MATH 206 Applied Calculus for Business

Spring 2016

Section 001 8:00 to 9:00 MTW F

Section 002 9:10 to 10:10 MTW F

Instructor: Dr. Chris Edwards          Phone: 948-3969       Office: Swart 123

Classroom: Swart 303     Text: Applied Calculus 5th edition, by Hughes-Hallett, Gleason, Lock, Flath, et al.

Required Calculator: TI-83, TI-83 Plus, or TI-84 Plus, by Texas Instruments. Other TI graphics calculators (like the TI-86) do not have the same commands we will be using and may cause you troubles.

Catalog Description: This course follows Mathematics 204. Topics include logarithmic and exponential functions, differential and integral calculus and their application to business problems. Prerequisite: Mathematics 104, 108 or 204 with a grade of C or better or placement.

Course Objectives: (Click here for full document.) Topics introduced in Math 206, such as marginal analysis, optimization, and finding total change, are used in subsequent Business and Economics courses. The ideas covered include function, derivative, and integral concepts. Upon completion of Math 206 students will be familiar with basic functions and be able to calculate and estimate derivatives and integrals using a variety of methods. A firm grounding in these topics will prepare students for success in later classes.

Liberal Arts Education: MATH 206 is part of the University Studies Program (USP) as an EXPLORE course in the NATURE category, and contributes to an education in the Liberal Arts. In this sense, ÒLiberalÓ means ÒbroadÓ, and ÒArtsÓ means ÒskillsÓ, so that someone educated in the Liberal Arts is able to think critically and make connections to a variety of disciplines and fields. Someone educated in the Liberal Arts is a responsible member of society, is engaged in the community, and is able to understand the issues of the day. They are problem solvers, and have learned how to learn new skills and knowledge. The field of Mathematics is vital to a Liberal Arts education, as quantitative data is collected and used to create mathematical models in virtually every discipline. Calculus is the study of how functions change and is guided by the symbolic manipulation of expressions. Being able to analyze data and draw conclusions from data through modeling is a vital component of an educated member of society.

Grading: Final grades are based on 410 points:

 

Topic

Points

Tentative Date

Exam 1

Functions

80 pts.

February 23

Exam 2

Derivatives

90 pts.

April 1

Exam 3

Integrals and Multivariate Derivatives

90 pts.

May 13

Group Presentations

20 Points Each

60 pts.

Before Exams

Homework

10 Points Each

90 pts.

Weekly

Attendance is a very important component of success in my class because many of the skills and lessons we will learn will be a direct result of classroom activities that cannot be reproduced easily. Please attend class as often as you can. You are responsible for any material you miss. The Day By Day notes will help you greatly in this regard.


Text Box: Final Grades:
Grade	Points (Percent)
A	369 (90 %)
A-	357 (87 %)
B+	340 (83 %)
B	328 (80 %)
B-	316 (77 %)
C+	299 (73 %)
C	287 (70 %)
C-	275 (67 %)
D+	258 (63 %)
D	246 (60 %)
D-	234 (57 %)
F	233 or fewer

Presentations: To demonstrate your competency in Calculus via oral communication, there will be three presentations, each worth 20 points. The descriptions of the presentations are in the Day By Day Notes. I will assign you to your groups for these presentations, as I want to avoid you having the same members each time. I expect each person in a group to contribute to the work; however, you may allocate the work in any way you like. If a group member is not contributing, see me as soon as possible so I can make a decision about what to do. Part of your presentation grade will be based on your own evaluations of how each person contributed to the presentation. The topics are: 1 – Modeling Population Growth (February 22). 2 – Describing Functions Using Derivatives (March 30). 3 – Multivariate Functions (May 11).

Homework: To demonstrate your competency in Calculus via written communication, I will collect several homework problems about once a week. The due dates are listed on the course outline below. While I will only be grading a few problems, I presume that you will be working on many more than just the ones I assign. I suggest that you work together in small groups on the homework for this class. I expect is a well thought-out, complete discussion of the problem. Please donÕt just put down a numerical answer; I want to see how you did the problem. (You wonÕt get full credit for just numerical answers.) The method you use and your description is much more important to me than the final numerical answer. Furthermore, as this is your opportunity to show me what you have learned, your submitted homework should be neatly written or typed, without crossed out sections or scribbles. Be professional and make your work products reflect your own professionalism. Important Grading Feature: If your homework percentage is lower than your exam percentage, I will replace your homework percentage with your exam percentage. Therefore, your final homework percentage cannot be lower than your exam percentage.

ePortfolio Information: Math 206 is part of the USP and is designated as an EXPLORE course in the NATURE category. Therefore, you will need to include several work products from the course in your ePortfolio. I have designated five problems in the attached Homework Problems as artifacts for your ePortfolio. You will need to scan each graded problem and include a paragraph written to your future self about describing how this problem fits into the overall purpose of Calculus as a discipline. You may want to compose these reflections paragraphs later in the course, after you have developed an appreciation of Calculus, instead of at the time the work is graded. Your CONNECT course that you take as the final part of the USP program will require you to look back on these artifacts, so you need to make sure you give yourself good products with which to work.

Office Hours: Office hours are times when I will be in my office to help you with the course. You may ask questions about your homework, about the text, about topics from class, or any other issues you may have. You will not be bothering me as I have set aside these times in my schedule solely for talking to students about coursework. There will be many other times when I am in my office. If I am in and not busy, I will be happy to help. My office hours for Spring 2016 semester are 10:20 to 11:00, Monday, Tuesday, Wednesday, and Friday, and 3:00 to 4:00 Monday, or by appointment.

Early Alert Information: To provide you with early feedback on your performance in the course, our class will participate in the Early Alert program. It is common for students to be unaware of or over-estimate their academic performance in classes, so this will help you be aware early on of your progress and provide strategies for success in the classroom. The registrarÕs office will send an email to students with academic and/or attendance issues during the 5th week of classes. If you receive such an email, be sure you read it carefully and arrange to meet with me or a counselor to develop an appropriate action plan.

Philosophy: I strongly believe that you, the student, are the only person who can make yourself learn. Therefore, whenever it is appropriate, I expect you to discover the mathematics we will be exploring. I do not feel that lecturing to you will teach you how to do mathematics. I hope to be your guide while we learn some mathematics, but you will need to do the learning. I expect each of you to come to class prepared to digest the dayÕs material. That means you will benefit most by having read each section of the text and the Day By Day notes before class.

My personal belief is that one learns best by doing. I believe that you must be truly engaged in the learning process to learn well. Therefore, I do not think that my role as your teacher is to tell you the answers to the problems we will encounter; rather I believe I should point you in a direction that will allow you to see the solutions yourselves. To accomplish that goal, I will find different interactive activities for us to work on. Your job is to use me, your text, your friends, and any other resources to become adept at the material. The Day By Day notes also include Skills that I expect you to attain.


 

Monday

Tuesday



Wednesday

Friday

February 1 Day 1
Introduction
Section 1.1

February 2 Day 2
Functions
Section 1.1

February 3 Day 3
Linear Functions
Section 1.2

February 5 Day 4
Rates
Section 1.3

February 8 Day 5
Economics Examples
Section 1.4

February 9 Day 6
Econ Examples
Section 1.4

February 10 Day 7
Exponential Functions
Section 1.5

February 12 Day 8
Homework 1 Due
Logarithms
Section 1.6

February 15 Day 9
Growth and Decay
Section 1.7

February 16 Day 10
Transformations
Section 1.8

February 17 Day 11
Polynomials
Section 1.9

February 19 Day 12
Homework 2 Due
Polynomials
Section 1.9

February 22 Day 13
Presentation 1

February 23 Day 14
Exam 1

February 24 Day 15
Instantaneous Change
Section 2.1

February 26 Day 16
Derivatives
Section 2.2

February 29 Day 17
Derivatives
Section 2.3

March 1 Day 18
Second Derivative
Section 2.4

March 2 Day 19
Econ Examples
Section 2.5

March 4 Day 20
Homework 3 Due
Polynomial Formulas
Section 3.1

March 7 Day 21
Exponential Formulas
Section 3.2

March 8 Day 22
Chain Rule
Section 3.3

March 9 Day 23
Product and Quotient Rules
Section 3.4

March 11 Day 24
Derivative Rules
Chapter 3

March 14 Day 25
Extrema
Section 4.1

March 15 Day 26
Homework 4 Due
Inflection Points
Section 4.2

March 16 Day 27
Global Extrema
Section 4.3

March 18 Day 28
Econ Examples
Section 4.4

March 28 Day 29
Econ Examples
Section 4.5

March 29 Day 30
Homework 5 Due
Logistic Growth and Surge Functions
Sections 4.7 and 4.8

March 30 Day 31
Presentation 2

April 1 Day 32
Exam 2

April 4 Day 33
Definite Integrals
Section 5.1

April 5 Day 34
Definite Integrals
Section 5.2

April 6 Day 35
Areas
Section 5.3

April 8 Day 36
Areas
Section 5.3

April 11 Day 37
Homework 6 Due
Fundamental Theorem
Section 5.4

April 12 Day 38
Fundamental Theorem
Section 5.5

April 13 Day 39
Antiderivatives
Section 6.1

April 15
NO CLASS

April 18 Day 40
Antiderivatives
Section 6.2

April 19 Day 41
Analyzing Antiderivatives
Section 6.3

April 20 Day 42
Homework 7 Due
Econ Examples
Section 6.4

April 22 Day 43
Econ Examples
Section 6.5

April 25 Day 44
Substitution
Section 6.6

April 26 Day 45
Substitution
Section 6.6

April 27 Day 46
Multivariate Functions
Section 8.1

April 29 Day 47
Homework 8 Due
Cross Sections
Section 8.1

May 2 Day 48
Contours
Section 8.2

May 3 Day 49
Partial Derivatives
Section 8.3

May 4 Day 50
Second Partial Derivatives
Section 8.4

May 6 Day 51
Optimization
Section 8.5

May 9 Day 52
Optimization
Section 8.5

May 10 Day 53
Homework 9 Due
Review

May 11 Day 54
Presentation 3

May 13 Day 55
Exam 3


 

Homework Assignments

 

Homework 1, Due February 12

 


1)   In the Andes mountains in Peru, the number, , of species of bats in a region is a function of the elevation, , in feet above sea level, so .

a)    Write a sentence interpreting the statement  in terms of bat species and elevation.

b)   Write sentences explaining the meaning of the vertical intercept, , and of the horizontal intercept, , in the graph below.

2)   You drive at a constant speed from Chicago to Detroit, a distance of 275 miles. About 120 miles from Chicago you pass through Kalamazoo, Michigan.

a)    Choose a specific speed (such as 60 mph) and sketch a graph of your distance from Kalamazoo as a function of time since your trip began.

b)   How does the shape of your graph change for a speed that is 10% faster than the one you chose for part a)? Explain.

3)   A company rents cars at a daily rate of $40 and also charges 15¢ per mile traveled. Its competitor has a daily rate of $50 but only charges 10¢ per mile.

a)    Give formulas for each companyÕs fee as a function of the distance traveled.

b)   Graph both functions on the same suitable axes. (Your intent is to use the graph to see which company is a better choice for various day-trip distances.)

c)    Use your graph in part b) to describe how you decide which company to choose.

4)   In a California town, the monthly charge for waste collection is $8 for 32 gallons of waste and $12.32 for 68 gallons of waste.

a)    Find a linear function for the cost, , of waste collection as a function of the number of gallons of waste, .

b)   Using appropriate units, write a sentence interpreting (in everyday English) the slope of your line.

c)    Using appropriate units, write a sentence interpreting  (in everyday English) the intercept of your line.

5)   Do you expect the average rate of change (in units per year) of each of the following to be positive, negative, or zero? Explain your reasoning.

a)    Number of acres of rain forest in the world.

b)   Population of the world.

c)    Number of polio cases each year in the US, since 1950.

d)    Height of a sand dune that is being eroded.

e)    Cost of living in the US.


 

Homework 2, Due February 19

 


1)   When the price, , charged for a boat tour was $25, the average number of passengers per week, , was 500. When the price was reduced to $20, the average number of passengers per week increased to 650. Find a formula for the demand function, as a function of price, assuming it is linear.

2)   Determine whether each of the following functions could be represented exactly with a linear function, an exponential function, or neither function. Explain how you know. Find a formula for the function if you determined it was either linear or exponential.

a)

0

1

2

3

 

10.5

12.7

18.9

36.7

 

b)

–1

0

1

2

 

50.2

30.12

18.072

10.8432

 

c)

0

2

4

6

 

27

24

21

18

 

3)   During a recession a firmÕs revenue declines continuously so that the revenue, , measured in millions of dollars, in  yearsÕ time is estimated by .

a)    Indicate the current revenue and calculate the estimated revenue in two yearsÕ time.

b)   After how many years will the estimated revenue have declined to $2.7 million?

4)   You win $38,000 in the state lottery, to be paid in two $19,000 installments: one now and one in a year. A friend offers you $36,000 now in return for your lottery ticket. Instead of accepting your friendÕs offer, you could take out a one-year loan at an interest rate of 8.25% per year, compounded annually. The loan will be paid back by a single payment of $19,000 (your second lottery check) at the end of the year. Which is a better deal for you, your friendÕs offer of $36,000, or your plan of taking the loan plus your first check of $19,000? [Hint: Figure out what the bank will let you borrow.]

5)   Complete the following table:

–3

–2

–1

0

1

2

3

0

1

2

3

2

1

0

3

2

2

0

–2

–2

–3

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 


 

Homework 3, Due March 5

 


1)   Allometry is the study of the relative size of different parts of a body as a consequence of growth. In this problem, you will check the accuracy of an allometric hypothesis: the weight of a fish is proportional to the cube of its length. The data below show the weight, , in gm, of plaice (a type of fish) to its length, , in cm. Does the data support the hypothesis that  (approximately)? If so, estimate the constant of proportionality, .

33.5

35.5

37.5

39.5

41.5

43.5

332

391

455

538

623

724

 

(à Artifact 1 à) 2)         The data below gives , the number of households, in millions, in the US with cable television  years since 1998.

0

2

4

6

8

10

64.65

66.25

66.73

65.73

65.14

64.87

 

a)    Does  appear to be positive or negative or zero? Write a sentence interpreting the importance of this result.

b)   Estimate  and  each time writing a sentence interpreting the value.

3)   Complete the following table:

0

5

10

15

20

100

70

55

46

40

 

 

 

 

 

 

4)   The average weight, , in pounds, of an adult is a function, , of the average number of calories per day, , consumed.

a)    Write a sentence interpreting the statements  and  in terms of daily caloric consumption and weight.

b)   What are the units of ?

5)   The data below shows , the number of Facebook subscribers, in millions, worldwide at 3-month intervals.

Mar. 2011

Jun. 2011

Sep. 2011

Dec. 2011

Mar. 2012

664.0

710.7

756.9

799.1

835.5

 

a)    Calculate the average rate of change of  per month for the time intervals shown. How do these values relate to ?

b)   What can you observe about the sign of  for the 12-month period between March, 2011 and March, 2012? [Hint: Calculate a few values of the second derivative using your values from part a).]


 

Homework 4, Due March 15

 


1)   A companyÕs cost of producing  liters of a chemical is  dollars and this quantity can be sold for  dollars. Suppose  and .

a)    What is the profit at a production level of 2000?

b)   If  and , what is the approximate change in profit if  is increased from 2000 to 2001? Should the company increase or decrease production from  and why?

c)    If  and , should the company increase or decrease production from  and why?

2)   The demand for a product is given by , where  is in dollars and  is in thousands of units.

a)    Find the - and -intercepts for this function and write a sentence for each interpreting them in terms of demand for this product.

b)   Find  and give units with your answer. Explain what it tells you about demand.

c)    Find  and give units with your answer. Explain what it tells you about demand.

(à Artifact 2 à) 3)         In 2009, the population of Hungary was approximated by , where  is in millions of people and  is in years since 2009. Assume the trend has continued and will continue in the future.

a)    What does this model predict for the population of Hungary in the year 2020?

b)   At what rate does this model predict the population of Hungary will be changing in the year 2020? Include units with your answer.

4)   A firm estimates that the total revenue, , received from the sale of  items is given by . Calculate the exact marginal revenue when  and then approximate the marginal revenue using the derivative.

5)   Find the equation of the tangent line to the graph of  at . Check your answer by graphing the function and your equation on the same axes and verifying visually that it is indeed the tangent line at . [Hint: You must find  first to get the slope of the line.]

 


 

Homework 5, Due March 29

 


1)   Suppose  has a continuous derivative whose values are given in the following table.

0

1

2

3

4

5

6

7

8

9

10

5

2

1

–2

–5

–3

–1

2

3

1

–1

 

a)    Estimate the -coordinates of critical points of  for .

b)   For each critical point, indicate if it is a local maximum of , a local minimum, or neither and explain your reasoning.

2)   Find all critical points for the function . Sketch several members of the family on the same axes. Discuss the effect of the parameter  on the graph.

3)   Using the figure below, indicate approximately where the inflection points are if the graph shows:

a)    The graph given is a graph of .

b)   The graph given is a graph of the derivative .

c)    The graph given is a graph of the second derivative .

4)   The function  is positive everywhere, continuous everywhere, and has a global maximum at the point . Sketch a possible graph of  if  and  have the same sign for , but opposite signs for . [Hint: The global maximum might not have a derivative value of zero.]

(à Artifact 3 à) 5)         If you have 100 feet of fencing and want to enclose a rectangular area up against an existing straight wall (longer than 100 feet), what is the largest area you can enclose? (Assume you use all of your fencing.) How does your answer change if you require the enclosure to be square instead of just rectangular? [Hint for both parts: First write the area as a function of one side. Then use derivatives.]

 


 

Homework 6, Due April 11

 


(à Artifact 4 à) 1)         A demand function for a product is , where  is the quantity sold for price .

a)    Find an expression for the total revenue, , as a function of . [Hint: Recall that in general .]

b)   Differentiate  with respect to  to find the approximate marginal revenue, , as a function of . Estimate the marginal revenue when .

c)    Calculate the change in total revenue when production increases from  to  units. Confirm that the estimated  from part b) is approximately equal to the change in revenue from a one-unit increase in production, the actual .

2)   You are the manager of a firm that produces slippers that sell for $20 a pair. You are producing 1,200 pairs of slippers each month, at an average cost of $10 each. The marginal cost at a production level of 1,200 is $12 per pair.

a)    Are you making or losing money?

b)   Will increasing production increase or decrease your average cost? Your profit?

c)    Would you recommend that production be increased or decreased?

3)   An old rowboat has sprung a leak. Water is flowing into the boat at a rate, , given in the table below.

 minutes

0

5

10

15

 liters / minute

12

20

24

16

 

a)    Compute upper and lower estimates for the volume of water that has flowed into the boat during the 15 minutes.

b)   Draw a graph that illustrates the lower estimate you calculated in part a).

4)   Use the graph below to estimate . (If you count boxes, be sure to be clear about how you dealt with partial boxes.)

5)   Using the graph below, rank the following five integrals in order from smallest value to largest. (Recall that –10 is smaller than –5.) Also indicate which integrals are negative and which are positive. Explain your reasoning.

I.        II.             III.

IV.      V.

 


 

Homework 7, Due April 20

 


1)   The birth rate, , in births per hour, of a bacteria population is given in the figure below. The other curve,  gives the death rate of the same population, in deaths per hour.

a)    Explain what the shape of each of these graphs tells you about the population.

b)   Use the graphs to find the time at which the net rate of increase of the population is at a maximum. Explain your reasoning.

c)    At time  the population has size . Sketch the graph of the total number born by time . Also sketch the graph of the number alive at time . Estimate the time at which the population is a maximum.

2)   The figure below shows the rate of change of the quantity of water in a water tower, in liters per day, during the month of April. If the tower had 12,000 liters of water in it on April 1, estimate the quantity of water in the tower on April 30.

3)   The graph below shows the derivative . If , find the values of , , and . Graph . [Note: Make sure you graph the critical points of  fairly; i.e., donÕt draw it flat if the derivative is not zero.]

4)   Find the indefinite integral: .

5)   A firmÕs marginal cost function is . Find the total cost function if the fixed costs are 200.

 


 

Homework 8, Due April 29

 


1)   Evaluate using the FTC: .

(à Artifact 5 à) 2)         The supply and demand curves for a product are given in the figure below.

a)    Estimate the equilibrium price and quantity.

b)   Estimate the consumer surplus and the producer surplus. [Warning: Using a triangle for demand will over-estimate the value.]

c)    Estimate the consumer surplus and the producer surplus if the price is artificially set low at  dollars per unit. Compare your answers to part b) above. Do the new values make sense?

3)   Your company needs $500,000 in two yearsÕ time for renovations and can earn 9% on investments.

a)    What is the present value of the renovations?

b)   If your company deposits money continuously at a constant rate throughout the two-year period, at what monthly rate should the money be deposited so that you have the $500,000 when you need it?

4)   Evaluate using substitution: .

5)   Evaluate using substitution: .

 


 

Homework 9, Due May 10

 


1)   An airport can be cleared of fog by heating the air. The amount of heat required,  (in calories per cubic meter of fog), depends on the temperature of the air,  (in ¡C), and the wetness of the fog  (in grams per cubic meter of fog). The figure below shows several cross-sections of  against  with  fixed.

a)    Estimate  and explain what information it gives us.

b)   Make a table of values for . Use  and , and , and .

2)   The contour diagram below shows your happiness as a function of love and money.

a)    Describe in words your happiness as a function of money, with love at a fixed level.

b)   Describe in words your happiness as a function of love, with money at a fixed level.

c)    Graph two different cross-sections with love fixed and two different cross-sections with money fixed.

3)   The figure below shows contours of  with values of  on the contours omitted. There are two points labeled on the figure,  and . Assume that .

a)    What is the sign of ? Explain.

b)   What is the sign of ? Explain.

c)    What is the sign of ? Explain.

4)   Calculate all four second-order partial derivatives and confirm that the mixed partials are equal, using .

5)   A missile has a guidance device which is sensitive to both temperature, , in ¡C, and humidity, , in percent. The range, , in km over which the missile can be controlled is given by . What are the optimal atmospheric conditions for controlling the missile, and what is the maximum range  over which it can be controlled?


 


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