Question 1

Find the average of the function $f ( x ) = 9 x ^ { 2 } - 8 x$ over $[ 0,8 ]$ .

Accepted Answer

The answer of Find the average of the function \(f...

Question 2

Evaluate the integral.   $\int _ { 1 } ^ { 2 } x \ln ( 5 x ) \mathrm { d } x$

Accepted Answer

A)  $\frac { 3 \ln 5 } { 2 } - \frac { 3 } { 4 }$ 
B)  $2 \ln 10 - \frac { \ln 5 } { 2 } - \frac { 3 } { 4 }$ 
C)    $2 \ln 10 - \frac { 3 \ln 5 } { 2 } - \frac { 15 } { 4 }$ 
D)  $- \ln 100 + \frac { \ln 5 - 1 } { 2 }$ 
E)  $\ln 100 + \frac { \ln 5 + 1 } { 2 }$ 
A)  $\frac { 3 \ln 5 } { 2 } - \frac { 3 } { 4 }$ 
B)  $2 \ln 10 - \frac { \ln 5 } { 2 } - \frac { 3 } { 4 }$ 
C)    $2 \ln 10 - \frac { 3 \ln 5 } { 2 } - \frac { 15 } { 4 }$ 
D)  $- \ln 100 + \frac { \ln 5 - 1 } { 2 }$ 
E)  $\ln 100 + \frac { \ln 5 + 1 } { 2 }$

Question 3

Find the average of the function $f ( x ) = 9 x ^ { 2 } - 6 x$ over $[ 0,2 ]$ .

Accepted Answer

A) Average = 5 
B)  Average = 6 
C)  Average = 12 
D)  Average = 10 
E)  Average = 7 
A) Average = 5 
B)  Average = 6 
C)  Average = 12 
D)  Average = 10 
E)  Average = 7

Question 4

Find the general solution of the differential equation.   $\frac { \mathrm { d } y } { \mathrm {~d} x } = ( x + 1 ) y ^ { 2 }$  
Solve for y as a function of x.

Accepted Answer

A)  $y = - ( x + 1 ) ^ { 2 } + C$ 
B)    $y = - \frac { 2 } { x ^ { 2 } + C }$ 
C)    $y = - \frac { 2 } { x ( x + 2 ) + C }$ 
D)    $y = \frac { C } { ( x + 1 ) ^ { 2 } }$ 
E)    $y = - \frac { 2 } { x + C }$ 
A)  $y = - ( x + 1 ) ^ { 2 } + C$ 
B)    $y = - \frac { 2 } { x ^ { 2 } + C }$ 
C)    $y = - \frac { 2 } { x ( x + 2 ) + C }$ 
D)    $y = \frac { C } { ( x + 1 ) ^ { 2 } }$ 
E)    $y = - \frac { 2 } { x + C }$

Question 5

Evaluate the integral.   $\int _ { 0 } ^ { 1 } ( x + 3 ) e ^ { x } \mathrm {~d} x$

Accepted Answer

A)  $3 e - 2$ 
B)  $4 e - 1$ 
C)  $3 e + 4$ 
D)  $3 e + 3$ 
E)  $3 e + 2$ 
A)  $3 e - 2$ 
B)  $4 e - 1$ 
C)  $3 e + 4$ 
D)  $3 e + 3$ 
E)  $3 e + 2$

Question 6

Calculate the 5-unit moving average of the given function.   $$f ( x ) = 8 x ^ { 3 }$$

Accepted Answer

A)  $\bar { f } ( x ) = 8 x ^ { 3 } - 60 x ^ { 2 } + 200 x + 250$ 
B)  $\bar { f } ( x ) = 8 x ^ { 3 } + 60 x ^ { 2 } + 200 x - 250$ 
C)  $\bar { f } ( x ) = 8 x ^ { 3 } - 60 x ^ { 2 } + 200 x - 250$ 
D)  $\bar { f } ( x ) = 8 x ^ { 3 } - 60 x ^ { 2 } - 250 x + 200$ 
E)  $\bar { f } ( x ) = 8 x ^ { 3 } - 60 x ^ { 2 } + 200 x$ 
A)  $\bar { f } ( x ) = 8 x ^ { 3 } - 60 x ^ { 2 } + 200 x + 250$ 
B)  $\bar { f } ( x ) = 8 x ^ { 3 } + 60 x ^ { 2 } + 200 x - 250$ 
C)  $\bar { f } ( x ) = 8 x ^ { 3 } - 60 x ^ { 2 } + 200 x - 250$ 
D)  $\bar { f } ( x ) = 8 x ^ { 3 } - 60 x ^ { 2 } - 250 x + 200$ 
E)  $\bar { f } ( x ) = 8 x ^ { 3 } - 60 x ^ { 2 } + 200 x$

Question 7

Evaluate the integral.
​ $\int w ^ { - \frac { 1 } { 7 } } \ln w d w$

Accepted Answer

The answer of Evaluate the integral.
​ \(\int w ^ {...

Question 8

Decide whether the integral converges. If the integral converges, compute its value.   $\int _ { 1 } ^ { + \infty } \frac { 1 } { x ^ { 1.5 } } \mathrm {~d} x$

Accepted Answer

A) 2 
B)  1 
C)  $\frac { 1 } { 2 }$ 
D)  -2 
E)  diverges 
A) 2 
B)  1 
C)  $\frac { 1 } { 2 }$ 
D)  -2 
E)  diverges

Question 9

The weekly demand for your company's Lo-Cal Mousse is modeled by the equation $q ( t ) = \frac { 18 e ^ { 2 t - 2 } } { 1 + e ^ { 2 t - 2 } }$ where t is time from now in weeks and q(t) is the number of gallons sold per week. Evaluate the integral $\int _ { - \infty } ^ { 0 } \frac { 18 e ^ { 2 t - 2 } } { 1 + e ^ { 2 t - 2 } } \mathrm {~d} t$ .

Accepted Answer

A) 0.254 gallons 
B)  5.639 gallons 
C)  2.285 gallons 
D)  1.142 gallons 
E)  2.819 gallons 
A) 0.254 gallons 
B)  5.639 gallons 
C)  2.285 gallons 
D)  1.142 gallons 
E)  2.819 gallons

Question 10

The Laplace transform F(x) of a function f(t) is given by the formula.   $F ( x ) = \int _ { 0 } ^ { + \infty } f ( t ) e ^ { - x t } \mathrm {~d} t$  
Find F(x) if $f ( t ) = e ^ { n t }$ (n constant).

Accepted Answer

A)  $\frac { 1 } { x }$ 
B)  $\frac { 1 } { x - n }$ 
C)  $\frac { n } { x }$ 
D)  $\frac { n } { x - \mathrm { n } }$ 
E)  $n x - n$ 
A)  $\frac { 1 } { x }$ 
B)  $\frac { 1 } { x - n }$ 
C)  $\frac { n } { x }$ 
D)  $\frac { n } { x - \mathrm { n } }$ 
E)  $n x - n$

Question 11

For the differential equation, find the particular solution $y = 0$ when $x = 0$ .   $\frac { \mathrm { d } y } { \mathrm {~d} x } = x ^ { 3 } - 14 x$

Accepted Answer

A)  $y = x ^ { 4 } - 14 x ^ { 2 }$ 
B)    $y = \frac { x ^ { 4 } } { 4 } - 7 x ^ { 2 }$ 
C)    $y = \frac { x ^ { 4 } } { 4 } + 7 x ^ { 2 }$ 
D)    $y = 4 x ^ { 4 } + 7 x ^ { 2 }$ 
E)    $y = \frac { x ^ { 4 } } { 2 } - 7 x$ 
A)  $y = x ^ { 4 } - 14 x ^ { 2 }$ 
B)    $y = \frac { x ^ { 4 } } { 4 } - 7 x ^ { 2 }$ 
C)    $y = \frac { x ^ { 4 } } { 4 } + 7 x ^ { 2 }$ 
D)    $y = 4 x ^ { 4 } + 7 x ^ { 2 }$ 
E)    $y = \frac { x ^ { 4 } } { 2 } - 7 x$

Question 12

Find the area of the indicated region.
​
Enclosed by $y = 2 \ln x$ and $y = x - 2$ . (Round answer to four significant digits.) [First use technology to determine approximately where the graphs cross.]

Accepted Answer

The answer of Find the area of the indicated region.
​
Enclosed...

Question 13

Find the total value of the given income stream and also find its present value (at the beginning of the given interval) using the given interest rate.
​ $R ( t ) = 10,000$ , $0 \leq t \leq 5$ , at 5%
​
Please enter your answer in the form TV = , PV = . Give the answer to the nearest cent if necessary.
​
TV = $__________ PV = $__________

Accepted Answer

The answer of Find the total value of the given...

Question 14

Evaluate the integral.   $\int \frac { x } { ( x - 3 ) ^ { 2 } } \mathrm {~d} x$

Accepted Answer

A)  $- \frac { x \ln | x - 3 | } { x - 3 } + C$ 
B)  $\frac { x } { x - 3 } - \ln | x - 3 | + C$ 
C)  $- \frac { x } { x - 3 } + \ln | x - 3 | + C$ 
D)  $- \frac { x + \ln | x - 3 | } { x - 3 } + C$ 
E)  $- \frac { x } { x - 3 } + \ln | x | + C$ 
A)  $- \frac { x \ln | x - 3 | } { x - 3 } + C$ 
B)  $\frac { x } { x - 3 } - \ln | x - 3 | + C$ 
C)  $- \frac { x } { x - 3 } + \ln | x - 3 | + C$ 
D)  $- \frac { x + \ln | x - 3 | } { x - 3 } + C$ 
E)  $- \frac { x } { x - 3 } + \ln | x | + C$

Question 15

Find the area of the region between $y = e ^ { x }$ and $y = \frac { 1 } { 2 } x$ for x in $[ 0,8 ]$ .

Accepted Answer

A)  $e ^ { 8 } - 18$ 
B)  $- e ^ { 8 } + 17$ 
C)  $- e ^ { 8 } + 18$ 
D)  $e ^ { 8 } - 17$ 
E)  $- e ^ { 8 } - 17$ 
A)  $e ^ { 8 } - 18$ 
B)  $- e ^ { 8 } + 17$ 
C)  $- e ^ { 8 } + 18$ 
D)  $e ^ { 8 } - 17$ 
E)  $- e ^ { 8 } - 17$

Question 16

Decide whether the integral converges. If the integral converges, compute its value.   $\int _ { 4 } ^ { + \infty } x \mathrm {~d} x$

Accepted Answer

A) converges to $\frac { 1 } { 4 }$ 
B)  converges to $\frac { 1 } { 8 }$ 
C)  converges to 8 
D)  converges to 0 
E)  diverges 
A) converges to $\frac { 1 } { 4 }$ 
B)  converges to $\frac { 1 } { 8 }$ 
C)  converges to 8 
D)  converges to 0 
E)  diverges

Question 17

Find the total value of the given income stream and also find its present value (at the beginning of the given interval) using the given interest rate.   $R ( t ) = 60,000 + 4,000 t$ , $0 \leq t \leq 5$ , at 11%

Accepted Answer

A)  $T V = \$ 240,025,000$ , $P V = \$ 269,133.35$ 
B)  $T V = \$ 250,000$ , $P V = \$ 2,079,283.09$ 
C)  $T V = \$ 354,000$ , $P V = \$ 265,705.99$ 
D)  $T V = \$ 80,000$ , $P V = \$ 195,803.31$ 
E)  $T V = \$ 350,000$ , $P V = \$ 265,705.99$ 
A)  $T V = \$ 240,025,000$ , $P V = \$ 269,133.35$ 
B)  $T V = \$ 250,000$ , $P V = \$ 2,079,283.09$ 
C)  $T V = \$ 354,000$ , $P V = \$ 265,705.99$ 
D)  $T V = \$ 80,000$ , $P V = \$ 195,803.31$ 
E)  $T V = \$ 350,000$ , $P V = \$ 265,705.99$

Question 18

Find the area of the region enclosed by $y = 16 x$ and $y = 2 x ^ { 4 }$ .

Accepted Answer

A)  $\frac { 101 } { 5 }$ 
B)  $\frac { 1 } { 10 }$ 
C)  $\frac { 96 } { 5 }$ 
D)  $- \frac { 96 } { 5 }$ 
E)  1 
A)  $\frac { 101 } { 5 }$ 
B)  $\frac { 1 } { 10 }$ 
C)  $\frac { 96 } { 5 }$ 
D)  $- \frac { 96 } { 5 }$ 
E)  1

Question 19

Find the area of the region between $y = e ^ { - x }$ and $y = - 4 x$ for x in $[ 0,3 ]$ .

Accepted Answer

A)  $18 - \frac { 1 } { e ^ { 3 } }$ 
B)  $19 - e ^ { 3 }$ 
C)    $18 + \frac { 1 } { e ^ { 3 } }$ 
D)  $19 - \frac { 1 } { e ^ { 3 } }$ 
E)  $18 + e ^ { 3 }$ 
A)  $18 - \frac { 1 } { e ^ { 3 } }$ 
B)  $19 - e ^ { 3 }$ 
C)    $18 + \frac { 1 } { e ^ { 3 } }$ 
D)  $19 - \frac { 1 } { e ^ { 3 } }$ 
E)  $18 + e ^ { 3 }$

Question 20

Find the areas of the indicated regions.

Choose the correct letter for each question
-256

Accepted Answer

A) between $y = 2 x ^ { 3 }$ and $y = 0$ for x in $[ - 4,4 ]$ 
B) between $y = x ^ { 2 }$ and $y = - 5$ for x in $[ - 4,4 ]$ 
C) between $y = - x$ and $y = x$ for x in $[ 0,5 ]$ 
A) between $y = 2 x ^ { 3 }$ and $y = 0$ for x in $[ - 4,4 ]$ 
B) between $y = x ^ { 2 }$ and $y = - 5$ for x in $[ - 4,4 ]$ 
C) between $y = - x$ and $y = x$ for x in $[ 0,5 ]$

Find the average of the function $f ( x ) = 9 x ^ { 2 } - 8 x$ over $[ 0,8 ]$ .

Evaluate the integral. $\int _ { 1 } ^ { 2 } x \ln ( 5 x ) \mathrm { d } x$

Find the average of the function $f ( x ) = 9 x ^ { 2 } - 6 x$ over $[ 0,2 ]$ .

Find the general solution of the differential equation. $\frac { \mathrm { d } y } { \mathrm {~d} x } = ( x + 1 ) y ^ { 2 }$ Solve for y as a function of x.

Evaluate the integral. $\int _ { 0 } ^ { 1 } ( x + 3 ) e ^ { x } \mathrm {~d} x$

Calculate the 5-unit moving average of the given function. $f ( x ) = 8 x ^ { 3 }$

Evaluate the integral. $\int w ^ { - \frac { 1 } { 7 } } \ln w d w$

Decide whether the integral converges. If the integral converges, compute its value. $\int _ { 1 } ^ { + \infty } \frac { 1 } { x ^ { 1.5 } } \mathrm {~d} x$

The Laplace transform F(x) of a function f(t) is given by the formula. $F ( x ) = \int _ { 0 } ^ { + \infty } f ( t ) e ^ { - x t } \mathrm {~d} t$ Find F(x) if $f ( t ) = e ^ { n t }$ (n constant).

For the differential equation, find the particular solution $y = 0$ when $x = 0$ . $\frac { \mathrm { d } y } { \mathrm {~d} x } = x ^ { 3 } - 14 x$

Find the area of the indicated region. Enclosed by $y = 2 \ln x$ and $y = x - 2$ . (Round answer to four significant digits.) [First use technology to determine approximately where the graphs cross.]

Evaluate the integral. $\int \frac { x } { ( x - 3 ) ^ { 2 } } \mathrm {~d} x$

Find the area of the region between $y = e ^ { x }$ and $y = \frac { 1 } { 2 } x$ for x in $[ 0,8 ]$ .

Decide whether the integral converges. If the integral converges, compute its value. $\int _ { 4 } ^ { + \infty } x \mathrm {~d} x$

Find the total value of the given income stream and also find its present value (at the beginning of the given interval) using the given interest rate. $R ( t ) = 60,000 + 4,000 t$ , $0 \leq t \leq 5$ , at 11%

Find the area of the region enclosed by $y = 16 x$ and $y = 2 x ^ { 4 }$ .

Find the area of the region between $y = e ^ { - x }$ and $y = - 4 x$ for x in $[ 0,3 ]$ .

Find the areas of the indicated regions. Choose the correct letter for each question -256

Functions and Applications

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Exam 14: Further Integration Techniques and Applications of the Integral

Find the average of the function f(x)=9x2−8xf ( x ) = 9 x ^ { 2 } - 8 xf(x)=9x2−8x over [0,8][ 0,8 ][0,8] .

Evaluate the integral. ∫12xln⁡(5x)dx\int _ { 1 } ^ { 2 } x \ln ( 5 x ) \mathrm { d } x∫12​xln(5x)dx

Find the average of the function f(x)=9x2−6xf ( x ) = 9 x ^ { 2 } - 6 xf(x)=9x2−6x over [0,2][ 0,2 ][0,2] .

Find the general solution of the differential equation. dy dx=(x+1)y2\frac { \mathrm { d } y } { \mathrm {~d} x } = ( x + 1 ) y ^ { 2 } dxdy​=(x+1)y2 Solve for y as a function of x.

Evaluate the integral. ∫01(x+3)ex dx\int _ { 0 } ^ { 1 } ( x + 3 ) e ^ { x } \mathrm {~d} x∫01​(x+3)ex dx

Calculate the 5-unit moving average of the given function. f(x)=8x3f ( x ) = 8 x ^ { 3 }f(x)=8x3

Evaluate the integral. ​ ∫w−17ln⁡wdw\int w ^ { - \frac { 1 } { 7 } } \ln w d w∫w−71​lnwdw

Decide whether the integral converges. If the integral converges, compute its value. ∫1+∞1x1.5 dx\int _ { 1 } ^ { + \infty } \frac { 1 } { x ^ { 1.5 } } \mathrm {~d} x∫1+∞​x1.51​ dx

The Laplace transform F(x) of a function f(t) is given by the formula. F(x)=∫0+∞f(t)e−xt dtF ( x ) = \int _ { 0 } ^ { + \infty } f ( t ) e ^ { - x t } \mathrm {~d} tF(x)=∫0+∞​f(t)e−xt dt Find F(x) if f(t)=entf ( t ) = e ^ { n t }f(t)=ent (n constant).

For the differential equation, find the particular solution y=0y = 0y=0 when x=0x = 0x=0 . dy dx=x3−14x\frac { \mathrm { d } y } { \mathrm {~d} x } = x ^ { 3 } - 14 x dxdy​=x3−14x

Find the area of the indicated region. ​ Enclosed by y=2ln⁡xy = 2 \ln xy=2lnx and y=x−2y = x - 2y=x−2 . (Round answer to four significant digits.) [First use technology to determine approximately where the graphs cross.]

Evaluate the integral. ∫x(x−3)2 dx\int \frac { x } { ( x - 3 ) ^ { 2 } } \mathrm {~d} x∫(x−3)2x​ dx

Find the area of the region between y=exy = e ^ { x }y=ex and y=12xy = \frac { 1 } { 2 } xy=21​x for x in [0,8][ 0,8 ][0,8] .

Decide whether the integral converges. If the integral converges, compute its value. ∫4+∞x dx\int _ { 4 } ^ { + \infty } x \mathrm {~d} x∫4+∞​x dx

Find the total value of the given income stream and also find its present value (at the beginning of the given interval) using the given interest rate. R(t)=60,000+4,000tR ( t ) = 60,000 + 4,000 tR(t)=60,000+4,000t , 0≤t≤50 \leq t \leq 50≤t≤5 , at 11%

Find the area of the region enclosed by y=16xy = 16 xy=16x and y=2x4y = 2 x ^ { 4 }y=2x4 .

Find the area of the region between y=e−xy = e ^ { - x }y=e−x and y=−4xy = - 4 xy=−4x for x in [0,3][ 0,3 ][0,3] .

Find the areas of the indicated regions. Choose the correct letter for each question -256

Functions and Applications

Nonlinear Functions and Models

The Mathematics of Finance

Systems of Linear Equations and Matrices

Matrix Algebra and Applications

Linear Programming

Sets and Counting

Probability

Random Variables and Statistics

Introduction to the Derivative

Techniques of Differentiation

Applications of the Derivative

The Integral

Functions of Several Variables

Trigonometric Models

Filters

Find the average of the function $f ( x ) = 9 x ^ { 2 } - 8 x$ over $[ 0,8 ]$ .

Evaluate the integral. $\int _ { 1 } ^ { 2 } x \ln ( 5 x ) \mathrm { d } x$

Find the average of the function $f ( x ) = 9 x ^ { 2 } - 6 x$ over $[ 0,2 ]$ .

Find the general solution of the differential equation. $\frac { \mathrm { d } y } { \mathrm {~d} x } = ( x + 1 ) y ^ { 2 }$ Solve for y as a function of x.

Evaluate the integral. $\int _ { 0 } ^ { 1 } ( x + 3 ) e ^ { x } \mathrm {~d} x$

Calculate the 5-unit moving average of the given function. $f ( x ) = 8 x ^ { 3 }$

Evaluate the integral. $\int w ^ { - \frac { 1 } { 7 } } \ln w d w$

Decide whether the integral converges. If the integral converges, compute its value. $\int _ { 1 } ^ { + \infty } \frac { 1 } { x ^ { 1.5 } } \mathrm {~d} x$

The Laplace transform F(x) of a function f(t) is given by the formula. $F ( x ) = \int _ { 0 } ^ { + \infty } f ( t ) e ^ { - x t } \mathrm {~d} t$ Find F(x) if $f ( t ) = e ^ { n t }$ (n constant).

For the differential equation, find the particular solution $y = 0$ when $x = 0$ . $\frac { \mathrm { d } y } { \mathrm {~d} x } = x ^ { 3 } - 14 x$

Find the area of the indicated region. Enclosed by $y = 2 \ln x$ and $y = x - 2$ . (Round answer to four significant digits.) [First use technology to determine approximately where the graphs cross.]

Evaluate the integral. $\int \frac { x } { ( x - 3 ) ^ { 2 } } \mathrm {~d} x$

Find the area of the region between $y = e ^ { x }$ and $y = \frac { 1 } { 2 } x$ for x in $[ 0,8 ]$ .

Decide whether the integral converges. If the integral converges, compute its value. $\int _ { 4 } ^ { + \infty } x \mathrm {~d} x$

Find the total value of the given income stream and also find its present value (at the beginning of the given interval) using the given interest rate. $R ( t ) = 60,000 + 4,000 t$ , $0 \leq t \leq 5$ , at 11%

Find the area of the region enclosed by $y = 16 x$ and $y = 2 x ^ { 4 }$ .

Find the area of the region between $y = e ^ { - x }$ and $y = - 4 x$ for x in $[ 0,3 ]$ .