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.
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 |
February 2 Day
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February 3 Day
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February 5 Day
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February 8 Day
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February 9 Day
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February 10
Day 7 |
February 12
Day 8 |
February 15
Day 9 |
February 16
Day 10 |
February 17
Day 11 |
February 19
Day 12 |
February 22
Day 13 |
February 23
Day 14 |
February 24
Day 15 |
February 26
Day 16 |
February 29
Day 17 |
March 1 Day 18 |
March 2 Day 19 |
March 4 Day 20 |
March 7 Day 21 |
March 8 Day 22 |
March 9 Day 23 |
March 11 Day
24 |
March 14 Day
25 |
March 15 Day
26 |
March 16 Day
27 |
March 18 Day
28 |
March 28 Day
29 |
March 29 Day
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March 30 Day
31 |
April 1 Day 32 |
April 4 Day 33 |
April 5 Day 34 |
April 6 Day 35 |
April 8 Day 36 |
April 11 Day
37 |
April 12 Day
38 |
April 13 Day
39 |
April 15 |
April 18 Day
40 |
April 19 Day
41 |
April 20 Day
42 |
April 22 Day
43 |
April 25 Day
44 |
April 26 Day
45 |
April 27 Day
46 |
April 29 Day
47 |
May 2 Day 48 |
May 3 Day 49 |
May 4 Day 50 |
May 6 Day 51 |
May 9 Day 52 |
May 10 Day 53 |
May 11 Day 54 |
May 13 Day 55 |
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 |
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0 |
1 |
2 |
3 |
2 |
1 |
0 |
|
3 |
2 |
2 |
0 |
–2 |
–2 |
–3 |
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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?
Managed by chris
edwards
Last updated January 25, 2016