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Practice Exam # 4 (4.9-5.5) (6968250)
Due:
Tue May 12 2015 10:40 AM PDT
Question
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Question Details
SCalcET7 4.9.064. [1835869]
-
A particle is moving with the given data. Find the position of the particle.
a(t) = t2 − 7t + 5,
s(t) =
s(0) = 0,
s(1) = 20
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SCalcET7 5.1.005. [2524208]
-
(a) Estimate the area under the graph of f(x) = 1 + 4x2 from x = −1 to x = 2 using three rectangles and right endpoints.
R3 =
23
Then improve your estimate by using six rectangles.
R6 =
18.5
Sketch the curve and the approximating rectangles for R3.
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Sketch the curve and the approximating rectangles for R6.
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(b) Repeat part (a) using left endpoints.
L3 =
11
L6 =
12.5
Sketch the curve and the approximating rectangles for L3.
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Sketch the curve and the approximating rectangles for L6.
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(c) Repeat part (a) using midpoints.
M3 =
14
M6 =
14.75
Sketch the curve and the approximating rectangles for M3.
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Sketch the curve and the approximating rectangles for M6.
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(d) From your sketches in parts (a)-(c), which appears to be the best estimate?
M6
R6
L6
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3.
Question Details
SCalcET7 5.2.026. [1535245]
(a) Find an approximation to the integral
R8 =
-
(x2 − 10x) dx using a Riemann sum with right endpoints and n = 8.
-19.3125
(b) If f is integrable on [a, b], then
evaluate
2
2
(x2
b
n
f(xi) Δx, where Δx =
f(x) dx = lim
n→∞
i=1
b−a
and xi = a + i Δx. Use this to
n
− 10x) dx.
-52/3
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SCalcET7 5.2.034.MI. [1912810]
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SCalcET7 5.2.037. [1865616]
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The graph of g consists of two straight lines and a semicircle. Use it to evaluate each integral.
10
(a)
g(x) dx
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30
(b)
g(x) dx
10
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35
(c)
g(x) dx
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5.
Question Details
Evaluate the integral by interpreting it in terms of areas.
0
1
5+
1 − x2
dx
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Write as a single integral in the form
2
4
f(x) dx +
5
f(x) dx −
SCalcET7 5.2.047.MI. [1535297]
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SCalcET7 5.2.053. [1776364]
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b
f(x) dx.
−1
4
f(x) dx
5
f(x) dx
-1
Solution or Explanation
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7.
Question Details
Each of the regions A, B, and C bounded by the graph of f and the x-axis has area 3. Find the value of
2
4
[f(x) + 2x + 3] dx.
3
Solution or Explanation
I=
2
4
[f(x) + 2x + 3] dx =
2
4
f(x) dx + 2
I1 = −3 [area below x-axis] + 3 − 3 = −3
1
I2 = − (4)(4) [area of triangle, see figure]
2
I3 = 3 [2 − (−4)] = 3(6) = 18
2
4
x dx +
+
2
4
3 dx = I1 + 2I2 + I3
1
(2)(2) = −8 + 2 = −6
2
Thus, I = −3 + 2(−6) + 18 = 3.
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SCalcET7 5.2.063. [1835818]
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If m ≤ f(x) ≤ M for a ≤ x ≤ b, where m is the absolute minimum and M is the absolute maximum of f on the interval [a, b],
then
m(b − a) ≤
b
f(x) dx ≤ M(b − a).
Use this property to estimate the value of the integral.
6
3xe−x dx
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9.
Question Details
SCalcET7 5.3.503.XP. [1835804]
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SCalcET7 5.4.505.XP. [1770491]
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Use Part 1 of the Fundamental Theorem of Calculus to find the derivative of the function.
g(y) =
g '(y) =
y
t2 sin 5t dt
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10.
Question Details
Evaluate the integral.
9
t (1 + t) dt
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SCalcET7 5.4.511.XP. [1776322]
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SCalcET7 5.5.059. [1865682]
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SCalcET7 5.5.062. [1865659]
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SCalcET7 5.5.063. [1865595]
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Evaluate the integral.
16
3x − 8
x
dx
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12.
Question Details
Evaluate the definite integral.
2
e 1/x
x5
4
dx
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13.
Question Details
Evaluate the definite integral.
π/10
cos 5x sin(sin 5x) dx
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14.
Question Details
Evaluate the definite integral.
dx
49
3
(27 + 2x)2
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SCalcET7 5.5.069. [1865744]
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SCalcET7 5.5.071. [1865751]
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SCalcET7 5.4.060. [1535355]
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Evaluate the definite integral.
e64
dx
x
e16
ln x
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16.
Question Details
Evaluate the definite integral.
1
ez + 27
dz
ez + 27z
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17.
Question Details
The velocity function (in meters per second) is given for a particle moving along a line.
v(t) = t2 − 2t − 24,
1≤t≤7
(a) Find the displacement.
-78 m
(b) Find the distance traveled by the particle during the given time interval.
266/3 m
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