Math 142 WIR, copyright Angie Allen, Spring 2013
(Problems 1b, 2, 8, 11, 16, 25-27, 29-32, 34, 36, 37, 39, and 41 courtesy of Heather Ramsey)
Math 142 Week-in-Review #11 (Final Exam Review: All previous sections as well as section 8.3)
Note: This review is intended to highlight the topics covered on the Final Exam (with emphasis on section 8.3),
but it should not be used as your sole source of practice. When studying, you should also rework your notes, the
previous week-in-reviews for this material, as well as your suggested and online homework in order to recall and
study the variety of formulas/questions for each topic.
1. Find the locations of all local extrema and saddle points of each of the following functions:
(a) f (x, y) = y3 − 9x2 + 18xy − 21y + 5
(b) g(x, y) = 2x2 − 2x2 y + 6y3
1
Math 142 WIR, copyright Angie Allen, Spring 2013
(Problems 1b, 2, 8, 11, 16, 25-27, 29-32, 34, 36, 37, 39, and 41 courtesy of Heather Ramsey)
2. A rectangular box is to have a square base and a volume of 20 cubic feet. If the material for the base costs
30 cents per square foot, the material for the sides costs 10 cents per square foot, and the material for the
top costs 20 cents per square foot, determine the dimensions of the box that can be constructed at minimum
cost. (source: #8, pg. 365 of Applied Calculus for the Managerial, Life, and Social Sciences, 5th ed., by Tan)
3. Find the equation of g(x) if its graph is obtained by taking the graph of f (x) = ln x and shifting it 4 units to
the left, stretching it vertically by a factor of 3, and then shifting it up 12 units.
2
Math 142 WIR, copyright Angie Allen, Spring 2013
(Problems 1b, 2, 8, 11, 16, 25-27, 29-32, 34, 36, 37, 39, and 41 courtesy of Heather Ramsey)
4. If we know f (2.5) = −0.25 and f ′ (2.5) = 0, and we also know that f ′′ is continuous everywhere and
f ′′ (2.5) = 2, then what (if anything) can we conclude about the behavior of f at x = 2.5?
5. Find the horizontal asymptotes of the function f (x) =
6. If y = 3t 4 − et and t = ln(h) + 4h , find dy/dh.
3
2e−6x − 7e−2x
, if any exist.
e−2x − 3e−x − 14
Math 142 WIR, copyright Angie Allen, Spring 2013
(Problems 1b, 2, 8, 11, 16, 25-27, 29-32, 34, 36, 37, 39, and 41 courtesy of Heather Ramsey)
7. The following table shows the number of heartbeats after t minutes of a patient after a surgery.
t (min)
Heartbeats
36
2256
38
2232
40
2264
42
2352
44
2496
(a) Find the average heart rate between 38 and 42 minutes. Interpret your answer.
(b) The data above can be modeled by the function h(t) = 7t 2 − 530t + 12, 264 heartbeats after t minutes,
where 36 ≤ t ≤ 44. Use this model to find the patient’s heart rate after 42 minutes. Interpret your answer.
(c) Find the average number of heartbeats between 38 and 42 minutes.
(d) Find the average heart rate between 37 and 41 minutes.
4
Math 142 WIR, copyright Angie Allen, Spring 2013
(Problems 1b, 2, 8, 11, 16, 25-27, 29-32, 34, 36, 37, 39, and 41 courtesy of Heather Ramsey)
8. Compute each of the following by hand.
(a)
Z a
1
(b)
(7x−1 + x0.2 )dx
Z 21
9
9. Find
where a > 1
√
(t + 1) t − 5dt
g ′ (t)
if g(t) =
Z 2et p
3
1 + u2 du.
lnt
5
Math 142 WIR, copyright Angie Allen, Spring 2013
(Problems 1b, 2, 8, 11, 16, 25-27, 29-32, 34, 36, 37, 39, and 41 courtesy of Heather Ramsey)
10. Find the domain of each of the following functions.
√
x+2
(a) f (x) = √
3 2
x − 16
x3
e x−7
(b) f (x) =
15 log2 (x − 4)
(c) f (x, y) =
ln(y − x)
2xy
11. Find the equation of the tangent line to f (x) = 0.5e2x at x = 1.
6
Math 142 WIR, copyright Angie Allen, Spring 2013
(Problems 1b, 2, 8, 11, 16, 25-27, 29-32, 34, 36, 37, 39, and 41 courtesy of Heather Ramsey)
12. ABC company sells gadgets and widgets. The gadgets sell at p = 1492 − 5x − 2y dollars, and the widgets
sell at q = 1116 − 4x − 3y dollars, where x is the number of gadgets sold and y is the number of widgets sold.
Find the number of gadgets and widgets ABC company must sell to obtain maximum revenue. What is the
maximum revenue?
13. Classify each of the following as a power function, rational function, polynomial function (state its degree
and leading coefficient), or none of these (state why).
(a) f (x) =
2x
4x3 + 5x
(b) g(x) = x1/4
7
(c) h(x) = x13
Math 142 WIR, copyright Angie Allen, Spring 2013
(Problems 1b, 2, 8, 11, 16, 25-27, 29-32, 34, 36, 37, 39, and 41 courtesy of Heather Ramsey)
14. For the function f whose graph is shown below, find each of the following.
(a) State the domain of f .
(Graph courtesty of Heather Ramsey)
f(x)
8
(b) Where is f continuous?
(c) For what values of c does lim f (x) not exist?
x→c
7
−7
−8
(d) Which condition in the definition of continuity fails
first at x = −3?
(e) For what values of a does f ′ (a) not exist? For each value of a,
explain why the derivative does not exist.
(f) Find the absolute maximum and minimum values of f on
[−6, −1), if they exist.
8
x
Math 142 WIR, copyright Angie Allen, Spring 2013
(Problems 1b, 2, 8, 11, 16, 25-27, 29-32, 34, 36, 37, 39, and 41 courtesy of Heather Ramsey)
15. Consider the function f (x) =
ex
.
x
(a) Use calculus to find any critical values of f (x), as well as the intervals where the function is increasing/decreasing and the values of any local extrema.
(b) Find the absolute minimum of f (x) on the interval [0.5, 3], if it exists.
(c) Find the absolute minimum of f (x) on the interval [−1, 3], if it exists.
(d) Find the absolute maximum of f (x) on the interval [0.5, 3), if it exists.
9
Math 142 WIR, copyright Angie Allen, Spring 2013
(Problems 1b, 2, 8, 11, 16, 25-27, 29-32, 34, 36, 37, 39, and 41 courtesy of Heather Ramsey)
16. Find each of the following limits, if they exist.
x2 + 8x + 15
x→−3 x2 + x − 6
(a) lim
2x2 − 7
x→0 5x + 14
(b) lim
4x2 − 3x3 + 2
x→∞ 5 − 7x + 12x2
(c) lim
7x4 + 5x
x→−∞ 10x3 − 21x4
(d) lim
10
Math 142 WIR, copyright Angie Allen, Spring 2013
(Problems 1b, 2, 8, 11, 16, 25-27, 29-32, 34, 36, 37, 39, and 41 courtesy of Heather Ramsey)
17. When using a Riemann sum with left endpoints to approximate the area under the curve f (x) = x2 − 5x − 3
on the interval from [6, 10] using 16 rectangles, what is the height of the 4th rectangle? Show your formula.
18. Use calculus to find any holes and horizontal and vertical asymptotes of the function f (x) =
If there are vertical asymptotes, use limits to describe the behavior near each asymptote.
Hint: x4 − x3 − x2 − 5x − 30 = (x − 3)(x2 + 5)(x + 2).
11
x2 − 2x − 8
.
x4 − x3 − x2 − 5x − 30
Math 142 WIR, copyright Angie Allen, Spring 2013
(Problems 1b, 2, 8, 11, 16, 25-27, 29-32, 34, 36, 37, 39, and 41 courtesy of Heather Ramsey)
19. Consider the graph of a function shown below.
(a) If the graph is f (x), find the partition numbers
of f ′ (x) and the critical values of f (x).
y
(b) If the graph is f (x), find where f ′′ (x) < 0.
x
−9
(c) If the graph is f (x), find where f ′ (x) is increasing.
−7
−5
−3
2
3
4
6
8
x=3
(d) If the graph is f ′ (x), and f is continuous on its domain of (−∞, −9) ∪ (−9, 3) ∪ (3, ∞) , find the partition
numbers of f ′ (x) and the critical values of f (x).
(e) If the graph is f ′ (x), and f is continuous on its domain of (−∞, −9) ∪ (−9, 3) ∪ (3, ∞) , find where f (x)
is concave down.
(f) If the graph is f ′ (x), and f is continuous on its domain of (−∞, −9) ∪ (−9, 3) ∪ (3, ∞) , find where f (x)
has local extrema and classify the type of extrema.
(g) If the graph is f ′′ (x), and f is continuous on its domain of (−∞, −9) ∪ (−9, 3) ∪ (3, ∞) and f
continuous on its domain of (−∞, −9) ∪ (−9, 3) ∪ (3, ∞), find where f (x) has any inflection points.
′
is
(h) If the graph is f ′′ (x), and f is continuous on its domain of (−∞, −9) ∪ (−9, 3) ∪ (3, ∞) and f
continuous on its domain of (−∞, −9) ∪ (−9, 3) ∪ (3, ∞), find where f ′ (x) is decreasing.
′
is
12
Math 142 WIR, copyright Angie Allen, Spring 2013
(Problems 1b, 2, 8, 11, 16, 25-27, 29-32, 34, 36, 37, 39, and 41 courtesy of Heather Ramsey)
20. Find lim +
x→−7
x2 + 4x − 21
.
|x + 7|
21. Consider the function g(x) =
√
2 − 3x.
(a) Find an expression for the slope of the secant line through (x, g(x)) and (x + h, g(x + h)).
(b) Find an expression for the slope of the tangent line at the point (x, g(x)).
13
Math 142 WIR, copyright Angie Allen, Spring 2013
(Problems 1b, 2, 8, 11, 16, 25-27, 29-32, 34, 36, 37, 39, and 41 courtesy of Heather Ramsey)
22. Find lim (−ax3 + bx − c), where a > 0, b < 0, and c > 0.
x→−∞
23. A jewelry store is having a sale on bracelets this weekend. The first 2 bracelets cost $20 each, and each
additional bracelet after that costs $10. The maximum number of bracelets a customer can purchase is 15.
Write a function that gives the amount one would pay for x bracelets. Simplify your answer.
24. Use the definition of the derivative to find f ′ (x) for f (x) = x − 4x2 .
14
Math 142 WIR, copyright Angie Allen, Spring 2013
(Problems 1b, 2, 8, 11, 16, 25-27, 29-32, 34, 36, 37, 39, and 41 courtesy of Heather Ramsey)
25. Acme Music Company has determined the price-demand function for its home audio system to be p =
3080 − 10x dollars per system. Acme Music has fixed costs that amount to $56,250 and variable costs of
$580 per system.
(a) How many home audio systems must be sold to maximize revenue?
(b) Find the profit and marginal profit from the production and sale of 175 home audio systems, and
interpret your answers.
(c) Approximate the profit from the production and sale of 176 home audio systems.
(d) How many home audio systems should be produced and sold so that Acme breaks even?
15
Math 142 WIR, copyright Angie Allen, Spring 2013
(Problems 1b, 2, 8, 11, 16, 25-27, 29-32, 34, 36, 37, 39, and 41 courtesy of Heather Ramsey)
26. Find the derivative of each of the following functions. Simplify the function before taking the derivative
where possible.
2 +1)
(a) h(x) = e−4.5 ln (x
5
+ e−x + 72x − e−2
(b) f (x) = 3x2 − √
4
x
√
(c) g(x) = log8 (2x − 7) x + 3
√
x
(d) g(x) = (10x3 − 1)e
16
Math 142 WIR, copyright Angie Allen, Spring 2013
(Problems 1b, 2, 8, 11, 16, 25-27, 29-32, 34, 36, 37, 39, and 41 courtesy of Heather Ramsey)
p
5 8
2
(e) m(t) = 3 t + 1 −
t
(f) h(x) =
ln x + 4x7
2x + 9
2 · 3−5x
(g) f (x) =
27x+2
4
17
Math 142 WIR, copyright Angie Allen, Spring 2013
(Problems 1b, 2, 8, 11, 16, 25-27, 29-32, 34, 36, 37, 39, and 41 courtesy of Heather Ramsey)
27. Solve each of the following for x or b as indicated.
(a) logb 64 =
3
2
(b) log9 x = −3
2 −4x
(c) 5x
=
1
25x
(d) 3x · 8x + 3x+1 = 0
(e) log2 (x + 2) + log2 x = 2 log2 3 + log2 10 − log2 6
18
Math 142 WIR, copyright Angie Allen, Spring 2013
(Problems 1b, 2, 8, 11, 16, 25-27, 29-32, 34, 36, 37, 39, and 41 courtesy of Heather Ramsey)
28. Given the function f (x) below,
(a) find the domain of f (x).

−8
x<0






 p
13
x5 − x3 − 3ex 0 ≤ x < 9
f (x) =






 ln(x − 7)
x>9
x2 − 7x − 30
(b) determine where f (x) is continuous.
(c) find f (−4), if it exists.
(d) find lim f (x), if it exists.
x→10
(e) find lim f (x), if it exists.
x→0
19
Math 142 WIR, copyright Angie Allen, Spring 2013
(Problems 1b, 2, 8, 11, 16, 25-27, 29-32, 34, 36, 37, 39, and 41 courtesy of Heather Ramsey)
29. In 1965, Bob invested $300 in a savings account paying 4.2% per year compounded continuously.
(a) What was the average rate of change of this account’s value from 1972 to 1975?
(b) What was the average value (i.e., average balance) of this account from 1972 to 1975?
30. Find the intervals where f (x) = 3e2x − 8ex is concave upward and concave downward, and find the coordinates of any inflection points.
31. Find the area between f (x) = x + 4 and g(x) = (x + 2)2 on the interval [−2, 1].
20
Math 142 WIR, copyright Angie Allen, Spring 2013
(Problems 1b, 2, 8, 11, 16, 25-27, 29-32, 34, 36, 37, 39, and 41 courtesy of Heather Ramsey)
32. The management of Bob’s Auto Shop has determined the price-demand equation for its air filters to be
2x + 100p = 2000 where x is the number of filters sold when the unit price is $p.
(a) Find the elasticity of demand E(p).
(b) Find and interpret E(9).
(c) If the current price of $9 per air filter is increased by 3%, what would be the approximate change in
demand?
(d) If the current price per air filter is $13, should Bob increase or decrease this price to produce an increase
in revenue?
33. Sketch the graph of the derivative of f (x) shown below.
y
x
21
Math 142 WIR, copyright Angie Allen, Spring 2013
(Problems 1b, 2, 8, 11, 16, 25-27, 29-32, 34, 36, 37, 39, and 41 courtesy of Heather Ramsey)
√
34. A company’s marginal profit can be modeled by m(x) = 18 x + 5 dollars per item, where x is the number
of items produced and sold. This company’s profit is $11,475 when 95 items are produced and sold.
(a) Find the profit when 120 items are produced and sold.
(b) Find the average marginal profit when the number of items sold is between 100 and 120.
(c) Find the change in profit when the number of items sold increases from 100 to 120.
(d) Find the average profit when the number of items sold is between 100 and 120.
(e) Estimate the profit from the sale of the 90th item.
22
Math 142 WIR, copyright Angie Allen, Spring 2013
(Problems 1b, 2, 8, 11, 16, 25-27, 29-32, 34, 36, 37, 39, and 41 courtesy of Heather Ramsey)
35. Use the graph of g ′ (x) below to answer the following:
Note: Each grid mark represents one unit on the graph below.
(a) Estimate
Z 7
1
g ′ (x) dx using a left Riemann sum with
three rectangles.
y
6
g’(x)
3
(b) Find
Z 8
5
g ′ (x) dx.
x
−6
−3
3
−3
−6
(c) Find
Z 1
3
g ′ (x) dx.
(d) Find g(3) if g(0) = 25.
23
6
Math 142 WIR, copyright Angie Allen, Spring 2013
(Problems 1b, 2, 8, 11, 16, 25-27, 29-32, 34, 36, 37, 39, and 41 courtesy of Heather Ramsey)
36. Compute each of the following.
(a)
Z
(b)
Z
4t 3
(c)
Z
x6
dx
4x7 − 5
(d)
Z
(4x + 2)ex
4ex − 7x3 + 4x−1 − e2 dx
p
t 4 − 7dt
2 +x
dx
24
Math 142 WIR, copyright Angie Allen, Spring 2013
(Problems 1b, 2, 8, 11, 16, 25-27, 29-32, 34, 36, 37, 39, and 41 courtesy of Heather Ramsey)
(e)
Z
2
√
dx
x 5 + ln x
37. Let f (x, y) =
7x2 y4
.
x3 y − 7y
(a) Find f (2, 10).
(b) Find the first-order partial derivatives of f (x, y).
25
Math 142 WIR, copyright Angie Allen, Spring 2013
(Problems 1b, 2, 8, 11, 16, 25-27, 29-32, 34, 36, 37, 39, and 41 courtesy of Heather Ramsey)
38. Reid is selling lemonade. When she charges $5 per glass, she can sell 36 glasses of lemonade. For every
$0.10 that she lowers the price, an additional 4 glasses of lemonade will be sold.
(a) Use optimization techniques from calculus to determine what price Reid should charge per glass to
maximize her revenue.
(b) Find the exact revenue from selling the 45th glass of lemonade.
39. Find the producers’ surplus at equilibrium price level for a product whose price-demand equation is given
by p = D(x) = 430 − 0.62x dollars per item and price-supply equation is given by p = S(x) = 0.004x2 + 25
dollars per item.
26
Math 142 WIR, copyright Angie Allen, Spring 2013
(Problems 1b, 2, 8, 11, 16, 25-27, 29-32, 34, 36, 37, 39, and 41 courtesy of Heather Ramsey)
40. The productivity of a manufacturing company is given by f (x, y) = 40x0.25 y0.75 units, where x and y represent
the number of units of labor and capital utilized, respectively. Find the marginal productivity of capital when
60 units of labor and 40 units of capital are utilized.
41. Find all second-order partial derivatives of f (x, y) = 2xexy .
27
Math 142 WIR, copyright Angie Allen, Spring 2013
(Problems 1b, 2, 8, 11, 16, 25-27, 29-32, 34, 36, 37, 39, and 41 courtesy of Heather Ramsey)
42. Sketch the graph of a function that satisfies all of the following conditions:
• f ′ (−3) = f ′ (−5) = 0
• f ′ (x) < 0 if −5 < x < −3
• f ′ (x) > 0 if x < −5, −3 < x < 1, or x > 1
• f ′′ (x) < 0 if x < −4 or x > 1
• f ′′ (x) > 0 if −4 < x < 1
• lim+ f (x) = −∞
x→1
• lim− f (x) = ∞
x→1
• lim f (x) = 2
x→∞
43. Bob has the option of investing in an account that pays 7.0% per year compounded quarterly.
(a) If Bob invests $10,000, what will be the value of the account in 5 years?
(b) How much should Bob invest now if he wants to have $50,000 in the account in 20 years?
(c) If Bob has the option of investing in another account that pays 5.5% per year compounded continuously,
should he choose that account instead?
28
Math 142 WIR, copyright Angie Allen, Spring 2013
(Problems 1b, 2, 8, 11, 16, 25-27, 29-32, 34, 36, 37, 39, and 41 courtesy of Heather Ramsey)
44. How long (in years) will it take a piece of equipment to be worth one third of its current value if it is
depreciating at a relative rate of 4.3% per year?
45. Find
Z
π
dt.
(3 − e−t )et
29