Math 181
Final Exam Review
In addition to the exam review, you need to go back and review all old
tests, notes, book homework, assigned homework, and in class work.
The exam is not limited to what is on the review. Make sure that you
show all your work, even on the final!
3 x3 − 12 x 2 + 13 x − 52
using a table of
x→4
x−4
1. Use a calculator to numerically compute lim
values.
Answer: 61
3x3 + 6 x 2 + 4 x + 8
2. Use a calculator to numerically compute lim
using a table of values.
x →−2
x+2
Answer: 16
x 2 − 11x + 24
3. Evaluate lim
x →3
x −3
Answer: -5
x −5
4. Evaluate lim
x →3 x + 4
Answer: -2/7
x−4
5. Evaluate lim
x→4
x −2
Answer: 4
x− 3
6. Evaluate lim
x →3
x −3
3
Answer:
6
 2x2 + 3 
7. What is lim 

x →∞ 3 + 4 x + 5 x 2


Answer: 2/5
 4 x 2 + 3x − 5 
8. What is lim 

x →∞ 

7 − 4x


Answer: -1/2


x+7
9. What is lim 

2
x →∞
 10 + 6 x + 9 x 
Answer: 1/3
 tan( x + h) − tan x 
10. Evaluate: lim 

h →0
h

2
Answer: sec x
 ( x + h) 2 − x 2 
11. Evaluate: lim 

h →0
h


Answer: 2x
 9sec( x + h) − 9sec x 
12. Evaluate: lim 

h →0
h

Answer: 9sec x tan x
13. Determine if the function is continuous everywhere.
a. f ( x)= x − 1
b. f =
( x)
x2 + 1
x +1
c. f ( x) =
5 − x2
Answer: a. yes, b. yes, c. no
14. Differentiate f(x) = x 2 tan x
Answer:=
f ′( x) x 2 sec 2 x + 2 x tan x
15. Differentiate f ( x) = 7 x csc x
Answer: f ′( x) =
−7 x csc x cot x + 7 csc x
16. Differentiate f ( x) = 8cos x csc x
Answer: f ′( x) = −8csc 2 x
3x − 5
17. Differentiate g(x) =
x+7
26
Answer: g ′( x) =
2
( x + 7)
18. Differentiate y = cos 2 ( 3x )
Answer: y′ =−2 ⋅ 3x cos ( 3x ) sin ( 3x ) ⋅ ln ( 3)
19. Differentiate y = x ln ( 3 x )
Answer: y′ = 1 + ln ( 3 x )
20. Differentiate y = e x tan x
Answer:
=
y′ e x tan x ( tan x + x sec 2 x )
21. Differentiate y = cot −1 2 x
−1
Answer: y′ =
2 x (1 + 2 x )
x+3
x−7
22. Differentiate f ( x) =
Answer: f ′( x) =
−10
( x − 7)
23. Differentiate h(x) =
2
3
4 + 3x 2
Answer: h′( x) =
2x
( 4 + 3x )
24. Differentiate g(x) = ( 3 x + 7 )
Answer:
=
g ′( x) 24 x ( 3 x + 7 )
2
2
3
2
2
4
3
25. Differentiate f(x) = sec 2 (4 x + 5)
Answer: f ′( x) = 8sec 2 (4 x + 5) tan(4 x + 5)
26. Differentiate h(x) = sin 2 x cos 3 x
Answer: h′( x) 2sin x cos x cos 3 x − 3sin 3 x sin 2 x
=
27. Use implicit differentiation to find dy/dx for the curve x3 y − 9 y 2 =3 x + 7
Answer:
dy 3 − 3 x 2 y
=
dx x3 − 18 y
28. Use implicit differentiation to find dy/dx for the curve x 2 ( x 2 + y 2 ) =
y2
2
2
dy x ( 2 x + y )
Answer:
=
dx
y (1 − x 2 )
29. Use implicit differentiation to find
Answer:
dy
2 − y2
=
dx 2 xy + cos y
30. Use implicit differentiation to find
Answer:
dy
for the curve 2 x − sin y =
xy 2 .
dx
dy
for the curve y = sin( xy ) .
dx
dy
y cos( xy )
=
dx 1 − x cos( xy )
31. Use implicit differentiation to find
dy
for the curve e y = xy .
dx
dy
y
y
= y
or
dx e − x
xy − x
32. Find the equation of the tangent line to the curve y = 4 x3 − 6 x at x = 3
Answer: y=102x - 216
x +1
33. Find the equation of the tangent line to the curve y =
at x = -2.
x −1
Answer: y= -2/9x – 1/9
34. Find the equation of the tangent line to the curve arctan
=
( xy ) arcsin ( x + y ) at x = 0.
Answer: y = -x
Answer:
35. Find the slope of a line tangent
to y cos
=
=
2 x at x
Answer: -1
π
12
.
36. Find the slope of a line tangent
to y x=
cos x at x
=
Answer:
π
4
.
−π 2
2
+
8
2
37. Find the critical numbers of y =
x
x + 16
2
Answer: -4 and 4
38. Find the critical numbers of y =
2x
x +1
2
Answer: -1 and 1
39. Find the vertical and horizontal asymptotes of y =
7 x + 2 x2
4 x 2 − 49
Answer: VA: x = 7/2 and x = -7/2, HA: y = ½
40. Find the vertical and horizontal asymptotes of y =
9 + 2x2
9 x 2 − 25
Answer: VA: x = 5/3 and x = -5/3, HA: y = 2/9
4
41. Let f be defined by =
f ( x) x 3 (5 − 7 x) . f is increasing on the interval.
Answer: (0, 20/49)
1
42. Let f be defined by =
f ( x) x 3 (8 − 3 x) . f is decreasing on the interval.
(
)
Answer: 2 , ∞
3
43. The function f ( x=
) x 5 − 10 x 3 has a relative minimum at x =
Answer: 6
44. The function f ( x=
) x 4 − 2 x 2 has a relative maximum at x =
Answer: 0
45. The function f ( x ) = e x − x3 + 1 has a relative minimum at x =
Answer: 3.73
46. When a circular plate of metal is heated in an oven, its radius increases at the rate of
0.02 cm/min. At what rate is the plate’s area increasing when the radius is 40 cm?
Answer: 5.03 cm2/min
47. The edges of a cube are expanding at a rate of 5 centimeters per second. How fast is
the surface area changing when each edge is 4.5 centimeters?
Answer: 270 cm2/sec
For problem 48 and 49.
A rectangular storage container with an open top is to have a volume of 20 m3. The
length of its base is twice the width. Material for the base costs $15 per square meter.
Material for the sides costs $7 per square meter.
48. Find the cost function for the container.
Answer: C = 30x2 + 420/x
49. Find the cost of materials for the cheapest such container.
Answer: $329.34
50. A ladder 30 feet long is leaning against the wall of a building. The base of the ladder
is pulled away from the wall at a rate of 5 feet per second. How fast is the top of the
ladder moving down the wall when the base of the ladder is 18 feet from the wall?
Answer: -3.75 ft/second
51. An airplane is flying at an altitude of 5 miles and passes directly over a radar antenna.
When the plane is 10 miles away, the radar detects that the distance is changing at a rate
of 240 miles per hour. What is the speed of the plane?
Answer: 277.13 mph
52. The position of a particle is given by the equation s (t ) = 2t 3 − 5t 2 + 3t , where s is
measured in meters, and t is measured in seconds. Find the velocity v(t) and acceleration
a(t) when t = 3 seconds.
Answer: v(3) = 27 m/s and a(3) = 26 m/s2
53. At time t = 0, a diver jumps from a platform diving board that is 32 feet above the
water. The position of the diver is given by s (t ) =
−16t 2 + 16t + 32 where s is measured in
feet and t is measured in seconds. Find when the diver will hit the water and what is the
velocity of the diver at impact?
Answer: t = 2 seconds and -48 ft/s
For problems 54 and 55.
The altitude h (in meters) of a rocket as a function of time t (in seconds) after launching is
given by
=
h 350t − 4.8t 2 .
54. What is the maximum altitude that the rocket reaches?
Answer: 6380.21 meters
55. Find the time that will produce this maximum altitude.
Answer: 36.46 seconds
56. Evaluate the Riemann sum for f(x) = 65 − 3x 2 , 1 ≤ x ≤ 4 , using 6 rectangles of equal
width and left endpoints.
Answer: 142.875
x +1
57. Evaluate the Riemann sum for f(x) =
, 1 ≤ x ≤ 5 , using 4 rectangles of equal
x+3
width using right endpoints and midpoints.
Answer: RH: 2.7310 Midpt: 2.6176
5.7
58. Evaluate the integral
∫ ( 5 + 2 x − 3x )dx
2
1.2
Answer: -129.915
1
59. Evaluate the integral
∫ (t
2
+ 2 )dt
−1
Answer: 14/3
3
60. Evaluate the integral
∫ ( sin x + 3cos x ) dx
1
Answer: -0.5708
π
61. Evaluate the integral
−
4
∫
π
sec 2 tdt
4
Answer: 2
62. Evaluate the integral
Answer:
∫ ( tan
7
x sec 2 x ) dx
1 8
tan x + c
8
63. Evaluate the integral
cosx
dx
sinx
∫
Answer: 2 sin x + c
64. Evaluate the integral ∫ 12 x 2 (2 x 3 − 15) 4 dx
Answer:
5
2
2 x3 − 15 ) + c
(
5
65. Evaluate the integral
Answer:
∫x
2
x 3 + 3 dx
3
2 3
x + 3) 2 + c
(
9
4
66. Evaluate the integral
∫
3 − xdx
4
x −5
dx
2
−4
−1
Answer: Does not exist.
67. Evaluate the integral
∫x
−1
Answer: Does not exist.
π
e tan x
∫0 cos2 x dx
4
68. Evaluate the integral
Answer: e – 1.
69. Evaluate the integral
Answer:
1
ln 7 x − 2 + c
7
70. Evaluate the integral ∫
Answer:
1
∫ 7 x − 2 dx
1
tan −1 ( 4 x ) + c
4
1
dx
1 + 16 x 2
Use the graph to answer the next 4 questions.
Graph of f’
71. Over what intervals is the function f increasing and decreasing?
Answer: Increasing: (-∞, -1.2) and (3, ∞) and Decreasing: (-1.2, 3)
72. At what values of x does the function f have max/min?
Answer: maximum occurs at x = -1.2 and minimum occurs at x = 3
73. Over what intervals is the function f concave up or concave down?
Answer: Concave up: (-0.5, 0.5) and (2.1, ∞) and Concave down: (-∞, -0.5) and (0.5, 2.1)
74. At what value(s) of x does the function f have inflection points?
Answer: x = -0.5, 0.5, and 2.1
Using the graph to answer the next 2 questions
Graph of g”
75. Over what is interval is the function g concave up or concave down?
Answer: Concave up: (-∞, -0.3) and (2.3, ∞) and Concave down: (-0.3, 2.3)
76. Determine the value of x that will produce an inflection point for the function g.
Answer: x = -0.3 and 2.3
2


t
77. Given the cost function for a particular paper product as C ( t ) =+
2500 12sin   + 51 ,
5


when C(t) represents cost in dollars and t is measured in weeks. Determine at what rate is the
cost of paper product is decreasing when t = 40.
Answer: $43.91 dollars per day
78. The sale of steel K (in thousands of square feet) for the years 1990 to 2000 is modeled by the
function=
K ( t ) 0.61cos(0.45t + 3.85) + 1.3 , where t is time in years with t = 0 corresponding
to the beginning of 1990. During what year would the number of steel sales be at the greatest
rate?
Answer: 1995