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Kathryn
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Final Exam Review (Section 8.3 and Review of Other Sections)
Note: This collection of questions is intended to be a brief overview of the material covered throughout the semester (with an emphasis at the beginning on material from Section 8.3, which I have not
previously reviewed). This is not intended to represent an actual exam. When studying you should
also rework your notes, quizzes and exams, the previous week-in-reviews, and be familiar with your
suggested and online homework problems.
1. Find all of the critical points for f (x, y) = 3x2 + 5y 2 − 8xy + 2x + 6y + 3
2. Locate any critical points of the following functions, and, if possible, identify each as relative
extrema or a saddle point.
(a) f (x, y) = 2x2 + 5y 2 + 6x − 2y + 12
(b) f (x, y) = −3x2 + xy − y 2 − 4x − 3y
(c) f (x, y) = xy − x3 − y 2
3. A firm manufactures and sells two products, X and Y , that sell for $15 and $10 each, respectively. The cost of producing x units of X and y units of Y is
C(x, y) = 400 + 7x + 4y + 0.01(3x2 + xy + 3y 2 )
Find the values of x and y that maximize the firm’s profit.
4. Find the area bounded by f (x) = 8 − x2 and g(x) = −x − 4.
5. A baker sells one dozen donuts when a dozen of donuts sells for $2.00. For each 2 cent decrease
in price, the baker sells an additional dozen of donuts. It costs the baker 25 cents to make a
dozen of donuts. Let x be the number of dozen of donuts sold. Find the
(a) price-demand function, p(x).
(b) cost function, C(x).
(c) revenue function, R(x).
(d) profit function, P (x).
6. Find
√
dy
if y = t3 + 4 and t = 3x4 .
dx
7. Find the derivative of f (x) = 24x ln |5x2 + 2x|
8. What transformation(s) would you apply to the graph of f (x) in order to obtain the graph
of g(x) = −2f (x + 3) − 7?
9. Find the instantaneous rate of change of f (x) = log3 x4 at x = 2.
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10. Given f (x) =

4x − 3 , x < 1



7
,x=1

x+2


6−x
,x>1
(a) find lim f (x).
x→1
(b) find lim f (x).
x→∞
(c) Is f (x) continuous at x = 1? Why or why not?
11. Evaluate
Z
b
a
2e3 + π
e6
!
dx.
12. Find all asymptotes and holes of f (x) =
13. Find the first derivative of f (x) =
q
−(x − 6)(x + 5)
.
(2x + 1)(x2 − 3x − 18)
8
e(x ) + (ln (x6 + 4) + 12)3
14. Find the absolute extrema of f (x) = 2x3 + 3x2 − 45 on [0, 4].
x2 + 2x
as an equivalent piecewise-defined function.
|x + 2|
15. Rewrite f (x) =
16. Use the given graph to answer the questions which follow.
−4
2
(a) If the given graph is f (x), find any absolute extrema of f (x).
(b) If the given graph is f (x), where is f ′ (x) < 0?
(c) If the given graph is f ′ (x), where is f (x) concave up?
(d) If the given graph is f ′′ (x), where is f (x) concave down?
(e) If the given graph is f ′ (x), where does f (x) have any local extrema?
17.
10ex + 12
x→−∞ 9 + 5ex
lim
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Kathryn
Bollinger, April 29, 2014
18. Find
Z
(8x + 16)ex
2 +4x+5
dx
19. Mr. Barker is adding onto his dog kennel. He needs to fence in 12 equally-sized yard lots
(all in a row, connected side-by-side). If Mr. Barker has 400 feet of fencing, what should the
dimensions of each dog yard be in order to maximize the total yard area?
4x3 − 1
x→∞ x + 3
20. Evaluate lim
21. When using a Riemann sum to approximate the area under f (x) = x2 + x + 5 on the interval
[−3, 8], using 10 equally spaced rectangles, what is the area of the last rectangle if you use
(a) right endpoints?
(b) left endpoints?
22. Suppose water is being pumped out of a well at a rate given by y = 300e−0.3t , where t is the
number of years since the pumping began and y is measured in millions of gallons/year. At
this rate, how much water will be pumped out during the fourth year?
23. How long will it take for money in an account to quadruple if the account pays 5% annual
interest compounded continuously?
x2 + x − 42
x→6
x−6
24. Evaluate lim
25. Find
Z √
√
x+ 6x
√
dx
9
x
26. Evaluate lim
x→∞
x+5
x2 − 2x − 3
27. Sketch a graph of a function with the following properties:
f ′ (x) > 0 on (−∞, −3) and (0, ∞)
f ′ (x) < 0 on (−3, −1) and (−1, 0)
f ′′ (x) < 0 on (−∞, −1)
f ′′ (x) > 0 on (−1, ∞)
VA: x = −1
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Kathryn
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28. Given g(x) = x2 + 4 and f (x) =
√
x + 20, find the following:
(a) (g ◦ f )(x)
(b) (f ◦ g)(x)
(c) (g ◦ g)(x)
29. Find the first derivative of f (x) =
(3x3
+
4x2
6
− 2x + 10)3
30. Solve for x: 2 · 155x = 26
31. Evaluate the end behavior of f (x) = 12x5 + Kx3 − Lx2 + Bx − 100
32. Given f (x) = 4x5 − x4 , find all values of x where there is a horizontal tangent line to f (x).
33. If we know that f (3) = 4, f ′ (3) = 0 and f ′′ (x) is continuous everywhere with f ′′ (3) = −5,
then what (if anything) can we conclude about f (x) at x = 3?
34. The rate at which the fanbase of a certain band is growing is given by f = 3.7x + 3 for
1 ≤ x ≤ 10, where x is the number of years since the band began touring in 1993 and f (x) is
measured in tens of fans/year. Evaluate and interpret
Z
5
f (x) dx
1
35. Given demand d(p) = (225 − 5p)1/2 , determine the point elasticity of demand at p = 10. At
this point is demand inelastic, elastic, or unitary? Should the price be lowered, raised or kept
the same to increase revenue?
36. Find the exact value of
37.
Z
5
x2
dx
+5
1
0 3 x3
(a) Which of the following is NOT a function?
(b) Which of the functions are one-to-one functions?
a.
b.
38. Find the domain of f (x) =
c.
ln (3x + 8)
.
x
d.
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Kathryn
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√
f (x + h) − f (x)
if f (x) = x + 4
h→0
h
39. Find lim
4x2 + 3x + 48
x→−4 6x4 + 3x2 + 20
40. Evaluate lim
41. Suppose a puppy grows at a rate of y = 0.16et , where t is the number of years since the
beginning of 1999 when the puppy was born and y is measured in inches/year. If the puppy
reaches a maximum height of 22 inches at 3 years of age, what is the puppy’s height on his
first birthday?
42. Solve for x: log5 (log3 (log 2x)) = 3
43. Evaluate
Z
2
(x3 + 2x2 + 1) dx and determine whether the value represents the total amount
−3
of area between a curve and the x-axis or if it represents a net area.
44. Given that
Z
5
2f (x) dx = 20 and
0
Z
5
f (x) dx = 15, find
2
Z
0
3f (x) dx.
2
45. Billy’s parents want to open an account which pays 6.06% annual interest compounded
monthly. They need to have $15,000 for Billy’s first year of college tuition. If they open
the account exactly 18 years before they plan to withdraw the money, how much should they
invest when they open the account to ensure they can pay for Billy’s first year of college
tuition?
46. Rewrite as a single logarithm:
4
(log 2x + log x3 − (log y 2 + log 3y 2 ))
5
47. Find the first derivative of f (x) =
6x3
e1/x
48. “Krissy’s Kosmetics” has determined its profit to be given by P (x) = 500x − x2 where
0 ≤ x ≤ 500 and x measures the number of tubes of lipstick produced and sold.
Find P ′ (250) and interpret.
(x2 − 1)3
.
49. Find the domain of f (x) = √
x+1
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Kathryn
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50. For f (x) =
(
x
,x≥2
,
2
−3 + x , x < 2
(a) find f (−2), f (2), and f (5).
(b) graph f (x).
(c) where is f (x) discontinuous? Non-differentiable? Explain your answers.
51. Solve for x: log7 (2x + 1) + log7 x = 4
52. Find the equation of the line tangent to the graph of f (x) = (4x2 + 6x)(2x − 5x3 ) at x = 2.
53. Find the exact value of
Z
1
5−
√
x2
2
dx
54. Compute the average rate of change of f (x) = x5 + x4 − x3 − x2 + 5 − x5 − x4 + x3 over [0, 5].
1
55. Given f (x) = x3 − x2 − 8x, find
3
(a) all critical values and relative extrema.
(b) all inflection points.
56. Where is f (x) =

 x + 12
x−6

x+1
,x≤2
,x>2
discontinuous?
57. A new moped costs $3500.00 in 2002 and its value depreciates at a rate of $450.00 per year.
(a) Find an equation for the value of the moped as a function of time.
(b) In what year will the moped be worth $1250.00?
58. What is the effective rate of interest of an account earning interest at a annual rate of 9.25%
compounded weekly?
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Kathryn
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59.
(a) Write the limit that would indicate that the graph of f (x) has
a vertical asymptote of x = 7.
(b) Write the limit that would indicate that the right-hand side of the graph of f (x) has
a horizontal asymptote of y = −5.
60. Find the domain of f (x, y) =
√
2x + 3y − 7.