Units 2 (just parts a,b,c) and 3 (All parts) Written Work Practice
8. For f (x) =
x−5
x+4
find :
(a) Find the domain (in interval notation).
(b) Find any Vertical asymptotes or holes. Provide a written explanation why you are choosing Vertical asymptote
or hole.
(c) Find any Horizontal Asymptotes.
(d) Find all x and y intercepts
(e) Graph, finding additional points as needed.
9. For f (x) =
(x+2)(x−3)
2x2
find :
(a) Find the domain (in interval notation).
(b) Find any Vertical asymptotes or holes. Provide a written explanation why you are choosing Vertical asymptote
or hole.
(c) Find any Horizontal Asymptotes.
(d) Find all x and y intercepts
10. For f (x) =
6x+8000
2x+300
find :
(a) Find the domain (in interval notation).
(b) Find any Vertical asymptotes or holes. Provide a written explanation why you are choosing Vertical asymptote
or hole.
(c) Find any Horizontal Asymptotes.
(d) Find all x and y intercepts
(e) Graph, finding additional points as needed.
11. For f (x) =
x2 −6x+9
x−3
find :
(a) Find the domain (in interval notation).
(b) Find any Vertical asymptotes or holes. Provide a written explanation why you are choosing Vertical asymptote
or hole.
(c) Find any Horizontal Asymptotes.
(d) Find all x and y intercepts
(e) Graph, finding additional points as needed.
12. For f (x) =
x3 −4x
x−2
find :
(a) Find the domain (in interval notation).
(b) Find any Vertical asymptotes or holes. Provide a written explanation why you are choosing Vertical asymptote
or hole.
(c) Find any Horizontal Asymptotes.
(d) Find all x and y intercepts
(e) Graph, finding additional points as needed.
13. For f (x) =
3x
x2 +2
find :
(a) Find the domain (in interval notation).
(b) Find any Vertical asymptotes or holes. Provide a written explanation why you are choosing Vertical asymptote
or hole.
(c) Find any Horizontal Asymptotes.
(d) Find all x and y intercepts
(e) Graph, finding additional points as needed.
Unit 4 Written Work Practice
14. For the function g(x) = −2x
(a) Graph
(b) Find Domain
(c) Are there any Horizontal or Vertical asymptotes? If so, which and what are they?
(d) Find x and y intercepts if they exist.
15. For the function g(x) = 3x+1
(a) Graph
(b) Find Domain
(c) Are there any Horizontal or Vertical asymptotes? If so, which and what are they?
(d) Find x and y intercepts if they exist.
16. For the function g(x) = −3 + e1+x
(a) Graph
(b) Find Domain
(c) Are there any Horizontal or Vertical asymptotes? If so, which and what are they?
(d) Find x and y intercepts if they exist.
17. For the function g(x) = log2 x
(a) Graph
(b) Find Domain
(c) Are there any Horizontal or Vertical asymptotes? If so, which and what are they?
(d) Find x and y intercepts if they exist.
18. For the function g(x) = ln x
(a) Graph
(b) Find Domain
(c) Are there any Horizontal or Vertical asymptotes? If so, which and what are they?
(d) Find x and y intercepts if they exist.
19. A company is trying to introduce a new product to as many people as possible through television advertising in a
large metropolitan area with 2 million possible viewers. A model for the number of people N (in millions) who are
aware of the product after t days of advertising was found to be
N = 2(1 − e−0.037t )
Graph this function on [0, 50]. What value does N
(a) How many people are aware of the product after 5 days?
(b) When do 500,000 people become aware of the product?
(c) Graph the function on [0, 50].
(d) What value does N approach as t increases without bound? What does this limit mean?
20. In its first 10 years the Janus Flexible Income Fund produced an average annual return of 9.58%. Assume that
money invested n this fund continues to earn 9.58% compounded annually.
(a) Suppose you initially invest $10,000. How much money would you have after 5 years?
(b) How long (to the nearest year) will it take money invested in this fund to double?
21. Suppose MakeBelieve bank offers you a savings account rate of 5% compounded continuously.
(a) If you initially invest $1000, how much money would be in the account after 10 years?
(b) How long (to the nearest year), will it take money invested in this account to double?
Unit 6 Written Work Practice
22. Find the derivative of f (x) using the limit definition: f (x) = x + 4 (Note: This is really simple and a lot of stuff
cancels. You should get that f 0 (x) = 1.)
23. For f (x) = x2 + 4
(a) Find the derivative of f (x) using the limit definition.
(b) Find the instantaneous rate of change at x = 3.
(c) Find the equation of the tangent line at x = 3.
24. For f (x) = x2 + 2x − 4
(a) Find the derivative of f (x) using the limit definition.
(b) Find the instantaneous rate of change at x = 3.
(c) Find the equation of the tangent line at x = 3.
Unit 7 Written Work Practice
25. The revenue (in dollars) from the sale of x infant car seats is given by
R(x) = 60x − 0.025x2 ;
0 ≤ x ≤ 2400
(a) Find the *average* change in revenue if production is changed from 1,000 car seats to 1,050 car seats.
(b) Find R(x) or marginal Revenue.
(c) Find the revenue and the derivative of revenue at a production level of 1,000 car seats, and write a brief verbal
interpretation of these results in the context of the problem. A person with no calculus background should be
able to understand what you mean.
26. Malik, the business manager of a company that produces in-ground outdoor spas, determines that the cost of
producing x spas is C thousand dollars, where
C(x) = 0.04x2 + 2.1x + 60
(a) If Malik decides to increase the level of production from x=10 to x=11 spas, what is the corresponding average
rate of change of cost? Write your results in a sentence.
(b) Next, Malik computes the derivative of cost with respect to the level of production x. What is this rate when
x=10, and how does the derivative compare with the average rate found in part (a)? By raising the level of
production from x=10, should Malik expect the cost to increase or decrease? Write sentence conclusions to
explain your results.
27. Suppose the total revenue function for a blender is R(x) = 36x − 0.01x2 dollars where x is the number of units sold.
(a) What function gives marginal revenue?
(b) What is the marginal revenue when 600 units are sold, and what does it mean? Write a full sentence interpretation.
28. A company is planning to manufacture and market science exploration kits for kids. For this kit, the research
department’s estimates are a weekly demand of 1000 kits at $25 per kit or 2000 kits at $15 per kit. The financial
departments estimates are fixed weekly costs of $6500 and variable costs of $7 per kit.
(a) Assume that the relationship between price p and demand x is linear. Use the research department’s estimates
to express p as a function of x, and find the domain of this function
(b) Find the revenue function in terms of x and state its domain.
(c) Assume that the cost function is linear. Use the financial department’s estimates to express the cost function
in terms of x.
(d) Graph the cost function and the revenue function on the same coordinate system on the domain. Find the
break-even points and indicate regions of loss and profit.
(e) Find the profit function in terms of x.
(f) Evaluate the marginal profit at x = 1850 and interpret the results.
29. Supply Suppose that the supply of x units of a product at price p dollars per unit is given by
p = 10 + 50 ln(3x + 1)
(a) Find the rate of change of supply price with respect to the number of units supplied.
(b) Find the rate of change of supply price when the number of units is 33.
(c) Approximate the price increase associated with the number of units supplied changing from 33 to 34. Write a
sentence to explain your results.
30. Sales Decay After the end of an advertising campaign the sales of a product are given by
S = 100, 000e−0.5t
where S is weekly sales in dollars and t is the number of weeks since the end of the campaign.
(a) Find the rate of change of S(that is, the rate of sales decay)
(b) Write a sentence explaining your results.
31. An investment services company experienced dramatic growth in the last two decades. The following models for
the company’s revenue R and expenses or costs C (both in millions of dollars) are functions of the years past 1990.
R(t) = 21.4e0.131t
C(t) = 18.6e0.131t
(a) Use the models to predict the company’s profit in 2020.
(b) At what rate is profit changing in 2005? What do your results mean? Write a full explanation of your result
in the context of the problem.
Unit 8 Written Work Practice
32. Spread of disease Suppose that the spread of a disease through the student body at an isolated college campus can
be modeled by
y=
10, 000
1 + 9999e−0.99t
where y is the total number affected at time t(in days). Find the rate of change of y.
33. Population By using Social Security Administration data for selected years from 1950 and projected to 2050, the
population of Americans ages 20 to 64 can be modeled by the function
P (t) =
250
1 + 1.91e−0.0280t
where t is the number of years after 1950 and P (t) is in millions.
(a) Find and interpret P 0 (45).
(b) At what rate is this population projected to increase in 2040?
(c) What does this tell us about this population after 2040?
34. Demand The demand function for a product is given by p =
4000
ln(x+100) ,
where p is the price per unit in dollars when
x units are demanded.
(a) Find the rate of change of price with respect to the number of units sold when 40 units are sold. Interpret
your result.
(b) Find the rate of change of price with respect to the number of units sold when 90 units are sold. Interpret
your result.
35. Drug Concentration Suppose the concentration C(t), in mg/ml, of a drug in the bloodstream t minutes after an
injection is given by
C(t) = 20te−0.04t
(a) Find the instantaneous rate of change of the concentration after 10 minutes.
(b) Interpret your result from part a.
36. An inferior product with an extensive advertising campaign does well when it is released, but sales decline as people
discontinue use of the product. If the sales S (in thousands of dollars) after t weeks are given by
S(t) =
200t
, t≥0
(t + 1)2
What is the rate of change of sales when t = 9? Interpret your result in the context of the question.
Unit 9 Written Work Practice
37. Find y 0 for (y − 3)4 − x = y. Then find the equation of the tangent line to the graph of the equation above at the
point (−3, 4).
38. Find y 0 for exy − 2x = y + 1. Then find the equation of the tangent line to the graph of the equation above at the
point (0, 0).
39. Given the price demand equation x = f (p) = 3, 125 − 5p2 ,
(a) Find the elasticity of demand, E(p).
(b) Determine is demand is elastic, inelastic or unit elasticity for p = $25.
(c) If this price (p =$25) is increased by 10%, what is the approximate percentage change in demand?
40. Given the price demand equation 0.02x + p = 60,
(a) Express the demand x as a function of the price p.
(b) Find the elasticity of demand, E(p).
(c) Determine is demand is elastic, inelastic or unit elasticity for p = $10.
(d) If this price (p =$10) is increased by 10%, what is the approximate percentage change in demand?
Units 11 and 12 Written Work Practice
41. For the function f (x) = x4 − 2x3 , find
(a) Domain:
(b) Intercepts (if possible)
(c) Intervals of increasing/decreasing and Relative max/min and
(d) Intervals of concavity and Points of inflection
(e) End behavior
(f) Any vertical or horizontal asymptotes (if they apply)
(g) Use all of the above to create a detailed graph of the function (graphs should be at least 4” x 4” in size)
42. For the function f (x) = 13 x3 + x2 − 24x + 20, find
(a) Domain:
(b) Intercepts (if possible)
(c) Intervals of increasing/decreasing and Relative max/min and
(d) Intervals of concavity and Points of inflection
(e) End behavior
(f) Any vertical or horizontal asymptotes (if they apply)
(g) Use all of the above to create a detailed graph of the function (graphs should be at least 4” x 4” in size)
43. For the function f (x) =
x+3
x−3 ,
find
(a) Domain:
(b) Intercepts (if possible)
(c) Intervals of increasing/decreasing and Relative max/min and
(d) Intervals of concavity and Points of inflection
(e) End behavior
(f) Any vertical or horizontal asymptotes (if they apply)
(g) Use all of the above to create a detailed graph of the function (graphs should be at least 4” x 4” in size)
44. For the function f (x) = 5xe−.02x , find
(a) Domain:
(b) Intercepts (if possible)
(c) Intervals of increasing/decreasing and Relative max/min and
(d) Intervals of concavity and Points of inflection
(e) End behavior
(f) Any vertical or horizontal asymptotes (if they apply)
(g) Use all of the above to create a detailed graph of the function (graphs should be at least 4” x 4” in size)
45. For the function f (x) = x − ln x, find
(a) Domain:
(b) Intercepts (if possible)
(c) Intervals of increasing/decreasing and Relative max/min and
(d) Intervals of concavity and Points of inflection
(e) End behavior
(f) Any vertical or horizontal asymptotes (if they apply)
(g) Use all of the above to create a detailed graph of the function (graphs should be at least 4” x 4” in size)
Unit 13 Written Work Practice
46. A car rental agency rents 200 cars per day at a rate of $30 per day. For each $1 increase in rate, 5 fewer cars are
rented. At what rate should the cars be rented to produce the maximum income and how many cars are rented at
that rate? What is the maximum income?
47. A travel agency will plan a group tour for groups of size 25 or larger. If the group contains exactly 25 people, the
cost is $300 per person. Each person’s cost is reduced by $10 for each additional person above a group of 25.
(a) Find a function for Revenue, where x is the number of additional people over 25.
(b) Find the marginal revenue if the group contains 30 people. Interpret your result in a sentence.
48. A company handles an apartment building with 50 units. Experience has shown that if the rent for each of the
units is $720 per month, all of the units will be filled, but 1 unit will become vacant for each $20 increase in this
monthly rate. (a) At what price is revenue maximum? What is the maximum revenue and how many apartments
are rented to reach this max? (b) If the monthly cost of maintaining the apartment building is $12 per rented unit,
what rent should be charged per month to maximize the profit?
49. A company manufactures and sells x smartphones per week. The weekly price-demand and cost equations are,
respectively
p = 500 − 0.5x
C(x) = 20000 + 135x
(a) What price should the company charge for the phones, and how many phones should be produced to maximize
the weekly revenue? What is the maximum weekly revenue? (b) What is the maximum weekly profit? How much
should the company charge for the phones, and how many phones should be produced to realize the maximum
weekly profit?
50. The marketing research department of a computer company used a large city to test market the firm’s new laptop.
The department found that the relationship between price p (dollars per unit) and the demand x (units per week)
was given approximately by
p = 1, 296 − 0.12x2 , 0 < x < 80
(a) Find an equation that models weekly revenue. (b) Find the price at which revenue reaches a maximum, and
what that maximum revenue is. (c) On which intervals is the graph of revenue concave up? Concave down? (d)
Suppose the cost equation fr the company is C(x) = 830 + 396x. (a) Find local extrema for the profit function. On
which intervals is the graph of the profit function concave upward? Concave downward?
Unit 2 Written Work Quiz - Page 1 of 1
Name
Show all work for credit. Be sure to label work and follow my instructions and examples from the videos
”Expectations for work” (See written work instructions) and ”Graphing using point plotting” (Unit 1 Section 1.1).
If you don’t have a printer, you can just write this on paper. Be sure your write the entire problem and
your solution.
For the function ,
f (x) =
(x + 2)(x − 3)
2x2
Find :
a. Find the domain (in interval notation).
b. Find any Vertical asymptotes or holes. Provide a written explanation why you are choosing Vertical asymptote or
hole.
c. Find any Horizontal Asymptotes.
Unit 3 Written Work Quiz - Page 1 of 2
Name
Show all work for credit. Be sure to label work and follow my instructions and examples from the videos
”Expectations for work” (See written work instructions) and ”Graphing using point plotting” (Unit 1 Section 1.1).
If you don’t have a printer, you can just write this on paper. Be sure your write the entire problem and
your solution.
Some of this is a repeat from last week. Redo it and make sure you know what you’re doing. A question
like this **will** be on the exam. If you have questions, please set up a time to see me.
For the function ,
f (x) =
(x + 2)(x − 3)
2x2
Find :
a. Find the domain (in interval notation).
b. Find any Vertical asymptotes or holes. Provide a written explanation why you are choosing Vertical asymptote or
hole.
c. Find any Horizontal Asymptotes.
d. Find all x and y intercepts
e. Graph, finding additional points as needed.
Unit 3 Written Work Quiz - Page 2 of 2
Show all your work to graph the following. Graphs without supporting work will be given a grade of 0.
Be sure to label work and follow my instructions and examples from the videos ”Expectations for work”
(See written work instructions) and ”Graphing using point plotting” (Unit 1 - Section 1.1).
If you are working on your own paper, your graph needs to be as big as the grid I have given you here,
which is approximately 6 inches square. No tiny graphs!
Unit 4 Written Work Quiz - Page 1 of 3
Name
Show all work for credit. Be sure to label work and follow my instructions and examples from the videos
”Expectations for work” (See written work instructions) and ”Graphing using point plotting” (Unit 1 Section 1.1).
If you don’t have a printer, you can just write this on paper. Be sure your write the entire problem and
your solution.
Suppose that $3000 is invested at 2% compounded monthly. How much money will be in the account in 5 years?
Unit 4 Written Work Quiz - Page 2 of 3
Show all your work to graph the following. Graphs without supporting work will be given a grade of 0.
Be sure to label work and follow my instructions and examples from the videos ”Expectations for work”
(See written work instructions) and ”Graphing using point plotting” (Unit 1 - Section 1.1).
If you are working on your own paper, your graph needs to be as big as the grid I have given you here,
which is approximately 6 inches square. No tiny graphs!
For the function f (x) = 7x−5 :
a. Classify the function.
b. Graph. Show all your work to graph. Graphs without supporting work will be given a grade of 0.
c. Find Domain (interval notation)
d. Are there any Horizontal or Vertical asymptotes? If so, which and what are they?
Unit 4 Written Work Quiz - Page 3 of 3
Show all your work to graph the following. Graphs without supporting work will be given a grade of 0.
Be sure to label work and follow my instructions and examples from the videos ”Expectations for work”
(See written work instructions) and ”Graphing using point plotting” (Unit 1 - Section 1.1).
If you are working on your own paper, your graph needs to be as big as the grid I have given you here,
which is approximately 6 inches square. No tiny graphs!
Unit 6 Written Work Quiz - Page 1 of 1
Name
Show all work for credit. Be sure to label work and follow my instructions and examples from the videos
”Expectations for work” (See written work instructions) and ”Graphing using point plotting” (Unit 1 Section 1.1).
If you don’t have a printer, you can just write this on paper. Be sure your write the entire problem and
your solution.
For f (x) = x2 + 4
(a) Find the derivative of f (x) using the limit definition.
(b) Find the instantaneous rate of change at x = 3.
(c) Find the equation of the tangent line at x = 3.
Unit 7 Written Work Quiz - Page 1 of 1
Name
Show all work for credit. Be sure to label work and follow my instructions and examples from the videos
”Expectations for work” (See written work instructions) and ”Graphing using point plotting” (Unit 1 Section 1.1).
Marginal Cost Suppose that the total cost(in dollars) for a product is given by
C(x) = 1500 + 200 ln(2x + 1)
where x is the number of units produced.
a. Find the marginal cost function.
b. Find the marginal cost when 200 units are produced.
c. Interpret your result from part b. That means write a sentence that would make sense to someone who has not
taken calculus and has not heard of marginal cost.
Unit 8 Written Work Quiz - Page 1 of 1
Name
Show all work for credit. Be sure to label work and follow my instructions and examples from the videos
”Expectations for work” (See written work instructions) and ”Graphing using point plotting” (Unit 1 Section 1.1).
Taken from Mathematical Applications for the Management, Life and Social Sciences, 11th Edition by Ronald J Harshbarger and James J Reynolds
U.S. debt For selected years from 1900 to 2014, the national debt d, in billions of dollars, can be modeled by
d = 1.60e0.083t
where t is the number of years past 1900.
(Source: Bureau of Public Debt, U.S. Treasury).
(a) What function describes how fast the national debt is changing?
(b) Find the instantaneous rate of change of the national debt model d(t) in 1950 and 2025.
(c) Explain the meaning of the instantaneous rate of change in 1950 in the context of the problem to someone who
doesn’t know calculus. Don’t use the word rate in your sentence.
Unit 9 Written Work Quiz - Page 1 of 2
Name
Show all work for credit. Be sure to label work and follow my instructions and examples from the videos
”Expectations for work” (See written work instructions) and ”Graphing using point plotting” (Unit 1 Section 1.1).
Find y 0 for
(y − 3)4 − x = y
Now find the equation of the tangent line to the graph of the equation above at the point (−3, 4).
Unit 9 Written Work Quiz - Page 2 of 2
Given the price demand equation x = f (p) = 3, 125 − 5p2 ,
(a) Find the elasticity of demand, E(p).
(b) Determine is demand is elastic, inelastic or unit elasticity for p = $15.
(c) If this price (p =$15) is increased by 10%, what is the approximate percentage change in demand?
Unit 11 Written Work Quiz - Page 1 of 1
Name
Show all work for credit. Be sure to label work and follow my instructions and examples from the videos
”Expectations for work” (See written work instructions) and ”Graphing using point plotting” (Unit 1 Section 1.1).
For the function f (x) = x4 − 2x3 , find
(a) (1 points) Domain:
(b) (1 points) Intercepts (if possible)
(c) (4 points) Intervals of increasing/decreasing and Relative max/min
(d) (4 points) Intervals of concavity and Points of inflection
Unit 12 Written Work Quiz - Page 1 of 2
Name
Show all work for credit. Be sure to label work and follow my instructions and examples from the videos
”Expectations for work” (See written work instructions) and ”Graphing using point plotting” (Unit 1 Section 1.1).
For the function f (x) = 3x − x3 , find
(a) (1 points) Domain:
(b) (1 points) Intercepts (if possible)
(c) (2 points) Intervals of increasing/decreasing and Relative max/min
(d) (2 points) Intervals of concavity and Points of inflection
(e) (1 points) End behavior
(f) (1 points) Any vertical or horizontal asymptotes
(g) (2 points) Use all of the above to create a detailed graph of the function (on the given grid)
Graphs must be at least 3/4 of a page in size (see next page for an example). Graphs smaller than that will have
significant deductions.
Unit 12 Written Work Quiz - Page 2 of 2
Unit 13 Written Work Quiz - Page 1 of 1
Name
Show all work for credit. Be sure to label work and follow my instructions and examples from the videos
”Expectations for work” (See written work instructions) and ”Graphing using point plotting” (Unit 1 Section 1.1).
A 315 room hotel is filled when the room rate is $80 per day. For each $2 increase in rate, 4 fewer rooms are rented.
(a) Find the room rate that maximizes daily revenue.
(b) What is the maximum revenue? How many rooms are rented to achieve this revenue?
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