Practice Problems for TEST III
1.
Find an anti-derivative of each of the following functions. Show your work!
(a) (1  3x) 2.9
(b)
sinh x
13  cosh x
(c )
sin x 8 cos 3 x
Hint: Express this as a function of sin x multiplied by cos x.
(d )
cos x sin 3 x Hint: Similar to (c).
2. Find the indefinite integral of each of the following functions. Show your work!
(a)
(b)
(c )
1  4 ln x
x
x 3e58 x
4
arctan x
1 x2
3. Using de l’Hôpital’s rule, compute the following limit:
lim
x 0
e2 x  1  2x  2x2
x3
2
4. Solve the following initial value problem:
dy 3
 t cos(t 4 )  t  4
dt
given that y = 3 when t = 0.
5. We wish to approximate the value of
5
x
(
1

2
) dx.

1
(a) Sketch the curve g(x) = 1 – 2-x over the interval [1, 5].
Using two rectangles of equal base length, compute each of the following
estimates:
(b) left-hand endpoints
(c) right-hand endpoints
(d) midpoints
6.
Verify the following anti-differentiation formula:

1
1
x sin 2 x  dx   x cos( 2 x)  sin( 2 x)  C
2
4
7. Using only the area interpretation of the definite integral, compute each of the
following. (Sketch!)
1
(a)

| 1  2 x | dx
2
7
(b)
 1  3x  dx
1
3
7
(c )

49  t 2 dt
0
8. Compute each of the following. Simplify your answers as much as possible.
2
(a)

k 0
k
k 1
 i
2
(b)
4
 2i

i 1
5
(c )
 ln j
j 1
2015
(d )
m
m 1
2015
(e)

j 1789
1
1

j j 1
2015
(f)
k
(

1
)

k 1
9. Evaluate
3

x

5  x 4  9  x 2 dx
3
by interpreting this definite integral in terms of areas. Sketch!
4
10.
Consider the graph of y = f (x) drawn below. (Note: This is not the graph of f.)
(a)
On which intervals, if any, is f increasing?
(b) At which values of x does f have a critical point?
(c) On which intervals, if any, is f concave up?
(d) Which values of x, if any, correspond to inflection points on the graph
of f?
(e) Assume that f(0) = 0. Sketch a graph of f. (Your graph need only have the
right general shape. You do not need to put units on the vertical axis.)
11. State the two versions of the Fundamental Theorem of Calculus.
12. Albertine has purchased a Chevy Volt that can accelerate from 0 ft/sec to 88 ft/sec
in 5 seconds. The car’s velocity is given below:
t (seconds)
v(t) (ft/second)
0
0
1
30
2
52
3
68
4
80
5
88
Using five rectangles, find upper and lower bounds (that is, over and underestimates) for the distance traveled by Albertine’s car in 5 seconds.
5
13. Using the FTC, evaluate each of the following. You need not simplify your
answers.

4
(a)
 tan x 
4
sec 2 x dx
0
1
(b)
e4 x
dx
5  3e 4 x

0
1
(c )

t
4
1  2t 2 dt
0
14. Using the method of judicious guessing evaluate:
1
 arcsin z 
15.
1 z2
dz
Using the FTC, compute g'(x) given that
x
g ( x)   t 5 (5  4 ln t )13 dt
0
What is the value of g'(e) ?
16.
Find an anti-derivative of each of the following functions. Show your work!
(a)
tan x
(b)
tan2 x
(c)
sin2 x
Hint: Recall that sin 2  
1  cos 2
2
(d) cos2 x
Hint: Recall that cos 2  
1  cos 2
2
(e) sec4 x
Hint: Write sec4 x = (sec2 x)(sec2 x) and then apply a basic identity.
6
17. Solve the initial value problem:
dy 
   2 ln x
 sec 2  x  
dx 4
x
4 
given that y(1) = 13.
18.
19.
Evaluate each of the following indefinite integrals:
(a)
cos x
 sin x  13 dx 
(b)
 x sin( x
(c)
x
2
2
 5) dx 
(11x 3  99) 51 dx 
Evaluate each of the following definite integrals using only the geometric
interpretation of the definite integral. Explain your solution.
2
(a)
 | 1  3x | dx
0
7
(b)

49  x 2 dx
0
7
(c)
x5
71  x 2  cos x dx
 3  x (cos x) e  dx
5
x2
(d)
5
7
20. Calculate each of the following sums. Simplify each answer.
 
4
(a)
 sin  j 2 
j 3
4
(b)

k 2
1
k 1
5
(c)

m2
m 1
2
21. Given that

5

f ( x) dx  4 and
2
5
f ( x) dx  3 and
 g ( x) dx  2
2
2
5
compute the value of
  f ( x)  2 g ( x)  5 dx.
2
22. Find the average value of the function f(x) = 4 sin 2x over the interval [0, ].
23. Which values of a and b minimize the value of
b
 (x
6
 25 x 4 ) dx
a
24. Find dy/dx given that
4 x 2 1
y
 sin
t dt
1
25. Find the area bounded by the x-axis and the curve y = x(x – 3)2.
26. Find the area between the curves y = 5 – x and y = 7 – x2.
8
27. Find the area of the region in the first quadrant that is bounded above by y = x 1/2
and below by the x-axis and the line y = x – 2.
28. Compute the average value of the curve y = cos2 4x over the interval [0, ].
29. Let f(x) be a continuous function. Express
lim
n
1 1
2
3
 n 
 f    f    f    ...  f   
n n
n
n
 n 
as a definite integral.
30. Compute the value of the following limit
115  215  315  ...  n15
lim
n
n16
by expressing this limit as a Riemann integral and then evaluating the integral.
31. Evaluate each of the following.
e  1  x3
lim
x 0
x6
x3
(a)
3
(b)
j
j 3

j 0
4
(c)

k 0
(and simplify)
ln( k  1)
(and simplify)
9
(d)

0
4 81  x 2 dx
9
32. Consider the following table of data:
x
-1.00 -0.25 0.50
1.25
2.00
F(x) 0.0000 2.6522 4.8755 6.8328 8.6790
Approximate the area below the graph of y = F(x) above the interval [-1, 2] using:
(a) left endpoints. Sketch.
(b) right endpoints. Sketch.
33. Suppose that Madame Verdurin wishes to find a solution to the equation
ln x = x – 5.
(a) Sketch the graphs of the two curves (y = ln x and y = x – 5) on the same pair
of axes. How many solutions would you expect to find?
(b) Define a function, G(x), whose roots will provide Verdurin with the answer
to her quest.
(c) Assume that Verdurin’s first guess is x0 = 6. Verdurin uses Newton’s
method to compute x1 and x2. Which values should she obtain! Show
your computation.
34. Solve the following initial value problem:
dy
2
3


dt 1  4t 2 t  1
given that y(0) = 11.
35. Verify the following integration formula:



x2
1
dx 
2 x  3 6  2 x  x2  C
5
2x  3
10
36. At time t, in seconds, the velocity, v, in miles per hour, of Albertine’s new Prius is
given by
v(t) = 5 + 0.8t2 for 0 < t < 8.
Use Δt = 2 to estimate the distance traveled during this time. Find the left endpoint
and right endpoint sums, and the average of the two. Sketch!
37. Using the method of judicious guessing, find an antiderivative for each of the
following functions. Be certain to show your reasoning!
x  5(2 x  1)
(a)
(b)
x3
(ln x)99
x
cosh x
(c)
(d)
5  3 sinh x
sec( 4 x)(tan( 4 x)
1  3 sec( 4 x)
3
38. Suppose that
  f ( x)  3x dx  0.
2
  f ( x)  x dx.
3
Evaluate
3
2
39. Find the area of the region enclosed by y = x(a2 – x2)1/2 , where a is a positive
constant and the y-axis.
40. Find the area of the region enclosed by y = (|x|)1/2 and 5y = x + 6.
41. Find the area of the region enclosed by x = y3 – y2 and x = 2y.
42. Find the area of the region enclosed by y = 2 sin x and y = sin (2x), 0 ≤ x ≤ .
11
x
43. Find dy / dx given that y 
1
dt

t

1
sec3 x 
2
44. Evaluate the following limit:
lim
x 0
4 x 2
x
45. Evaluate the following limit
lim x 3 / x
x 
46. Evaluate the following limit
lim
x 0
1  2x  1  x
x2
47. Evaluate the following limit
cos(4 x)  1  8 x 2
lim
x 0
x4
48. Evaluate the following limit
x1357  1
lim
x1
x 1
3
49. Explain why the value of
 cos
2
( x 4 ) dx cannot equal 4.5.
0
50. [U. Michigan] Using the graph of y = h(x) shown below, compute each of the
following quantities. If there is not enough information to compute the given quantity
write “not enough information.” Justify each answer!
12
40
y = h(x)
20
Area = 12
1
2
3
4
5
Area = 25
20
Area = 5
40
0
( A)
 h( x)  3 dx
2
5
( B)
 3h( y) dy
0
9
(C )
 h( x  4) dx
8
(D) The average value of h(x) on the interval [-2, 2], assuming that h(x) is an
even function.
(E)
H(2), where H is an anti-derivative of h.
(F)
H(2) – H(0), where H is an anti-derivative of h.
51. (a) State Rolle’s Theorem.
(b) Using Rolle’s Theorem, prove that the function
g(x) = (x – 2) ln (x + 1) + x sin(4x)
13
has at least one critical point between x = 0 and x = 2?
Explain!
52. (a) State the Mean Value Theorem.
(b) Show how the Mean Value Theorem applies to the function
f(x) = 4 + ln x on the interval [1, e3]. Sketch! Find explicitly the c value.
53. Explain why any two anti-derivatives of a function F(x) must differ by a constant.
54. Find dy/dt given that
sinh t
y
 cosh
v dv
1
55. (a) State Rolle’s Theorem.
(b) Using Rolle’s Theorem, prove that the function
g(x) = (x – 2) ln (x + 1) + x sin(4x)
has at least one critical point between x = 0 and x = 2?
Explain!
56. (U. Michigan) A car, initially going 100 feet per second, brakes at a constant rate
(constant negative acceleration), coming to a stop in 8 seconds. Let t be the time
in seconds after the car started to brake.
(a) Sketch a graph of the velocity of the car from t = 0 to t = 8, being sure to
include labels.
(b) Exactly how far does the car travel? Make it clear how you obtained your
answer.
57. (U. Michigan) Suppose dg/dx > 0 on the interval [3, 5], g(3) = 12, and g(5) = 20.
We want to use a Riemann sum with equal-size subdivisions to approximate
14
5
 g ( x) dx
3
If we want to guarantee that the error in our estimate is no larger than ¼, then what
is the minimum number of subdivisions that we must use?
58. (U. Michigan) If
4
 2 f ( x)  7 dx  31
1
find the value of
4

f ( x) dx
1
59. State three corollaries to the Mean Value Theorem.
60. (Apostol) A function f is continuous everywhere and satisfies the equation
x

1
1
1
f (t ) dt    x 2  x sin 2 x  cos 2 x
2
2
for all x. Compute f(/4) and df/dx at x =  /4.
61. (Apostol) Find a function f and a value of the constant c such that, for all real x,
x

f (t ) dt  cos x 
c
1
2
62. Can the following limit be solved using L’Hopital’s rule? Explain.
3x 2  1
lim
x 1 2 x  3
63. Can the following limit be solved using only L’Hopital’s rule? Explain.
15
sin x
x 0 x 2  3x
lim
64. Find:

1 1/ n
lim e  e 2 / n  ...  e ( n 1) / n  e n / n
n  n

65. Can the following limit be solved using L’Hopital’s rule? Explain.
cos 2 x  1
x 0 x3  5 x 2
lim
66. Can the following limit be solved using only L’Hopital’s rule? Explain.
ex
lim 13
x x
67. Can the following limit be solved using only L’Hopital’s rule? Explain.
x
x  0 ln x
lim
68. (MIT 18.01 final)
Use L’Hopital’s rule to compute the following limits:
a x  bx
(i ) lim
where 0  a  b
x 0
x
4 x3  5 x  1
(ii ) lim
x 1
ln x
16
69. (U. Michigan) The rate at which a coal plant releases CO2 into the atmosphere t
days after 12:00 am on January 1, 2015 is given by the function E(t) measured in tons
31
per day. Suppose that
 E (t ) dt  223.
0
59
(a) Give a practical interpretation of  E (t ) dt.
31
(b) Give a practical interpretation of E(15) = 7.1.
(c) The plant is upgrading to “clean coal” technology which will cause its July 2015
CO2 emissions to be one fourth of its January 2015 CO2 emissions. How much
CO2 will the coal plant release into the atmosphere in July?
(d) Using a left-hand sum with four subdivisions, write an expression which
59
approximates  E (t ) dt.
31
70.
(U. Michigan) Shown below is a graph of a function f(t). The graph consists of a
straight line between t = 0 and t = 2 and a quarter circle between t = 2 and t = 3.
2
1
0.5
1.0
1.5
2.0
2.5
3.0
1
2
Calculate the following using the graph and the properties of integrals.
17
3
(a)
3 (2  f (t )) dt
0
(b) The average value of f on the interval [1, 3].
71. (U. Michigan) A car, initially going 100 feet per second, brakes at a constant rate
(constant negative acceleration), coming to a stop in 8 seconds. Let t be the time in
seconds after the car started to brake.
(a) Sketch a graph of the velocity of the car from t = 0 to t = 8, being sure to
include labels.
(b) Exactly how far does the car travel? Make it clear how you obtained your
answer.
72. (U. Michigan) Suppose dg/dx > 0 on the interval [3, 5], g(3) = 12, and g(5) = 20.
We want to use a Riemann sum with equal-size subdivisions to approximate
5
 g ( x) dx
3
If we want to guarantee that the error in our estimate is no larger than ¼, then what is
the minimum number of subdivisions that we must use?
Learning without thought is labor lost; thought without learning is perilous.
- Confucius