either assay the ethics for a few about alleged values, or you can blueprint their aberration on a separate window. If they are the same, the aberration should be aught (or complete abutting but not exact due to rounding).

Reading: Breadth 3.2.

Day 21

Activity: Using Exponential acquired formulas.

Exponential rule: The exponential chic of functions is absolutely unique. They are their own derivatives. Action 2 beneath will hopefully argue you of this. The altercation gives a simpler adaptation of this rule, but I like to alpha with the added complicated version, which is absolutely a aftereffect of the chain aphorism (Day 12). . Aback f(x) is a beeline function, is the slope, so that connected is assorted in front. Example: . Aback the base is e, ln(e) = 1, so it is alike simpler. .

Logarithmic rule: The logarithmic aphorism is complete simple: . I will appearance a simple affidavit of this in chic based on the exponential rule.

Activity 1: Advertent the altered appearance of the exponential functions.

Graph and its acquired in the aforementioned window. What is the dispatch time for ? What is the dispatch time for its derivative? These two dispatch times betoken an important result. Use this aftereffect to deduce the formula for the acquired of .

Activity 2: Practice.

For anniversary of the afterward functions, artifice the action in Y1, its nDeriv( in Y2, and your applicant acknowledgment in Y4. Appliance trace, assay to see if your acknowledgment is right. (Compare Y2 to Y4.) (Note in botheration 2 you will acquire to achieve up ethics for k and a. This array of botheration is why alive algebra is still important.)

1)

2)

Goals: Learn and use the basal rules for adverse shortcuts.

Skills:

· Know the Rules for Exponential Functions. . This rule absolutely uses the alternation rule, advancing up tomorrow, but I like to use the more accepted aphorism now.

· Know the Rules for Logarithmic Functions. .

· Realize that your nDeriv( action will verify that you acquire a absolute derivative. By graphing the afterwards acquired on your calculator (nDeriv), forth with what you anticipate the acknowledgment is, you can verify if your acknowledgment is correct. You can either assay the ethics for a few about alleged values, or you can blueprint their aberration on a separate window. If they are the same, the aberration should be aught (or complete abutting but not exact due to rounding).

Reading: Breadth 3.3.

Day 22

Activity: Practicing the Product, and Caliber Rules.

Today we convenance formulas. I will show you how the artefact and caliber rules work. Afresh we will absorb time practicing.

Activity: Practicing the Artefact and Caliber Rules.

Calculate the derivatives of the afterward functions. Be abiding to aboriginal adjudge whether the action requires the artefact rule, the alternation rule, the accession rule, etc. Afresh assay your answers on your calculator appliance nDeriv.

1)

2)

3)

4)

5)

Goals: Become accustomed with the artefact and caliber rules.

Skills:

· Know the Aphorism for Products. .

· Know the Aphorism for Quotients. . To bethink this rule, I’ve memorized the little carol “Low dee high, beneath aerial dee low, aboveboard the basal down below”. I’ve never abandoned the caliber aphorism because of it!

Reading: Section 3.4.

Day 23

Activity: Practicing the Alternation Rule. Quiz 4 today.

Today we abide practicing formulas. I will appearance you how the alternation aphorism works. Afresh we will absorb time practicing.

Activity: Practicing the Alternation Rule.

Calculate the derivatives of the afterward functions. Be abiding to aboriginal adjudge whether the action requires the artefact rule, the alternation rule, the accession rule, etc. Afresh assay your answers on your calculator appliance nDeriv.

1)

2)

3)

4)

5)

Goals: Become accustomed with the alternation rule.

Skills:

· Know the Aphorism for Compositions (Chain Rule). . You allegation acquire composed functions to use this rule correctly. If you cannot identify what g(x) is, you can’t get the absolute acquired in front. You additionally allegation to be able to “replace” g(x) with x in the f action to get the able acquired there, then afresh “replace” x with g(x).

Reading: Chapter 3.

Day 24

Activity: Practicing the Derivative Formulas.

Description.

Activity: Practicing the Rules.

Calculate the derivatives of the afterward functions. Be abiding to aboriginal adjudge whether the action requires the artefact rule, the alternation rule, the accession rule, etc. Afresh assay your answers on your calculator appliance nDeriv.

1)

2)

3)

4)

5)

Goals: Goal.

Skills:

· Identify the authentic acquired aphorism bare for a problem. For abounding functions, abandoned one of the acquired rules we acquire abstruse is absolutely used. (Of course, for some functions, more than one blazon of aphorism ability be present.) Your task, then, is to be able to assay which authentic aphorism or rules are needed. This accomplishment will arise with practice. It is up to you to put in the time so that you acquire the acquaintance to acquire the proper rules. There are a lot of problems on folio 173 for you to convenance on.

Reading: Breadth 4.1.

Day 25

Activity: Exploring Belted Extrema.

Now that we acquire abstruse some formulas, we can achieve use of this advice algebraically to acquisition absorbing places on curves. In particular, we can find peaks and valleys, or added formally, belted maxima and minima (together alleged extrema). You may anticipate at aboriginal with our able calculator that we don’t allegation algebra any more. While our machines advice us in abounding circumstances, there is still a use for analytic results. For example, the calculator will help us acquisition the turns in a polynomial, but abandoned if we acquire a acceptable window already. Algebraic after-effects advice us acquisition the able window. We will explore this in Action 1.

Another accident breadth algebra is all-important is aback the ambit of a archetypal are unspecified. Action 2 today addresses this situation.

Activity 1: Allegory polynomial turns.

Without accomplishing calculus, try to acquisition the allotment of this cubic breadth the analytic credibility are. . Now, acquisition the analytic points algebraically. Use the first acquired assay to allocate the analytic points. Use the added acquired assay to allocate the critical points.

Activity 2: Award the altitude on a cubic so it has two turns.

The accepted anatomy for a cubic polynomial is . However, we apperceive that some cubics acquire no extrema, such as . What altitude on the ambit cause a cubic to acquire none, one, or two analytic points? Hint: you will allegation to use the boxlike blueprint and note breadth the discriminant is negative, zero, or positive.

Activity 3: Analytic credibility of a non-polynomial.

Without accomplishing calculus, try to acquisition the allotment of this action breadth the analytic credibility are. . Now, acquisition the analytic points algebraically. Use the first acquired assay to allocate the analytic points. Use the added acquired assay to allocate the critical points.

Activity 4: Appliance a table of acquired ethics to acquisition the best and minimum.

Problem 20 folio 181.

Goals: Understand analytic points, and how to allocate them.

Skills:

· Know the analogue of a Analytic Point. Locations on the blueprint of a action breadth the acquired is either zero, or undefined, are critical points. Places with aught abruptness ability be maxima, minima, or neither. Important examples to accumulate in apperception are the cubic ability function, which has a analytic point that is neither a best nor a minimum, and the complete bulk function, which has a analytic point with an amorphous derivative.

· Be able to use the Aboriginal and Added Acquired Tests for classifying extrema. The added acquired assay is advantageous for free whether a analytic point is a best or a minimum. Simply appraise the added acquired at the applicant point, and allocate it as a maximum, a minimum, as it is abrogating or positive. If the added acquired is zero, we allegation resort to the aboriginal acquired test. To perform this test, we appraise whether the aboriginal acquired is complete or negative about the analytic point. If it is abrogating to the larboard and complete to the right, we acquire a minimum. If it is complete to the larboard and abrogating to the right, we acquire a maximum. If it is complete on both abandon of a analytic point or if it is negative on both abandon of the analytic point, afresh we acquire a “saddle point”, which is neither a best nor a minimum.

Reading: Breadth 4.2.

Day 26

Activity: Exploring Inflection Points. Arrangement 4 due today.

We acquire apparent that the analytic credibility of a action alarm that functions extrema, if any exist. The analytic credibility of the acquired action represent places breadth the concavity of the aboriginal action changes sign. These credibility are alleged articulation points. We ascertain them in aloof the aforementioned way we activate analytic points, but alive with the added acquired instead of the aboriginal derivative. Remember that it is accessible for the added acquired to be aught and yet the concavity doesn’t change. The fourth degree ability action is one example.

Activity: Award articulation points.

Find articulation points for these functions.

1) [Hint: Use the TI-83.]

2) [Hint: The added acquired algebraically is tough; persevere, or use the TI-83.]

3) [Hint: The added acquired algebraically is tough; persevere, or use the TI-83.]

4)

Goals: Be able to acquisition and adapt credibility of inflection.

Skills:

· Know the analogue of Articulation Points. Credibility on a blueprint breadth the concavity changes assurance are articulation points. We commonly ascertain these credibility by analytic breadth the added acquired is zero. However, we still allegation check on anniversary ancillary of such candidates, as artlessly equaling aught is not the aforementioned as alteration sign.

Reading: Breadth 4.3.

Day 27

Activity: Anecdotic All-around Extrema.

In accession to free inflection credibility and analytic points, we additionally appetite to actuate all-around extrema. We acquire already talked about about (or local) extrema. The all-embracing best (or minimum) allegation be either one of the analytic points, great (such as with a vertical asymptote), or one of the endpoints (if the arena is bounded). Aback you attending for global extrema, I acclaim authoritative a annual of the analytic points, and endpoints. Then, afterwards attractive at the blueprint for places breadth the blueprint goes off to aloft in either direction, acquire the bigger for the maximum, and the aboriginal for the minimum. If the blueprint does go to infinity, the best byword to use is “There is no all-around best (or minimum).”

Asymptotes are beeline ambit that a blueprint approaches. y = 1/x is an archetype with two asymptotes; there is a accumbent asymptote on the x-axis and a vertical asymptote on the y-axis. We will mostly be anxious with abandoned accumbent and vertical asymptotes, but they could additionally be angled (diagonal). Generally, aback the denominator of a rational action is zero, we acquire a accessible vertical asymptote. While these are not analytic points, they are important to identify, as a action is great at an asymptote.

Activity 1: Anecdotic the absorbing credibility in a function.

For anniversary of the afterward functions, acquisition all the absorbing points/features, including analytic points, extrema, articulation points, asymptotes, accretion and abbreviating intervals, and complete and abrogating concavity.

1) [Hint: Use the TI-83.]

2)

3)

4)

5) , x ≥ 0

Goals: Understand the aberration amid about extrema and all-around extrema.

Skills:

· Understand the all-embracing action for allegory and abstraction functions. Our all-embracing action is to acquisition the analytic points, the asymptotes (both vertical and horizontal), articulation points, intercepts, and any added easy-to-find points. You should be able to make both quantitative and qualitative descriptions of functions/graphs/equations.

· Sketch graphs for equations with bearding parameters. Appliance the abilities acquired in classifying analytic points, and appliance the abilities for award asymptotes, you should be able to annual curves with bearding parameters. You may allegation to be told whether the ethics of the coefficients are complete or negative, or over what ambit the coefficients can acquire values. For example, for parabolas we apperceive the acme has an x-coordinate of , breadth . It may be difficult or absurd to blueprint the ancestors of curves in general, as different coefficients may crop abnormally shaped curves, but in best cases you can assemble an able sketch.

· Be able to acquisition all-around extrema. To acquisition the all-embracing extrema (global extrema), we appraise all the analytic points, as able-bodied as any endpoints of the breadth or any credibility with undefined derivative. Caution: abandoned looking at the analytic credibility will not be sufficient, as abounding functions acquire no global best or minimum due to the function’s ethics abutting infinity.

Reading: Breadth 4.4.

Day 28

Activity: Bread-and-butter Examples. Quiz 5 today.

Today we acknowledgment to economic applications. Anamnesis that profit is the aberration amid acquirement and cost. The aboriginal affair to apprehension is that the abundance that maximizes acquirement is not consistently the aforementioned abundance that maximizes the profit. We acquire several approaches to solving the best accumulation problem. We can artlessly annual accumulation at all quantities and afresh acquire the maximum. This can be annoying and time consuming. It ability be abundant easier to use the calculus rules we acquire learned. Specifically, we apperceive that aback a acquired is zero, the action has a about best or minimum. Because accumulation is a aberration (), we can use the formulas from Chapter 3 to appearance that . Now, if we set the acquired to aught and solve, we acquisition . We use this action now to break problems with abandoned a graph, or a table of marginal values: acquisition breadth bordering bulk and bordering acquirement are equal. We will convenance with all three approaches (tabular, graphical, algebraic) today.

Activity: Accumulation maximization.

We will aerate profit appliance three sets of information: tabular, graphical, and algebraic.

Tabular: botheration 8 page 200.

Graphical: botheration 13 folio 200.

Algebraic: botheration 16 folio 201.

Goals: Understand some uses of the acquired in economics and business.

Skills:

· Know that best (or minimum) accumulation occurs breadth bordering bulk equals marginal revenue. Because accumulation is bulk subtracted from revenue, and because best accumulation occurs aback its acquired is zero, we can achieve that profit is a best aback bordering bulk equals bordering revenue. In blueprint form: , , . We don’t have a agreement that such spots are maxima; we allegation assay to achieve abiding appliance the aboriginal acquired test, for example.

Reading: Breadth 4.5.

Day 29

Activity: Bread-and-butter Examples.

Another archetype appliance derivatives in economics is boilerplate cost. By abacus the bulk action by quantity, we acquire the blueprint for average cost. Appliance the caliber rule (which I will do in class), we ascertain that the minimum bulk occurs where boilerplate bulk equals bordering cost. If we acquire the formulas, this will aloof be an algebra problem. If we acquire graphs, it will be easier, as there is a accessible geometric band-aid (see folio 204 bulk 4.60).

Activity: Exploring Boilerplate Cost.

Find the minimum boilerplate bulk for the afterward problems.

Graphical: botheration 13 folio 200.

Algebraic: botheration 16 folio 201.

Goals: Understand some uses of the acquired in economics and business.

Skills:

· Know that boilerplate bulk is a minimum aback boilerplate bulk equals bordering cost. By appliance the caliber aphorism to acquisition the acquired of the boilerplate cost, we find that boilerplate bulk is minimized aback boilerplate bulk equals bordering cost.

Reading: Sections 4.7 and 4.8.

Day 30

Activity: Logistic Growth, Surge Functions. Arrangement 5 due today.

We saw that some accompaniment populations grow about exponentially for periods of time. However, we additionally apperceive that this exponential advance cannot action forever, due to complete apple constraints, such as accessible amplitude and resources. A added astute archetypal would account for this closing high bound. The logistic action is such a model. Today we will assay this function, by demography its derivatives, award its absorbing points, and abstraction graphs for its assorted parameters.

The billow action is about acclimated to archetypal biologic concentration problems. We will assay this action today also. I will work out the derivatives and the blueprint during class; afresh you will practice yourself.

Activity 1: Revisiting accompaniment populations.

Pick one of the 50 states and fit a logistic corruption ambit appliance the TI-83. You will acquisition the action in the STAT CALC card of the calculator, at the basal of the menu. There are some abstracts sets for which the TI-83 will abort to acquisition a acceptable fit. I haven’t figured out aback it will and will not work; it may acquire to do with the appearance of the abstracts not attractive “logistic” enough.

Activity 2: Billow action example.

Problem 8, page 226. In accession to answering the questions asked, try to arise up with estimates of the formulas.

Goals: Examine two added examples of derivatives, the logistic action in citizenry growth and the billow action in biologic concentrations.

Skills:

· Know the anatomy of the logistic advance function. One conception of the logistic action is . This curve models citizenry advance realistically. The breadth is all complete numbers, and the ambit is 0 to L.

· Know facts about the logistic advance function. Through our calculus results, we acquisition that there are no analytic points, but there is an articulation point breadth P = L / 2, additionally alleged the point of abbreviating returns. L is the accustomed capacity, or the value of the accumbent asymptote as x approaches infinity.

· Know the anatomy of the billow function. The billow action is . The breadth is all complete absolute numbers, and the ambit is 0 to 1 / be.

· Know facts about the billow function. The billow action begins at the origin, increases to a aiguille at x = 1 / b, afresh decreases to a accumbent asymptote at zero. The ambit is about acclimated to archetypal biologic absorption curves.

Reading: Chapters 2, 3, and 4.

Day 31

Activity: Presentation 2.

Pick one of these functions (first come, aboriginal served): 1), 2), 3), 4), 5), 6), 7). Completely alarm the absorbing behavior, afterwards graphing. Be abiding to accommodate analytic points, articulation points, all-around extremes, endpoint behaviors, etc. Afterwards your description, appearance us a blueprint with an adapted window that demonstrates the definiteness of your analysis.

Reading: Chapters 2, 3, and 4.

Day 32

Activity: Exam 2.

This added assay is on Derivatives and Applications, Chapters 2, 3, and 4. Some of the questions ability be assorted choice. Others will crave you to appearance your formed out solution.

Reading: Section 5.1.

Day 33

Activity: Introduction to Definite Integrals, appliance horse speeds.

We will abject our antecedent altercation on the blueprint “Distance equals dispatch times time”. In abounding cases, we will not apperceive the dispatch at any almost time, but at anchored intervals. Thus we allegation assumption the ethics in between. We usually acquire smoothness, and accordingly pretend our functions are monotonic, or either abandoned accretion or abandoned decreasing. So, for anniversary interval, we will acquire an high and lower appraisal of the ambit covered, depending on whether we use the dispatch afore or afterwards the accepted time period.

In the afterward data, we acquire the time of a horse race, and the dispatch of the horse at that moment. Use this advice to appraisal the complete ambit the horse has traveled.

Time (sec)

0

30

60

90

120

Speed (mpg)

0

40

38

35

37

Goals: Acquire how ambit can be estimated by alive speed.

Skills:

· Be able to appraisal ambit accustomed speed. By alive that “Distance equals dispatch times time”, we can annual ambit catholic over an breach with ability of the speed. This fact is the abject for all of our ambit calculations, alike for speeds that are not constant, as we shall see in the accessible material.

· Know there are high and lower bound for the ambit estimate. Because dispatch changes over an interval, and we do not apperceive the ethics in amid two time points, we allegation achieve assumptions about how dispatch varies. About we will acquire that the speed does not go aloft or beneath the two ethics that bracket a time interval. This leads to two estimates of distance in one time interval, an “upper” and a “lower” estimate. We add all the lower estimates and all the high estimates over an complete set of intervals to acquisition the accumulated ambit traveled.

Reading: Breadth 5.2.

Day 34

Activity: Audible Integrals.

How could we advance this appraisal of distance? The best important cessation we will achieve today is that the abstraction of ambit turns out to be an area, not a length. It is analytic that you acquire this point in the accessible material.

We will about allocution about larboard and adapted sums, but these represent the lower and high estimates abandoned on banausic intervals. If the dispatch bounces up and bottomward (as in the horse race example) afresh we will acquire to be authentic about which appraisal is the lower one and which is the high one.

To acquisition the bulk of the definite integral, we booty abate and smaller intervals (if we can) and eventually the complete as this breach amplitude approaches zero. These Riemann sums are mostly a conceptual notion; in convenance we will use a altered admission (antiderivatives in Breadth 7.1, Day 42).

Activity 1: Did they hit the skunk?

Jan and Pat are driving forth a country alley at 45 afar per hour (about 66 ft/sec). As the car circuit a curve, Jan sees a bunco in the boilerplate of the alley about 100 anxiety ahead. Jan anon applies the brakes, and Pat notices that the dispatch of the car drops from 66 ft/sec to 51 ft/sec to 34 ft/sec to 0 ft/sec over the abutting three seconds. (Pat is a bit strange.) Does the car hit the skunk?

Goals: Explore Reimann sums and audible integrals.

Skills:

· Realize the ambit appraisal can be beheld as an breadth beneath a curve. A complete important ascertainment to achieve about our ambit calculations is that these distances can be anticipation of as areas beneath the ambit of the dispatch values. In general, aback we acquire a bulk function, and are interested in the accumulative change in the “distance” action for that rate, we will annual an area.

· Know that the audible basic is a complete of advancing high and lower estimates. If we acquire the affluence of “refining” our intervals (that is, authoritative them narrower), afresh we can force the lower and upper estimates to assemble to the authentic bulk of the ambit traveled. The bulk to which the estimates assemble is alleged the audible integral.

· Realize that if the action isn’t monotone, the high and lower estimates won’t be identical to adapted and larboard sums. If we use a blueprint and anxiously accumulate clue of which rectangle represents the lower appraisal and which represents the upper estimate, afresh we see that “upper” and “lower” are additionally “right” and “left” abandoned on an breach that is banausic (either consistently accretion or consistently decreasing).

Reading: Breadth 5.3.

Day 35

Activity: Exploring Areas and Integrals.

Obviously appliance abate and smaller intervals is annoying assignment by hand. Fortunately we acquire a calculator command that saves us. fnInt( (MATH 9) accomplishes the assignment for us. Accumulate in apperception that this command calculates the integral, not necessarily the breadth (due to the assurance on the y-values). Agenda the implications: There is a aberration amid complete and abrogating ethics on the integral. If we appetite breadth we allegation accumulate clue abandoned of regions aloft and beneath the x-axis.

Activity: Award areas beneath curves.

For anniversary of the afterward functions, acquisition the breadth indicated.

1) The breadth bounded amid , y = 0, x = 0, and x = 2.

2) The breadth enclosed amid and , amid x = 0 and x = 5.

3) The breadth amid the x-axis and between x = 0 and x = 2.

In the aforementioned card as nDeriv( is our arch apparatus for integration: fnInt( (MATH 9). The syntax is fnInt( expression, variable, start, end). expression is the blueprint for the acquired that we appetite the (signed) breadth underneath, capricious is usually x, aloof as in nDeriv(, and alpha and end are the boundaries of the breach we want. We can additionally admission this action from the CALC card while on the graphing awning (CALC 7 or ∫f(x)dx).

Goals: Apperceive the graphical estimation of the audible integral.

Skills:

· Be able to use integrals to acquisition areas belted by curves. Areas can be affected appliance integrals. However, you allegation be acquainted that integrals can be negative, if the function is negative. So to acquisition areas, we allegation ensure that all functions are positive. If we acquire to, we accumulate by -1 to achieve a function positive. This amounts to abacus a bare assurance to an basic to acquisition the agnate area. If we are ambidextrous with the breadth between two curves, we abatement the lower ambit from the college curve, and the consistent basic is the breadth amid them. If they cantankerous and accordingly about-face roles, we about-face the subtraction.

· Know the calculator commands to acquisition areas. We can annual audible integrals (or areas beneath curves) with fnInt( or ∫f(x)dx. fnInt( requires able syntax while ∫f(x)dx requires the breadth be currently on the graphing window.

Reading: Breadth 5.3.

Day 36

Activity: Exploring Areas and Integrals.

Description.

Activity 1: Affection pumping rate.

If r(t) represents the bulk at which the affection is pumping blood, in liters per second, and t is time in seconds, give the units and acceptation of the afterward integral: .

Activity 2: Advance of a population.

Assume gives the rate of change of the citizenry of a city, in bodies per year, at time t years aback 2000. If the citizenry of the burghal is 5,000 bodies in 2000, what is the citizenry in 2009?

Goals: Know the graphical estimation of the audible integral.

Skills:

· Know the calculator commands to acquisition areas. We can annual audible integrals (or areas beneath curves) with fnInt( or ∫f(x)dx. fnInt( requires able syntax while ∫f(x)dx requires the breadth be currently on the graphing window.

Reading: Breadth 5.4.

Day 37

Activity: Adapt the Fundamental Assumption of Calculus in complete apple settings. Arrangement 6 due today.

The Fundamental Assumption of Calculus lets us allocution about accumulated change of a action appliance its derivative information. This is appealing much what we acquire been accomplishing the aftermost few sessions. Today we will assignment on some examples breadth we achieve abiding we’re putting the advice in context. I will alpha with a academic bike cruise (Problem 26 page 267). Afresh you will assignment on several problems yourself.

Activity 1: Bike trip.

Problem 30 folio 261.

Activity 2: Balloon flight.

Problem 38 folio 262.

Goals: Using the Fundamental Assumption of Calculus in complete apple settings.

Skills:

· Know the Fundamental Assumption of Calculus. The Fundamental Assumption of Calculus relates basic and acquired as inverses. To acquisition the integral, we use the derivative, but for a action we ability not yet know. Fortunately, the basic can be interpreted as an area, so we don’t allegation to apperceive the aboriginal action absolutely if we can almost it using areas.

Reading: Sections 6.1 and 6.2.

Day 38

Activity: Adapt the Fundamental Assumption of Calculus in complete apple settings. Quiz 6 today.

Description.

Activity 2: Amphitheater line.

Here is the blueprint of the bulk (in arrivals per hour) at which assemblage access at the amphitheater to get blitz seats for the black performance. The aboriginal bodies access at 8 a.m. and the admission windows accessible at 9 a.m. Acquire that already the windows open, bodies can be served at an boilerplate bulk of 200 per hour. Use the blueprint to approximate:

1) The breadth of the band at 9 a.m. aback the windows open.

2) The breadth of the band at 10 a.m. and 11 a.m.

3) The bulk at which the band is growing at 10 a.m.

4) The time aback the band is longest.

5) The breadth of time a actuality who arrives at 9 a.m. has to angle in line.

6) The time the line disappears.

7) Acquire you were accustomed a blueprint for r in agreement of t. Explain how you would acknowledgment the above.

Goals: Using the Fundamental Assumption of Calculus in complete apple settings.

Skills:

· Be able to almost areas beneath curves appliance graphs. In the examples today we affected areas accustomed graphs. This is usually best able with a acceptable filigree on blueprint paper, and counting boxes. But if other approximations assignment (like triangles) you should use them. The important allotment is actuality able to acquisition a acceptable acknowledgment for how abundant breadth is belted by the curve. Afterwards (Chapter 7) we will focus added on formulas.

Reading: Breadth 6.1.

Day 39

Activity: Examples appliance integrals.

If we could alter a action over an breach with a constant, so that the areas are equal, afresh we would acquire the average bulk over that interval. The key abstraction is that the areas are equal. Because a connected function makes a ellipsoidal area, all we allegation to annual boilerplate breadth is the amplitude of the breach and the breadth (the audible integral). In Action 1 you will convenance this rephrasing.

Activity: Boilerplate Value.

Is the boilerplate of the best and minimum over an breach according to the boilerplate bulk over the interval? Assignment Problems 12 and 18 on folio 279, which abode this.

Goals: Use integrals in economics settings.

Skills:

· Know the Boilerplate Bulk of a action over an interval. Graphically, we can adapt the audible basic as the breadth of a rectangle over an interval. The acme of this rectangle represents the boilerplate bulk of the action over the interval.

Reading: Breadth 6.2.

Day 40

Activity: Examples appliance integrals.

Consumer surplus is the bulk of money not spent that would acquire been spent at higher prices. This is altered for each consumer, as there are abounding altered “demand” levels. So, for anniversary bulk level, we determine how abundant money was “saved” from the complete bulk against the accommodating price, as bent by the appeal curve, and complete this over all prices (down to the accepted price). Similarly, we can bulk a ambassador surplus, but appliance the accumulation curve. Afresh the acumen is that if the bulk were lower, beneath items would be made, and therefore sold. It is important to agenda that at equilibrium, both producers and consumers are “gaining” from the transaction.

Geometrically, the customer surplus is the breadth belted by the bulk (horizontal line) and the appeal ambit (integral breadth if the appeal ambit isn’t linear). The ambassador surplus is the breadth beneath the bulk band belted by the accumulation curve.

The absorbing assignment comes aback we (perhaps the government) appoint non-equilibrium prices. What aftereffect does this acquire on the bread-and-butter interpretations? We will assay this abstraction in Action 2. In class, I will assignment on Botheration 10 folio 285.

Activity: Customer and Ambassador surplus.

Problem 9, page 285. I acclaim graphing the curves in accession to appliance fnInt(.

Now acquire a bulk greater than calm is imposed. (Invent one.) Annual the change in the two surpluses.

Now acquire a bulk lower than calm is imposed. (Invent one.) Annual the change in the two surpluses.

Goals: Use integrals in economics settings..

Skills:

· Understand the Customer and Aftermath Surplus examples. Calm bulk is lower than abounding consumers are accommodating to pay. The aberration amid what they would acquire paid and what they are advantageous is alleged the customer surplus. Similarly, the calm bulk is college than abounding producers are accommodating to produce. The aberration in the calm bulk and the supplier’s accommodating bulk is the ambassador surplus.

Reading: Breadth 6.3.

Day 41

Activity: Description.

Description.

Activity: Description.

Activity

Goals: Use integrals in economics settings..

Skills:

· Skills. Description

· Skills. Description

Reading: Breadth 7.1.

Day 42

Activity: Antiderivatives. Quiz 7 today.

We acquire explored how to interpret audible integrals. The techniques we’ve been appliance absorb ciphering areas beneath curves. The Fundamental Assumption of Calculus guided us, but it additionally shows us accession admission to the solution, if we acquire a blueprint for the bulk action (the derivative). The FTC says all we acquire to do is arise up with a formula whose acquired is the blueprint we’ve started with. This sounds easier than it about is.

However, aback such a blueprint does exist, the band-aid to a audible basic is then artlessly the aberration of two ethics in this new function, which, because it is an changed function, is alleged an antiderivative. It is important to agenda adapted abroad that antiderivatives are not unique functions. We apperceive from Chapter 3 that aback we add a connected to a action we don’t change the acquired at all. So there are abounding antiderivatives to any problem, but they only alter by abacus a altered constant.

To acquisition antiderivatives we allegation to admit a few appearance of the rules we abstruse in Chapter 3. First, derivatives add together, so we can assignment on anniversary allotment in a sum separately. Second, derivatives of ability functions are themselves power functions. So we aloof allegation to work backwards. Exponential functions are additionally their own derivatives. Composed functions are accession matter. They may or may not acquire simple solutions. We will use barter to see if we can ascertain the answers to them.

Activity: Alive “backwards”.

Work as abounding of the problems on folio 304 as you can.

Goals: Realize that antiderivatives are the inverses of derivatives.

Skills:

· Realize that an antiderivative is a action whose acquired is the original expression. The Fundamental Assumption of Calculus shows us that accumulated change in a action is an breadth beneath the acquired curve. Conversely, if we apperceive the original function’s formula, we can artlessly abatement two ethics to acquisition the definite integral.

· Know how to acquisition antiderivatives of simple functions. Ability functions, exponential functions, constants, and the alternate action 1/x are all simple functions that acquire simple antiderivatives. Breadth 7.1 enumerates them in the assorted boxed formulas.

Reading: Breadth 7.2.

Day 43

Activity: Affiliation by Substitution. Arrangement 7 due today.

Composed functions do not acquire simple antiderivatives. We will use barter to see if we can ascertain their antiderivatives. But barter doesn’t consistently work; we allegation acquire functions that bout the alternation aphorism exactly. Barter gives us a adventitious at least. It ability crave balloon and absurdity to acquisition the adapted barter to make. The action is to try w as an “inner” function; afresh . Replacing what we can, we see if we acquire fabricated the botheration into article simpler and solvable. I will assignment problems in chic to appearance you the basis of it. Afresh you will convenance with simple functions today and the composed functions tomorrow.

Activity: Affiliation by Substitution.

Work as abounding of the problems on folio 308 as you can.

Goals: Know how to do affiliation by substitution.

Skills:

· Know how to acquisition antiderivatives appliance substitution. The antiderivatives for composed functions can sometimes be activate appliance substitution. This abode abandoned works if the acquired of the proposed barter appears in the blueprint in aloof the adapted way. If the barter is alleged well, afresh the botheration afterwards barter will be of a simpler nature.

Reading: Breadth 7.3.

Day 44

Activity: Allegory Antiderivatives.

Definite integrals, as we acquire seen, are specific areas beneath a curve. If we acquire a blueprint for the acquired that we can acquisition an antiderivative for, we can use the Fundamental Assumption of Calculus to acquisition the breadth exactly. Today we will do assignment complete agnate to yesterday’s work, but we will move on to do the complete subtractions specified by the FTC.

In accession to apprenticed integrals, we can additionally try our duke at abnormal integrals, or those with aloft in either integrand. These problems will be apparent with limits, and accordingly may be catchy to achieve aggregation with (recall the harmonic alternation of Day 2). However, the FTC saves us, if we are able to appraise the antiderivative as x approaches infinity.

Activity: Evaluating audible integrals absolutely appliance the FTC.

For anniversary of the afterward functions, acquisition an antiderivative. Then, appraise the audible basic appliance your calculator (fnInt), and by appliance the Fundamental Assumption of Calculus. Assay answers. Which one is “right” and which one is abandoned an approximation?

1) . Integrate from 0 to 10.

2) . Note: you will acquire to adjudge how to “con” your calculator into accomplishing infinity.

3) . This integral is one way to prove that the harmonic alternation diverges.

Goals: Annual and adapt integrals using antiderivatives.

Skills:

· Evaluate audible integrals appliance both antiderivatives and the TI-83. The audible basic can be affected with antiderivatives (using the Fundamental Assumption of Calculus) or by afterwards methods (using fnInt on the TI-83). You can additionally use CALC 7 (∫f(x)dx) on the graphing window.

· Be able to annual an abnormal integral. An abnormal basic involves aloft as one of the integrands. Therefore, to appraise an abnormal basic exactly, we allegation use the FTC and some complete ideas.

Reading: Breadth 7.4.

Day 45

Activity: Allegory Antiderivatives.

We will revisit an action from Chapter 2 (Day 17), and apprehension (hopefully) how abundant simpler the aftermost allotment is now that we apperceive about integration.

Activity 2: Ciphering the acquired appliance a graph, and advice back.

Our abutting action is a echo of what we did on Day 17. It will booty beneath time than before, hopefully.

Each of you will sketch an almost action on a allotment of paper, labeling it “Original Curve” and putting your name on it. You will afresh canyon your blueprint to accession else; they will blueprint the acquired function on a abstracted breadth of paper, labeled with “Derivative Ambit for <insert name here>”. The actuality drawing the acquired will acquire to anxiously appraisal the slopes, so a calibration is needed. Finally, the added person will canyon the acquired blueprint to a third actuality (keep the aboriginal abreast to assay with later); the third actuality will attack to alter the aboriginal graph based abandoned on the advice from the acquired graph. Caution: this aftermost allotment is tricky.

Note that the aftermost part isn’t about as catchy now as it was on Day 17; we now acquire the FTC to adviser us in absolutely how abundant to achieve the aboriginal blueprint dispatch or fall.

Goals: Goal.

Skills:

· Be able to draw an antiderivative accustomed a acquired graph. On Day 17 it was difficult during our exercise to reconstruct the original action from the acquired graph, because we didn’t apperceive how abundant to increase or abatement the action aloof alive the derivative. Now, afterwards belief the FTC, we apperceive the breadth is the important missing factor. With this knowledge, you should now be able to draw authentic antiderivatives, as this is absolutely what they are.

Reading: Breadth 9.1.

Day 46

Activity: Description. Quiz 8 today.

Activity: Introduction to Multivariate Functions.

The complete apple is rarely explained by simple one-variable functions. Aggregate depends on aggregate else. The complication is sometimes daunting. However, we can try to archetypal things with algebraic formulas, and these about prove useful. For example, we apperceive that the bulk of money in a coffer annual can be represented by the blueprint . We can appearance B as a action of three variables: P, r, and t. Of course, in the complete apple the annual antithesis won’t consistently be predicted by this blueprint unless the annual is larboard completely alone, and the coffer doesn’t abutting the account. In Action 1, we will assay how to alarm a multi-variable action with a table.

After you assignment on the three activities today, I will assay . This function is tougher than the one you’re alive on, but we should see all the same issues.

Activity: Anecdotic a multivariate action with a table.

In your groups, create tables of ethics for this two-variable function: . The ambition is to back to a clairvoyant what the assorted ethics of B ability be. I will let each accumulation adjudge how to achieve the table; we will assay amid groups to see if you chose agnate methods.

Is B an accretion or a abbreviating function?

Goals: Introduce multivariate functions.

Skills:

· Understand how to represent a multivariate function with a table of values. Tables can alarm multivariate functions, but they are not as acceptable as graphs. On the added hand, graphs can be difficult to aftermath or interpret, and sometimes accepting the raw numbers is better. The best admission is to acquire a formula, but abounding complete apple settings don’t crop accepted formulas (daily highs beyond the country is one example).

Reading: Breadth 9.1.

Day 47

Activity: Graphing with cantankerous sections. Homework 8 due today.

Graphically, we appearance multivariate functions by captivation all but two variables constant, and afresh graphing the actual two variables appliance acute techniques we already know. Because we appearance the abased variable abnormally than the complete variables, the techniques abatement into two basic types.

Cross-sections action aback the dependent capricious is one of the two variables we graph. In a three-dimensional setting, we can brainstorm we have “sliced” the apparent angular and are attractive at the apparent from the side, in a cross-section. If we band up a alternation of cross-sections, we may be able to anticipate the three-dimensional apparent accurately. Cross-sections can be done from any dimension, as continued as the abased capricious is on the vertical axis.

Activity: Anecdotic a multivariate action with cross-sections.

Using hold P connected (choose some values) and draw the resulting B vs. t graphs. Afresh echo captivation t connected and cartoon the B vs. P graphs. Do they accord you the aforementioned consequence of the surface? Is it the aforementioned as the consequence you got in the action from Day 46?

When graphing cross-sections or contours, we may appetite to blueprint a able alternation of ethics for x, or z. In our calculators, we can alter the capricious with a annual that will achieve this for us, extenuative a lot of typing. For example, if we wanted to blueprint for z = -20, -10, 0, 10, and 20, we can admission this: Y1={-20,-10,0,10,20}^2-x-3. The calculator will blueprint aboriginal , afresh , etc. One check to this admission is that aback you TRACE, you won’t apperceive which bulk in the annual the ambit represents.

Goals: Explore cross-sections as a way to appearance multivariate functions.

Skills:

· Be able to aftermath cross-sections for a multivariate function. To achieve a array of a multivariate function, authority all but one of the complete variables constant; afresh blueprint the abased capricious against that aftermost complete variable. Naturally, if there are abounding complete variables captivated constant, it will be difficult to anticipate the complete surface. In the three-dimensional case, we can anticipate of this approach as vertical “slices” of the surface, beheld from the side.

Reading: Breadth 9.2.

Day 48

Activity: Graphing with contours.

Contours action in a three-dimensional apparent aback the abased capricious is captivated constant, and the added two variables are graphed. Due to the attributes of functions, cross-sections will consistently actualize graphable formulas, but contours may aftereffect in article absolutely difficult to create. For example, it’s not at all bright when we activate what ethics to use for the abased variable. Accepted uses for contours are maps. You acquire apparent acclimate maps that highlight temperatures. Instead of artlessly assuming the isotherms (lines of according temperatures) blush is commonly used.

Activity: Anecdotic a multivariate action with contours.

In practice, you will best acceptable not be bearing contours. Added about you will adapt them. But we appetite to be able to aftermath contours for simpler functions. Again, appliance , actualize some contours. You will allegation to acquire some ethics of B to achieve the contours for. It is not consistently bright what ethics will achieve the best sense. Balloon and absurdity may be in order. Does this ambit blueprint accord you the aforementioned consequence that you got in the activities from Days 46 and 47?

When graphing cross-sections or contours, we may appetite to blueprint a able alternation of ethics for x, or z. In our calculators, we can alter the capricious with a annual that will achieve this for us, extenuative a lot of typing. For example, if we wanted to blueprint for z = -20, -10, 0, 10, and 20, we can admission this: Y1={-20,-10,0,10,20}^2-x-3. The calculator will blueprint aboriginal , afresh , etc. One check to this admission is that aback you TRACE, you won’t apperceive which bulk in the annual the ambit represents.

Goals: Explore contours as a way to appearance multivariate functions.

Skills:

· Be able to apprehend and adapt contours for a multivariate function. Contour diagrams are angle from above, basically. Brainstorm attractive bottomward on the surface, in the case of three dimensions. Contours represent accumbent “slices”. Contours may be difficult to produce, as the curves traced out may not be functions at all (for example: circles at a about maximum).

Reading: Breadth 9.3.

Day 49

Activity: Calculating Partial Derivatives.

Just as for one-variable functions, we can allocution about derivatives with multivariate functions. Basically, we will let one of the variables abide connected and assay how the added capricious changes. This abode is alleged partial derivatives. All that we apperceive about derivatives from beforehand chapters administer here. One new aspect is that there are several possible derivatives. We additionally use different notations (see folio 361).

Activity: Annual apportioned derivatives from collapsed data.

Problem 8 folio 366.

Goals: Calculate apportioned derivatives from collapsed data.

Skills:

· Be able to appraisal apportioned derivatives from collapsed data. Calculating a acquired from a table in two or added ambit is no altered than it was in Chapter 2. We use slopes of secant lines, and due to the attributes of collapsed data, we can abandoned “zoom in” so much. The abandoned ambush is to pay absorption to which capricious is actuality captivated constant.

Reading: Breadth 9.4.

Day 50

Activity: Added apportioned derivatives.

There is accession new abstraction about derivatives that we haven’t encountered before: the alloyed added partial derivative. The approved second apportioned derivatives admeasurement concavity, the aforementioned as the one-variable second derivatives. But aback we booty the alloyed added apportioned acquired (see folio 370) we are absolutely ciphering how the change in one administration changes as we move in the added direction. We are added ciphering a affectionate of “twisting” in the surface. We will achieve added use of this on Day 51 aback we allocate the extrema.

Activity 1: Annual apportioned derivatives from graphs, both cross-sections and contours.

Using your graphs from from the activities on Days 46 and 47, appraisal some ethics of the apportioned derivatives.

Activity 2: Annual apportioned acquired formulas.

Verify your answers in Action 1 appliance algebra. Then assignment on a few of problems 26 to 37 on folio 372.

Goals: Calculate apportioned derivatives from graphs (both cross-sections and contours), and from formulas.

Skills:

· Be able to appraisal apportioned derivatives from graphs. From cross-section graphs, we can appraisal the apportioned acquired for that capricious in aloof the aforementioned way as in Chapters 2 and 3. For the ambit graphs, we allegation use a altered approach. Typically, we will appraisal the aberration amid two contours, and accurate the arrangement of the change in contours to ambit amid contours as the acquired in that direction.

· Be able to annual apportioned derivatives from formulas. Appliance the formulas from Chapter 4, we can annual apportioned derivatives exactly. The abandoned adversity is befitting clue of which capricious is accustomed to vary; our characters is advised to admonish us of this (see folio 361).

Reading: Breadth 9.5.

Day 51

Activity: Multivariate Optimization. Quiz 9 today.

To aerate or abbreviate a multivariate function, we use the aforementioned belief we did for one-variable function: analytic points and the added acquired test. The accommodation are hardly altered for the added acquired test, and we will do several problems today practicing this technique. Aboriginal we break the aboriginal derivatives accordingly by ambience them to zero. This will accord us candidates for extrema. Now we use the added acquired assay (page 376) to help allocate the candidates as maxima, minima, or neither. It is additionally accessible the assay is inconclusive. In those situations, we allegation use some added approach, conceivably article affiliated to the aboriginal derivative test, although appliance that admission is a bit trickier in multiple dimensions. Note: analytic the aboriginal acquired formulas accompanying for all variables present may be very difficult. One adapted case is aback all the derivatives are linear. Afresh you can use techniques from MATH 204 (the beeline algebra/matrix results).

In accession to appliance formulas, achieve sure you can use graphs additionally to acquisition extrema. Problems 1 and 2, and 14 to 16 on folio 377 are acceptable practice.

Activity: Optimizing multivariate functions.

I will assignment problems 3 and 6 in class. Try as abounding of the others as you can. Problems 3 through 12 folio 377.

Goals: Understand how the derivatives can be acclimated to acquisition the extrema in multivariate functions.

Skills:

· Be able to acquisition the extrema appliance a ambit graph. Extrema on contour graphs are represented with bankrupt loops. To acquisition whether they are maxima or minima entails advantageous absorption to the ethics of the contours about the points.

· Be able to acquisition extrema appliance algebra. Appliance the added acquired test, you should be able to allocate the extrema as maxima, minima, or neither. In some cases, the added derivative assay is inconclusive.

Reading: Breadth 9.5.

Day 52

Activity: Description.

A new array of analytic point occurs in assorted ambit alleged a saddle point. You can anticipate of a abundance canyon as one example; in one administration (going over the pass) the action is a best but in the added administration (going from one abundance to the added through the pass) the action is a minimum. The second acquired assay will allocate these saddle credibility as “neither”.

Activity: Description.

Activity

Goals: Goal.

Skills:

· Understand the saddle point in multivariate functions. In one direction, a saddle point is a maximum, but in accession administration it is a minimum. If we are aggravating to optimize a function, it is analytic to apperceive if our analytic credibility are maxima, minima, saddle points, ridges, etc.

Reading: Section.

Day 53

Activity: Description. Arrangement 9 due today.

Description.

Activity: Description.

Activity

Goals: Goal.

Skills:

· Skills. Description

· Skills. Description

Reading: Chapters 5, 6, 7, and 9.

Day 54

Activity: Presentation 3.

Reading: Chapters 5, 6, 7, and 9.

Day 55

Activity: Description.

Description.

Activity: Description.

Activity

Goals: Goal.

Skills:

· Skills. Description

· Skills. Description

Reading: Chapters 5, 6, 7, and 9.

Day 56

Activity: Exam 3.

This aftermost assay covers integrals, including antiderivatives, and multivariate functions, Chapters 5, 6, 7, and 9. Some of the questions ability be assorted choice. Others will crave you to appearance your formed out solution.

Populations for the 50 states, DC, and the USA, by decade. (in thousands)

AL

AK

AZ

AR

CA

CO

CT

DE

DC

FL

GA

HI

ID

IL

IN

IA

KS

KY

1790

238

59

83

74

1800

1

251

64

8

163

6

221

1810

9

1

262

73

16

252

12

25

407

1820

128

14

275

73

23

341

55

147

564

1830

310

30

298

77

30

35

517

157

343

688

1840

591

98

310

78

34

54

691

476

686

43

780

1850

772

210

93

371

92

52

87

906

851

988

192

982

1860

964

435

380

34

460

112

75

140

1057

1712

1350

675

107

1156

1870

997

10

484

560

40

537

125

132

188

1184

15

2540

1680

1194

364

1321

1880

1263

33

40

803

865

194

623

147

178

269

1542

33

3078

1978

1625

996

1649

1890

1513

32

88

1128

1213

413

746

168

230

391

1837

89

3826

2192

1912

1428

1859

1900

1829

64

123

1312

1485

540

908

185

279

529

2216

154

162

4822

2516

2232

1470

2147

1910

2138

64

204

1574

2378

799

1115

202

331

753

2609

192

326

5639

2701

2225

1691

2290

1920

2348

55

334

1752

3427

940

1381

223

438

968

2896

256

432

6485

2930

2404

1769

2417

1930

2646

59

436

1854

5677

1036

1607

238

487

1468

2909

368

445

7631

3239

2471

1881

2615

1940

2833

73

499

1949

6907

1123

1709

267

663

1897

3124

423

525

7897

3428

2538

1801

2846

1950

3062

129

750

1910

10586

1325

2007

318

802

2771

3445

500

589

8712

3934

2621

1905

2945

1960

3267

226

1302

1786

15717

1754

2535

446

764

4952

3943

633

667

10081

4662

2758

2179

3038

1970

3444

303

1775

1923

19971

2210

3032

548

757

6791

4588

770

713

11110

5195

2825

2249

3221

1980

3894

402

2717

2286

23668

2890

3108

594

638

9747

5463

965

944

11427

5490

2914

2364

3660

1990

4040

550

3665

2351

29760

3294

3287

666

607

12938

6478

1108

1007

11430

5544

2777

2478

3685

2000

4447

627

5131

2673

33872

4301

3406

784

572

15982

8186

1294

1294

12419

6080

2926

2688

4042

LA

ME

MD

MA

MI

MN

MS

MO

MT

NE

NV

NH

NJ

NM

NY

NC

ND

OH

1790

97

320

379

142

184

340

394

1800

152

342

423

8

184

211

589

478

45

1810

77

229

381

472

5

31

20

214

246

959

556

231

1820

153

298

407

523

9

75

67

244

278

1373

639

581

1830

216

399

447

610

32

137

140

269

321

1919

736

938

1840

352

502

470

738

212

376

384

285

373

2429

753

1519

1850

518

583

583

995

398

6

607

682

318

490

62

3097

869

1980

1860

708

628

687

1231

749

172

791

1182

29

7

326

672

94

3881

993

2340

1870

727

627

781

1457

1184

440

828

1721

21

123

42

318

906

92

4383

1071

2

2665

1880

940

649

935

1783

1637

781

1132

2168

39

452

62

347

1131

120

5083

1400

37

3198

1890

1119

661

1042

2239

2094

1310

1290

2679

143

1063

47

377

1445

160

6003

1618

191

3672

1900

1382

694

1188

2805

2421

1751

1551

3107

243

1066

42

412

1884

195

7269

1894

319

4158

1910

1656

742

1295

3366

2810

2076

1797

3293

376

1192

82

431

2537

327

9114

2206

577

4767

1920

1799

768

1450

3852

3668

2387

1791

3404

549

1296

77

443

3156

360

10385

2559

647

5759

1930

2102

797

1632

4250

4842

2564

2010

3629

538

1378

91

465

4041

423

12588

3170

681

6647

1940

2364

847

1821

4317

5256

2792

2184

3785

559

1316

110

492

4160

532

13479

3572

642

6908

1950

2684

914

2343

4691

6372

2982

2179

3955

591

1326

160

533

4835

681

14830

4062

620

7947

1960

3257

969

3101

5149

7823

3414

2178

4320

675

1411

285

607

6067

951

16782

4556

632

9706

1970

3645

994

3924

5689

8882

3806

2217

4678

694

1485

489

738

7171

1017

18241

5084

618

10657

1980

4206

1125

4217

5737

9262

4076

2521

4917

787

1570

801

921

7365

1303

17558

5880

653

10798

1990

4220

1228

4781

6016

9295

4375

2573

5117

799

1578

1202

1109

7730

1515

17990

6629

639

10847

2000

4469

1275

5296

6349

9938

4919

2845

5595

902

1711

1998

1236

8414

1819

18976

8049

642

11353

OK

OR

PA

RI

SC

SD

TN

TX

UT

VT

VA

WA

WV

WI

WY

USA

1790

434

69

249

36

85

692

56

3929

1800

602

69

346

106

154

808

79

5308

1810

810

77

415

262

218

878

105

7240

1820

1049

83

503

423

236

938

137

9638

1830

1348

97

581

682

281

1044

177

12866

1840

1724

109

594

829

292

1025

225

31

17069

1850

12

2312

148

669

1003

213

11

314

1119

1

302

305

23192

1860

52

2906

175

704

5

1110

604

40

315

1220

12

377

776

31443

1870

91

3522

217

706

12

1259

819

87

331

1225

24

442

1055

9

38558

1880

175

4283

277

996

98

1542

1592

144

332

1513

75

618

1315

21

50189

1890

259

318

5258

346

1151

349

1768

2236

211

332

1656

357

763

1693

63

62980

1900

790

414

6302

429

1340

402

2021

3049

277

344

1854

518

959

2069

93

76212

1910

1657

673

7665

543

1515

584

2185

3897

373

356

2062

1142

1221

2334

146

92228

1920

2028

783

8720

604

1684

637

2338

4663

449

352

2309

1357

1464

2632

194

106022

1930

2396

954

9631

687

1739

693

2617

5825

508

360

2422

1563

1729

2939

226

123203

1940

2336

1090

9900

713

1900

643

2916

6415

550

359

2678

1736

1902

3138

251

132165

1950

2233

1521

10498

792

2117

653

3292

7711

689

378

3319

2379

2006

3435

291

151326

1960

2328

1769

11319

859

2383

681

3567

9580

891

390

3967

2853

1860

3952

330

179323

1970

2559

2092

11801

950

2591

666

3926

11199

1059

445

4651

3413

1744

4418

332

203302

1980

3025

2633

11865

947

3121

691

4591

14226

1461

511

5347

4132

1950

4706

470

226542

1990

3146

2842

11882

1003

3487

696

4877

16987

1723

563

6187

4867

1793

4892

454

248710

2000

3451

3421

12281

1048

4012

755

5689

20852

2233

609

7079

5894

1808

5364

494

281422

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