Download Chapter 6 Practice Test

Chapter 6 PRACTICE test
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MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Use a finite approximation to estimate the area of the region enclosed between the graph of f and the x-axis for a ≤ x ≤
b.
1) f(x) = x2, a = 3, b = 7
1)
Use LRAM with four rectangles of equal width.
A) 117
B) 105
C) 126
D) 86
2) f(x) = 1 , a = 1, b = 5
x
2)
Use MRAM with two rectangles of equal width.
A) 3
B) 5
4
16
C) 5
24
D) 3
2
Use a calculator or computer program to solve the problem.
3) Use RAM to estimate the area of the region enclosed between the graph of f(x) = 8sin x and the
x-axis for 0 ≤ x ≤ π
A) 16
B) 8π
C) 0
D) 8
Estimate the value of the quantity.
4) The table shows the velocity of a remote controlled race car moving along a dirt path for 8
seconds. Estimate the distance traveled by the car using 8 subintervals of length 1 with left-end
point values.
Time Velocity
(sec) (in./sec)
0
0
1
10
2
18
3
14
4
24
5
27
6
29
7
12
8
5
A) 134 in.
B) 139 in.
C) 124 in.
1
D) 268 in.
3)
4)
Express the limit as a definite integral.
n
2
5) lim n→∞ ∑ (3 c k - 11ck + 19) △xk , [-5, 4]
k=1
n
A)
(6x - 11) dx
1
4
C)
(3x - 11) dx
-5
∫
B)
∫
D)
6) lim n→∞
A)
n
4
∑
k = 1 11 - 9 c 2
k
∫
5
∫
5
3
C)
5)
3
11 - 9x2
Evaluate the integral.
11
7)
(-50) dx
9
A) -541
-5
∫
4
(3x2 - 11x + 19) dx
4
(3x2 - 11x + 19) dx
-5
△xk , [3, 5]
6)
4
dx
11 - 9x
4
∫
B)
∫
3
∫
n
11 - 9x2
5
dx
D)
4
1
dx
4
dx
11 - 9x
∫
8)
∫
3.9
7)
B) 550
C) -100
D) -550
0.6 ds
-5.7
A) -5.76
8)
B) 5.76
C) 1.08
Graph the integrand and use areas to evaluate the integral.
4
9)
16 - x2 dx
-4
A) 16π
B) 16
D) -1.08
∫
10)
∫
10
9)
C) 8π
D) 4π
x dx
10)
-3
A) 109
2
C) 91
2
B) 13
2
D) 109
Use areas to evaluate the integral.
b
11)
7x dx ,
0<a<b
a
A) 7 (b 2 - a2)
2
∫
12)
∫
7a
x dx ,
11)
B) 7 (b - a)
2
C) 7(b - a)
D) 7(b 2 - a2)
a>0
12)
a
A) (
7 - 1)a
B)
7a2
C) 3a2
D) 6a2
Use NINT on a calculator to find the numerical integral of the function over the specified interval.
13) y = 6tan x ; from x = 0 to x = π
6
A) 1.86304469
B) -0.8630447
C) -1.1369553
13)
D) 0.86304469
Find the points of discontinuity of the integrand on the interval of integration, and use area to evaluate the integral.
2 x
14)
dx
14)
x
-7
A) 0, -14
B) 0, 9
C) 0, -5
D) 0, - 1
5
∫
Solve the problem.
15) Suppose that
∫
5
f(x) dx = -4. Find
3
5
f(x) dx and
5
A) 5; -4
16) Suppose that
∫
-4
-7
f(x) dx .
-4 g(x)
-7
dx and
- g(t) dt .
9
-7
-4
B) 1; -9
C) -1; -9
g(t) dt = 9. Find
A) 1; 9
3
5
C) -4; 4
B) 0; 4
∫
∫
∫
15)
D) 0; -4
∫
16)
D) 0; 9
USE NINT to find the average value of the function on the interval. At what point in the interval does the function
assume its average value?
17) y = -6x2 - 1, [0, 3.46410162]
17)
A) -73, at x = 3.46410162
C) 73, at x = 3.46410162
B) 25, at x = 2
D) -25, at x = 2
3
Find the average value of the function without integrating, by appealing to the geometry region between the graph
and the x-axis.
-6 ≤ x ≤ -1
18) f(x) = x + 6,
18)
-x + 4, -1 < x ≤ 4
y
10
8
6
4
2
-10 -8
-6
-4
A) 3
-2
2
4
6
8
10 x
B) 5
2
C) 5
D) 2
Interpret the integrand as the rate of change of a quantity and evaluate the integral using the antiderivative of the
quantity.
π
19)
9 sin x dx
19)
0
A) 9
B) 18
C) 162
D) 2
∫
20)
∫
5
ex dx
20)
-7
A) e12
C) e5 - 1
e7
B) e5 + e7
D) e5 - e7
Find the average value over the given interval.
21) y = 4x4 ; [-3, 3]
A) 162
5
22) y = 6x + 6; [3, 7]
A) 36
21)
B) 1944
5
C) 324
5
D) 0
22)
B) 66
C) 144
D) 6
SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.
Provide an appropriate response.
23) Use the max-min inequality to show that if f is integrable and f(x) ≤ 0 on [a, b], then
b
f(x) dx ≤ 0.
a
∫
4
23)
24) Show that the average value of a linear function L(x) on [a, b] is L(a) + L(b) .
2
24)
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Find dy/dx.
25)
∫
x
16t9 dt
25)
1
A) 8 x6 - 8
5
5
B) 16x9/2
Construct a function of the form y =
∫
x
D) 32 x6
3
C) 8x4
f(t) dt + C that satisfies the given conditions.
a
26)
dy = cot x, and y = 8 when x = -3
dx
A) y =
∫
x
26)
cot t dt + -3
B) y =
8
C) y = -
∫
x
csc 2 t dt + 8
D) y =
-3
Evaluate the integral.
-1
27)
2x dx
2
-7
A)
2 ln 2
∫
-3
x
x
∫
cot t dt + 8
cot t dt + 8
-3
∫
27)
B)
5
2 ln 2
C)
-3
2 ln 2
D)
-9
2 ln 2
Find the total area of the region between the curve and the x-axis.
28) y = x2 - 6x + 9; 2 ≤ x ≤ 4
A)
1
3
B)
2
3
C)
5
4
3
28)
D)
7
3
Find the area of the shaded region.
29)
29)
y = x3 - 4x
y
16
(3, 15)
12
8
4
(0, 0)
A)
1
2
41
4
3
4
B)
x
17
4
C)
Use NINT to solve the problem.
10
1
30) Evaluate
dx.
4 + 3 cos x
0
A) ≈ 0.022
B) ≈ 0.026
9
4
D)
33
4
∫
30)
C) ≈ 3.107
D) ≈ 4.064
Solve the problem.
31) Suppose that
∫
x
f(t) dt = 4x2 + 8x - 2. Find f(x).
31)
1
4
4
A) x3 + x2 - 2x - 10
3
3
B) 8x + 8
D) 4 x3 + 4x2 - 2x
3
C) 4x2 + 8x - 2
6
32) Suppose that f is the differentiable function shown in the graph and that the position at time t (in
t
seconds) of a particle moving along a coordinate axis is s =
f(x) dx feet.
0
y = f(x)
32)
∫
y
10
8
6
4
2
-10 -8 -6 -4 -2
-2
-4
-6
-8
-10
(8, 8)
(1, 2.4)
(2, 2)
2
4
6 8 10 x
(4, -2.4)
(6, -3.6)
What is the particle's velocity at time t = 1?
A) 2.4 ft/sec
B) 0 ft/sec
C) -2.4 ft/sec
Use the Trapezoidal Rule to estimate the integral.
8
33)
x dx , n = 4
0
A) 64
B) 16
D) 0.8 ft/sec
∫
34)
∫
0
1
33)
C) 40
D) 32
6 dx , n = 4
1 +x
A) 743
140
34)
B) 1747
840
C) 1171
280
D) 1171
140
Solve the problem.
35) Suppose that the accompanying table shows the velocity of a car every second for 8 seconds.
Use the Trapezoidal Rule to approximate the distance traveled by the car in the 8 seconds.
Time (sec) Velocity (ft/sec)
0
20
1
21
2
22
3
24
4
23
5
25
6
22
7
20
8
21
A) 355 feet
B) 177.5 feet
C) 269.5 feet
7
D) 198 feet
35)
Answer Key
Testname: CHAPTER 6 PRACTICE
1) D
2) D
3) A
4) A
5) D
6) C
7) C
8) B
9) C
10) A
11) A
12) C
13) D
14) C
15) B
16) A
17) D
18) B
19) B
20) C
21) C
22) A
23) If f(x) ≤ 0 on [a, b], then max f ≤ 0. So
∫
b
f(x) dx ≤ max f ∙ (b - a) ≤ 0.
a
24) Let L(x) = cx + d. Then the average value of L(x) on [a, b] is
b
av(L) = 1
(cx +d)dx
b-a a
∫
=
1
b-a
cb 2 + db - cb 2 + db
2
2
=
1
b-a
c(b 2 - a2) + db + d(b - a)
2
= c(b + a) + 2d
2
= (ca + d) + (cb + d)
2
= L(a) + L(b)
2
25) C
26) D
27) D
28) B
29) A
30) D
31) B
8
Answer Key
Testname: CHAPTER 6 PRACTICE
32) A
33) D
34) C
35) B
9