Question 1

Find the partial fraction expansion of the rational function: $\frac { 9 } { ( x + 1 ) ^ { 2 } ( 2 - x ) }$ .

Accepted Answer

A)  $\frac { - 1 } { x + 1 } + \frac { 3 } { ( x + 1 ) ^ { 2 } } + \frac { - 1 } { x - 2 }$ 
B)  $\frac { 1 } { x + 1 } - \frac { 1 } { x - 2 }$ 
C)  $\frac { - 1 } { x + 1 } + \frac { - 3 } { ( x + 1 ) ^ { 2 } } + \frac { - 1 } { x - 2 }$ 
D)  $\frac { 3 } { ( x + 1 ) ^ { 2 } } - \frac { 1 } { x - 2 }$ 
E)  $\frac { 1 } { x + 1 } + \frac { 3 } { ( x + 1 ) ^ { 2 } } + \frac { 1 } { x - 2 }$ 
F) $\frac { 1 } { x + 1 } - \frac { 3 } { ( x + 1 ) ^ { 2 } } - \frac { 1 } { x - 2 }$ 
G) $\frac { 1 } { x + 1 } + \frac { 3 } { ( x + 1 ) ^ { 2 } } - \frac { 1 } { x - 2 }$ 
H) $\frac { - 1 } { x + 1 } + \frac { - 3 } { ( x + 1 ) ^ { 2 } } + \frac { 1 } { x - 2 }$ 
A)  $\frac { - 1 } { x + 1 } + \frac { 3 } { ( x + 1 ) ^ { 2 } } + \frac { - 1 } { x - 2 }$ 
B)  $\frac { 1 } { x + 1 } - \frac { 1 } { x - 2 }$ 
C)  $\frac { - 1 } { x + 1 } + \frac { - 3 } { ( x + 1 ) ^ { 2 } } + \frac { - 1 } { x - 2 }$ 
D)  $\frac { 3 } { ( x + 1 ) ^ { 2 } } - \frac { 1 } { x - 2 }$ 
E)  $\frac { 1 } { x + 1 } + \frac { 3 } { ( x + 1 ) ^ { 2 } } + \frac { 1 } { x - 2 }$ 
F) $\frac { 1 } { x + 1 } - \frac { 3 } { ( x + 1 ) ^ { 2 } } - \frac { 1 } { x - 2 }$ 
G) $\frac { 1 } { x + 1 } + \frac { 3 } { ( x + 1 ) ^ { 2 } } - \frac { 1 } { x - 2 }$ 
H) $\frac { - 1 } { x + 1 } + \frac { - 3 } { ( x + 1 ) ^ { 2 } } + \frac { 1 } { x - 2 }$ G

Question 2

Let f (x) = $2 x + 1$ on the interval [ - 1,1]. Let the interval be divided into four equal subintervals. Find the value of the Riemann sum $\sum _ { i = 1 } ^ { 4 } f \left( x _ { i } ^ { * } \right) \Delta x _ { i } \text { if } x _ { i } ^ { * }$ is the left endpoint of its subinterval.

Accepted Answer

A) 1 
B) 2 
C)  $- 2$ 
D) 0 
E) 6
F)3
G)4
H) $- 1$ 
A) 1 
B) 2 
C)  $- 2$ 
D) 0 
E) 6
F)3
G)4
H) $- 1$ A

Question 3

Given the graph below, use 5 rectangles to estimate the area under the graph from x = 0 to x = 5. Compute L5 (sample points are left endpoints), R5 (sample points are right endpoints) and M5 (sample points are midpoints). Which of the estimates appears to give the best estimate? Justify your answer.

Accepted Answer

L₅ $\approx$ 44.8; R₅ $\approx$ 28.8; M₅ $\approx$ 36.2; Since the graph of the function is decreasing, using midpoints appears to give the best approximation.

Question 4

Let $$f ( x ) = \left\{ \begin{array} { l l l } 
0 & \text { if } & x < 0 \
2 x & \text { if } & 0 \leq x \leq 1 \text { and } g ( x ) = \int _ { 0 } ^ { x } f ( t ) d t . \
2 & \text { if } & 1 < x
\end{array} \right.$$ (a) Find an expression for g (x) similar to the one for f (x).(b) Where is f differentiable? Where is g differentiable?

Accepted Answer

(a) @#IMG-DLM& (b) f (x) is di

Question 5

Evaluate $\int _ { - 1 } ^ { 1 } \left( 2 - \sqrt { 1 - x ^ { 2 } } \right) d x$ by interpreting the integral in terms of area.

Accepted Answer

The answer of Evaluate \(\int _ { - 1 }...

Question 6

Find the value of the integral $\int _ { - 3 } ^ { 3 } \frac { x } { \sqrt { 1 + 3 x ^ { 2 } } } d x$

Accepted Answer

A)  $\frac { 2 } { 3 } ( \sqrt { 7 } - 1 )$ 
B)  $\sqrt { 7 } - 1$ 
C) 0 
D)  $\frac { 1 } { 3 }$ 
E) $\frac { 2 } { 3 }$
F) $\frac { 1 } { \sqrt { 7 } }$ 
G) $\sqrt { 7 }$ 
H)Does not exist 
A)  $\frac { 2 } { 3 } ( \sqrt { 7 } - 1 )$ 
B)  $\sqrt { 7 } - 1$ 
C) 0 
D)  $\frac { 1 } { 3 }$ 
E) $\frac { 2 } { 3 }$
F) $\frac { 1 } { \sqrt { 7 } }$ 
G) $\sqrt { 7 }$ 
H)Does not exist

Question 7

What is the value of the estimate using four approximating rectangles and taking sample points to be right-hand endpoints?

Accepted Answer

A)  $\frac { 1 } { 2 }$ 
B)  $\frac { 7 } { 4 }$ 
C)  $\frac { 21 } { 8 }$ 
D) 3 
E)  $\frac { 15 } { 4 }$ 
F) $\frac { 29 } { 8 }$

G) $\frac { 9 } { 2 }$ 
H)5 
A)  $\frac { 1 } { 2 }$ 
B)  $\frac { 7 } { 4 }$ 
C)  $\frac { 21 } { 8 }$ 
D) 3 
E)  $\frac { 15 } { 4 }$ 
F) $\frac { 29 } { 8 }$

G) $\frac { 9 } { 2 }$ 
H)5

Question 8

Given the graph of $y = h ( x )$ below:   (a) Find c in [0,6] which will maximize $\int _ { 0 } ^ { c } h ( x ) d x$ (b) Show that $\int _ { - 1 } ^ { 3 } h ( x ) d x$ is between 4 and 7.

Accepted Answer

(a) 4
(b) By simply

Question 9

Prove that $\frac { - x ^ { 3 } } { 3 } \cos x$ is not an antiderivative of $x ^ { 2 } \sin x$ .

Accepted Answer

The answer of Prove that \(\frac { - x ^...

Question 10

Evaluate the improper integral $\int _ { 0 } ^ { \infty } e ^ { - 3 x } d x$

Accepted Answer

A) 3 
B)  $\frac { 1 } { 3 }$ 
C)  $- \frac { 1 } { 3 }$ 
D) 1 
E)  $- 1$  
F) $- 3$ 
G)0
H)Divergent 
A) 3 
B)  $\frac { 1 } { 3 }$ 
C)  $- \frac { 1 } { 3 }$ 
D) 1 
E)  $- 1$  
F) $- 3$ 
G)0
H)Divergent

Question 11

A particle travels along a line. Its velocity in meters per second is given by $v ( t ) = 3 t ^ { 2 } - 12 t$ Find
(a) the displacement from t = 0 to t = 8.(b) the distance traveled by the particle from t = 0 to t = 8.

Accepted Answer

The answer of A particle travels along a line. Its...

Question 12

Suppose we wish to estimate the area under the curve y = x between x = 1 and x = 2 using 2 subintervals of equal length. What is the largest value the approximation could have?

Accepted Answer

A)  $\frac { 5 } { 16 }$ 
B) 0 
C)  $\frac { 1 } { 4 }$ 
D)  $\frac { 3 } { 8 }$ 
E)  $\frac { 5 } { 4 }$ 
F) $\frac { 7 } { 4 }$  
G) $\frac { 1 } { 2 }$  
H) $\frac { 3 } { 16 }$ 
A)  $\frac { 5 } { 16 }$ 
B) 0 
C)  $\frac { 1 } { 4 }$ 
D)  $\frac { 3 } { 8 }$ 
E)  $\frac { 5 } { 4 }$ 
F) $\frac { 7 } { 4 }$  
G) $\frac { 1 } { 2 }$  
H) $\frac { 3 } { 16 }$

Question 13

Intelligence Quotient (IQ) scores are assumed to be normally distributed in the population. The probability that a person selected at random from the general population will have an IQ between 100 and 120 is given by $\int _ { 0 } ^ { 2 } p ( x ) d x$ Use the graph of p (x) graphed below to answer the questions which follow:   
(a) Use Simpson's Rule with n = 4 to approximate $\int _ { 0 } ^ { 2 } p ( x ) d x$ 
(b) The probability that a person selected from the general population will have an IQ score between 80 and 120 is given by $\int _ { - 2 } ^ { 2 } p ( x ) d x$ What is the approximate value of $\int _ { - 2 } ^ { 2 } p ( x ) d x ?$ (c) Since p (x) represents a probability distribution, the entire area of the region under the graph is exactly 1. Using this information, what is the approximate probability that a person selected at random from the general population will have an IQ score over 120?

Accepted Answer

The answer of Intelligence Quotient (IQ) scores are assumed to...

Question 14

Use the Midpoint Rule with 2 equal subdivisions to get an approximation for ln 5.

Accepted Answer

The answer of Use the Midpoint Rule with 2 equal...

Question 15

Use (a) the Trapezoidal Rule with n = 8 and (b) Simpson's Rule with n = 8 to approximate $\int _ { 0 } ^ { 2 } e ^ { x } d x$ Round your answers to six decimal places.

Accepted Answer

(a) T = 6.

Question 16

Given functions of $f$ and $g$ , use the facts that $\int _ { 0 } ^ { 4 } f ( x ) d x = 4$ , $\int _ { 4 } ^ { 6 } f ( x ) d x = 3$ $\int _ { 0 } ^ { 6 } g ( x ) d x = - 5 \text {, and } \int _ { 2 } ^ { 6 } g ( x ) d x = - 4$ to evaluate the following integrals:
(a) $\int _ { 0 } ^ { 4 } ( 3 \cdot f ( x ) + 1 ) d x$ (b) $\int _ { 0 } ^ { 6 } f ( x ) d x$ (c) $\int _ { 0 } ^ { 6 } [ f ( x ) - 4 \cdot g ( x ) ] d x$ (d) $\int _ { 0 } ^ { 2 } g ( x ) d x$

Accepted Answer

(a) 16
(b)

Question 17

Let $f ( x ) = \int _ { x ^ { 2 } } ^ { 3 } \sqrt [ 3 ] { 4 - t ^ { 3 } } d t . \text { Find } f ^ { \prime } ( x )$

Accepted Answer

A) 0 
B)  $- 2 x \sqrt [ 3 ] { 4 - x ^ { 6 } }$ 
C)  $2 x \sqrt [ 3 ] { 4 - x ^ { 6 } }$ 
D)  $\sqrt [ 3 ] { 4 - x ^ { 3 } }$ 
E)  $- \sqrt [ 3 ] { 4 - x ^ { 6 } }$  
F)
$- 2 x \sqrt [ 3 ] { 4 - x ^ { 3 } }$  
G) $\sqrt [ 3 ] { 4 - x ^ { 6 } }$  
H) $- \sqrt [ 3 ] { 4 - x ^ { 3 } }$ 
A) 0 
B)  $- 2 x \sqrt [ 3 ] { 4 - x ^ { 6 } }$ 
C)  $2 x \sqrt [ 3 ] { 4 - x ^ { 6 } }$ 
D)  $\sqrt [ 3 ] { 4 - x ^ { 3 } }$ 
E)  $- \sqrt [ 3 ] { 4 - x ^ { 6 } }$  
F)
$- 2 x \sqrt [ 3 ] { 4 - x ^ { 3 } }$  
G) $\sqrt [ 3 ] { 4 - x ^ { 6 } }$  
H) $- \sqrt [ 3 ] { 4 - x ^ { 3 } }$

Question 18

What is the largest value of $\int _ { 0 } ^ { 3 } | f ( x ) | d x$ if $- 4 \leq f ( x ) \leq 2$ .

Accepted Answer

The answer of What is the largest value of \(\int...

Question 19

Sketch a plane region whose area is equal to $\lim _ { n \rightarrow \infty } \left\{ \sum _ { i = 1 } ^ { n } \left[ \sqrt { 4 - \left( \frac { 2 i } { n } \right) ^ { 2 } } \right] \frac { 2 } { n } \right\}$ then fine the limit.

Accepted Answer

The answer of Sketch a plane region whose area is...

Question 20

Evaluate $\int \frac { 3 x + 2 } { x - 1 } d x$

Accepted Answer

A)  $5 \ln | x - 1 | + 3 x + C$ 
B)  $5 \ln | x - 1 | + C$ 
C)  $3 \ln | x - 1 | + 3 x + C$ 
D)  $3 x + C$ 
E)  $3 \ln | x - 1 | + 5 x + C$ 
F) $5 x + C$ 
G) $2 \ln | x - 1 | + 3 x + C$ 
H) $2 \ln | x | + C$ 
A)  $5 \ln | x - 1 | + 3 x + C$ 
B)  $5 \ln | x - 1 | + C$ 
C)  $3 \ln | x - 1 | + 3 x + C$ 
D)  $3 x + C$ 
E)  $3 \ln | x - 1 | + 5 x + C$ 
F) $5 x + C$ 
G) $2 \ln | x - 1 | + 3 x + C$ 
H) $2 \ln | x | + C$

Find the partial fraction expansion of the rational function: $\frac { 9 } { ( x + 1 ) ^ { 2 } ( 2 - x ) }$ .

Let f (x) = $2 x + 1$ on the interval [ - 1,1]. Let the interval be divided into four equal subintervals. Find the value of the Riemann sum $\sum _ { i = 1 } ^ { 4 } f \left( x _ { i } ^ { * } \right) \Delta x _ { i } \text { if } x _ { i } ^ { * }$ is the left endpoint of its subinterval.

Evaluate $\int _ { - 1 } ^ { 1 } \left( 2 - \sqrt { 1 - x ^ { 2 } } \right) d x$ by interpreting the integral in terms of area.

Find the value of the integral $\int _ { - 3 } ^ { 3 } \frac { x } { \sqrt { 1 + 3 x ^ { 2 } } } d x$

What is the value of the estimate using four approximating rectangles and taking sample points to be right-hand endpoints?

Prove that $\frac { - x ^ { 3 } } { 3 } \cos x$ is not an antiderivative of $x ^ { 2 } \sin x$ .

Evaluate the improper integral $\int _ { 0 } ^ { \infty } e ^ { - 3 x } d x$

A particle travels along a line. Its velocity in meters per second is given by $v ( t ) = 3 t ^ { 2 } - 12 t$ Find (a) the displacement from t = 0 to t = 8.(b) the distance traveled by the particle from t = 0 to t = 8.

Suppose we wish to estimate the area under the curve y = x between x = 1 and x = 2 using 2 subintervals of equal length. What is the largest value the approximation could have?

Use the Midpoint Rule with 2 equal subdivisions to get an approximation for ln 5.

Use (a) the Trapezoidal Rule with n = 8 and (b) Simpson's Rule with n = 8 to approximate $\int _ { 0 } ^ { 2 } e ^ { x } d x$ Round your answers to six decimal places.

Let $f ( x ) = \int _ { x ^ { 2 } } ^ { 3 } \sqrt [ 3 ] { 4 - t ^ { 3 } } d t . \text { Find } f ^ { \prime } ( x )$

What is the largest value of $\int _ { 0 } ^ { 3 } | f ( x ) | d x$ if $- 4 \leq f ( x ) \leq 2$ .

Sketch a plane region whose area is equal to $\lim _ { n \rightarrow \infty } \left\{ \sum _ { i = 1 } ^ { n } \left[ \sqrt { 4 - \left( \frac { 2 i } { n } \right) ^ { 2 } } \right] \frac { 2 } { n } \right\}$ then fine the limit.

Evaluate $\int \frac { 3 x + 2 } { x - 1 } d x$

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Exam 5: Integrals

Find the partial fraction expansion of the rational function: 9(x+1)2(2−x)\frac { 9 } { ( x + 1 ) ^ { 2 } ( 2 - x ) }(x+1)2(2−x)9​ .

Evaluate ∫−11(2−1−x2)dx\int _ { - 1 } ^ { 1 } \left( 2 - \sqrt { 1 - x ^ { 2 } } \right) d x∫−11​(2−1−x2​)dx by interpreting the integral in terms of area.

Find the value of the integral ∫−33x1+3x2dx\int _ { - 3 } ^ { 3 } \frac { x } { \sqrt { 1 + 3 x ^ { 2 } } } d x∫−33​1+3x2​x​dx

What is the value of the estimate using four approximating rectangles and taking sample points to be right-hand endpoints?

Given the graph of y=h(x)y = h ( x )y=h(x) below: (a) Find c in [0,6] which will maximize ∫0ch(x)dx\int _ { 0 } ^ { c } h ( x ) d x∫0c​h(x)dx (b) Show that ∫−13h(x)dx\int _ { - 1 } ^ { 3 } h ( x ) d x∫−13​h(x)dx is between 4 and 7.

Prove that −x33cos⁡x\frac { - x ^ { 3 } } { 3 } \cos x3−x3​cosx is not an antiderivative of x2sin⁡xx ^ { 2 } \sin xx2sinx .

Evaluate the improper integral ∫0∞e−3xdx\int _ { 0 } ^ { \infty } e ^ { - 3 x } d x∫0∞​e−3xdx

A particle travels along a line. Its velocity in meters per second is given by v(t)=3t2−12tv ( t ) = 3 t ^ { 2 } - 12 tv(t)=3t2−12t Find (a) the displacement from t = 0 to t = 8.(b) the distance traveled by the particle from t = 0 to t = 8.

Suppose we wish to estimate the area under the curve y = x between x = 1 and x = 2 using 2 subintervals of equal length. What is the largest value the approximation could have?

Use the Midpoint Rule with 2 equal subdivisions to get an approximation for ln 5.

Use (a) the Trapezoidal Rule with n = 8 and (b) Simpson's Rule with n = 8 to approximate ∫02exdx\int _ { 0 } ^ { 2 } e ^ { x } d x∫02​exdx Round your answers to six decimal places.

Let f(x)=∫x234−t33dt. Find f′(x)f ( x ) = \int _ { x ^ { 2 } } ^ { 3 } \sqrt [ 3 ] { 4 - t ^ { 3 } } d t . \text { Find } f ^ { \prime } ( x )f(x)=∫x23​34−t3​dt. Find f′(x)

What is the largest value of ∫03∣f(x)∣dx\int _ { 0 } ^ { 3 } | f ( x ) | d x∫03​∣f(x)∣dx if −4≤f(x)≤2- 4 \leq f ( x ) \leq 2−4≤f(x)≤2 .

Evaluate ∫3x+2x−1dx\int \frac { 3 x + 2 } { x - 1 } d x∫x−13x+2​dx

Functions and Models

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Find the partial fraction expansion of the rational function: $\frac { 9 } { ( x + 1 ) ^ { 2 } ( 2 - x ) }$ .

Evaluate $\int _ { - 1 } ^ { 1 } \left( 2 - \sqrt { 1 - x ^ { 2 } } \right) d x$ by interpreting the integral in terms of area.

Find the value of the integral $\int _ { - 3 } ^ { 3 } \frac { x } { \sqrt { 1 + 3 x ^ { 2 } } } d x$

Prove that $\frac { - x ^ { 3 } } { 3 } \cos x$ is not an antiderivative of $x ^ { 2 } \sin x$ .

Evaluate the improper integral $\int _ { 0 } ^ { \infty } e ^ { - 3 x } d x$

A particle travels along a line. Its velocity in meters per second is given by $v ( t ) = 3 t ^ { 2 } - 12 t$ Find (a) the displacement from t = 0 to t = 8.(b) the distance traveled by the particle from t = 0 to t = 8.

Use (a) the Trapezoidal Rule with n = 8 and (b) Simpson's Rule with n = 8 to approximate $\int _ { 0 } ^ { 2 } e ^ { x } d x$ Round your answers to six decimal places.

Let $f ( x ) = \int _ { x ^ { 2 } } ^ { 3 } \sqrt [ 3 ] { 4 - t ^ { 3 } } d t . \text { Find } f ^ { \prime } ( x )$

What is the largest value of $\int _ { 0 } ^ { 3 } | f ( x ) | d x$ if $- 4 \leq f ( x ) \leq 2$ .

Evaluate $\int \frac { 3 x + 2 } { x - 1 } d x$