Question 1

Simplify the expression. Select the correct answer.
$e ^ { 3 \ln 2 }$

Accepted Answer

A)  9 
B)  5 
C)  6 
D)  8 
A)  9 
B)  5 
C)  6 
D)  8 D

Question 2

Suppose $g$ is the inverse function of a differentiable function $f$ and $G ( x ) = \frac { 1 } { g ( x ) }$.
If $f ( 4 ) = 3$ and $f ^ { t } ( 4 ) = \frac { 1 } { 16 }$, find $G ^ {t} ( 3 )$.

Accepted Answer

$$- 1$$

Question 3

Find $g ^ { t } ( x )$
$$g ( x ) = \int _ { 3 } ^ { \sqrt { x } } \frac { e ^ { s } } { s } d s$$

Accepted Answer

$$g ^ { t } ( x ) = \frac { e ^ { \sqrt { x } } } { 2 x }$$

Question 4

$$\text { If } f \text { is a one-to-one function such that } f ( 7 ) = 1 \text {, what is } f ^ { - 1 } ( 1 ) \text { ? }$$ Select the correct answer.

Accepted Answer

A)  12 
B)  1 
C)  7 
D)  11 
E)  5
F) 3 
A)  12 
B)  1 
C)  7 
D)  11 
E)  5
F) 3

Question 5

Differentiate the function.
$$f ( x ) = \cos ( \ln ( 6 x ) )$$

Accepted Answer

The answer of Differentiate the function.
\[f ( x ) =...

Question 6

Use logarithmic differentiation to find the derivative of the function.
$$y = ( x + 2 ) ^ { 7 / x }$$

Accepted Answer

The answer of Use logarithmic differentiation to find the derivative...

Question 7

Find the volume of the solid obtained by rotating the region bounded by the given curves about the specified line.
$y = \ln x , y = 1 , y = 5 , x = 0$; about the $y$ - axis

Accepted Answer

A)  $\frac { \pi } { 4 } \left( e ^ { 10 } - e \right)$ 
B)  $\frac { \pi } { 2 } \left( 1 - e ^ { 10 } \right)$ 
C)  $\frac { \pi } { 2 } \left( e ^ { 10 } - e ^ { 2 } \right)$ 
D)  $\frac { \pi e ^ { 10 } } { 2 }$ 
E)  $2 \pi \left( e ^ { 10 } - e ^ { 2 } \right)$ 
A)  $\frac { \pi } { 4 } \left( e ^ { 10 } - e \right)$ 
B)  $\frac { \pi } { 2 } \left( 1 - e ^ { 10 } \right)$ 
C)  $\frac { \pi } { 2 } \left( e ^ { 10 } - e ^ { 2 } \right)$ 
D)  $\frac { \pi e ^ { 10 } } { 2 }$ 
E)  $2 \pi \left( e ^ { 10 } - e ^ { 2 } \right)$

Question 8

Find the integral.
$$\int \sqrt { 1 + 8 e ^ { x } } e ^ { x } d x$$

Accepted Answer

The answer of Find the integral.
\[\int \sqrt { 1 +...

Question 9

Solve the inequality.
$$\ln \left( x ^ { 2 } - 6 x - 6 \right) \leq 0$$

Accepted Answer

The answer of Solve the inequality.
\[\ln \left( x ^ {...

Question 10

Evaluate the integral.
$$\int 5 ^ { \sin \theta } \cos \theta d \theta$$

Accepted Answer

The answer of Evaluate the integral.
\[\int 5 ^ { \sin...

Question 11

Find the limit.
$$\lim _ { x \rightarrow 0 ^ { + } } \frac { \ln 5 x } { x }$$

Accepted Answer

A)  $\infty$ 
B)  $\pi$ 
C)  0 
D)  5 
E)  $- \infty$ 
A)  $\infty$ 
B)  $\pi$ 
C)  0 
D)  5 
E)  $- \infty$

Question 12

Calculate $g ( x )$, where $g = f ^ { - 1 }$. State the domain and range of $g$. Calculate $g ^ { \prime } ( a )$.
$$f ( x ) = \frac { 1 } { x - 3 } , x > 3 ; a = 2$$

Accepted Answer

The answer of Calculate $g ( x )$, where \(g...

Question 13

Differentiate the function.
$$y = \ln \left( x ^ { 5 } \sin ^ { 2 } x \right)$$

Accepted Answer

The answer of Differentiate the function.
\[y = \ln \left( x...

Question 14

Find the limit. Select the correct answer.
$$\lim _ { x \rightarrow \infty } e ^ { 4 - x ^ { 4 } }$$

Accepted Answer

A)  $\infty$ 
B)  $- \infty$ 
C)  $e ^ { 4 }$ 
D)  1 
E)  0 
A)  $\infty$ 
B)  $- \infty$ 
C)  $e ^ { 4 }$ 
D)  1 
E)  0

Question 15

Find the absolute extrema of the function on the indicated interval.
$$f ( x ) = x e ^ { - 3 x } ; \quad [ - 1,2 ]$$

Accepted Answer

Absolute m

Question 16

A lighthouse is located on a small island, $3 \mathrm {~km}$ away from the nearest point $P$ on a straight shoreline, and its light makes four revolutions per minute. How fast is the beam of light moving along the shoreline when it is $1 \mathrm {~km}$ from $P$ ?

Accepted Answer

A)  $1900 \pi \mathrm { km } / \mathrm { h }$ 
B)  $2000 \pi \mathrm { km } / \mathrm { h }$ 
C)  $1800 \pi \mathrm { km } / \mathrm { h }$ 
D)  $1700 \pi \mathrm { km } / \mathrm { h }$ 
E)  $1600 \pi \mathrm { km } / \mathrm { h }$ 
A)  $1900 \pi \mathrm { km } / \mathrm { h }$ 
B)  $2000 \pi \mathrm { km } / \mathrm { h }$ 
C)  $1800 \pi \mathrm { km } / \mathrm { h }$ 
D)  $1700 \pi \mathrm { km } / \mathrm { h }$ 
E)  $1600 \pi \mathrm { km } / \mathrm { h }$

Question 17

Use the properties of logarithms to expand the quantity.
$$11 \ln \sqrt { a \left( b ^ { 2 } + c ^ { 2 } \right) }$$

Accepted Answer

The answer of Use the properties of logarithms to expand...

Question 18

Find $f ^ { - 1 } ( a )$ for the function $f$ and the real number $a$. $f ( x ) = x ^ { 3 } + x - 5 ; a = - 5$

Accepted Answer

A)  $- 1$ 
B)  2 
C)  1 
D)  0 
A)  $- 1$ 
B)  2 
C)  1 
D)  0

Question 19

Find the derivative of the function.
$$y = 3 \sin ^ { - 1 } \left( x ^ { 2 } \right)$$

Accepted Answer

A) 
$y ^ { t } = \frac { 6 x } { \sqrt { 1 + x ^ { 4 } } }$ 
B) 
$y ^ { t } = \frac { x } { \sqrt { 1 - x ^ { 2 } } }$ 
C) 
$y ^ { t } = \frac { 6 x } { \sqrt { 1 - x ^ { 4 } } }$ 
D) 
$y ^ {t } = \frac { 6 x } { \sqrt { 1 - x ^ { 2 } } }$ 
E)  None of these 
A) 
$y ^ { t } = \frac { 6 x } { \sqrt { 1 + x ^ { 4 } } }$ 
B) 
$y ^ { t } = \frac { x } { \sqrt { 1 - x ^ { 2 } } }$ 
C) 
$y ^ { t } = \frac { 6 x } { \sqrt { 1 - x ^ { 4 } } }$ 
D) 
$y ^ {t } = \frac { 6 x } { \sqrt { 1 - x ^ { 2 } } }$ 
E)  None of these

Question 20

Evaluate the expression.
$\log _ { 125 } 25$

Accepted Answer

A)  $\frac { 2 } { 3 }$ 
B)  $\frac { 3 } { 2 }$ 
C)  2 
D)  $\frac { 1 } { 2 }$ 
A)  $\frac { 2 } { 3 }$ 
B)  $\frac { 3 } { 2 }$ 
C)  2 
D)  $\frac { 1 } { 2 }$

Simplify the expression. Select the correct answer. $e ^ { 3 \ln 2 }$

Suppose $g$ is the inverse function of a differentiable function $f$ and $G ( x ) = \frac { 1 } { g ( x ) }$ . If $f ( 4 ) = 3$ and $f ^ { t } ( 4 ) = \frac { 1 } { 16 }$ , find $G ^ {t} ( 3 )$ .

Find $g ^ { t } ( x )$ $g ( x ) = \int _ { 3 } ^ { \sqrt { x } } \frac { e ^ { s } } { s } d s$

$\text { If } f \text { is a one-to-one function such that } f ( 7 ) = 1 \text {, what is } f ^ { - 1 } ( 1 ) \text { ? }$ Select the correct answer.

Differentiate the function. $f ( x ) = \cos ( \ln ( 6 x ) )$

Use logarithmic differentiation to find the derivative of the function. $y = ( x + 2 ) ^ { 7 / x }$

Find the volume of the solid obtained by rotating the region bounded by the given curves about the specified line. $y = \ln x , y = 1 , y = 5 , x = 0$ ; about the $y$ - axis

Find the integral. $\int \sqrt { 1 + 8 e ^ { x } } e ^ { x } d x$

Solve the inequality. $\ln \left( x ^ { 2 } - 6 x - 6 \right) \leq 0$

Evaluate the integral. $\int 5 ^ { \sin \theta } \cos \theta d \theta$

Find the limit. $\lim _ { x \rightarrow 0 ^ { + } } \frac { \ln 5 x } { x }$

Calculate $g ( x )$ , where $g = f ^ { - 1 }$ . State the domain and range of $g$ . Calculate $g ^ { \prime } ( a )$ . $f ( x ) = \frac { 1 } { x - 3 } , x > 3 ; a = 2$

Differentiate the function. $y = \ln \left( x ^ { 5 } \sin ^ { 2 } x \right)$

Find the limit. Select the correct answer. $\lim _ { x \rightarrow \infty } e ^ { 4 - x ^ { 4 } }$

Find the absolute extrema of the function on the indicated interval. $f ( x ) = x e ^ { - 3 x } ; \quad [ - 1,2 ]$

A lighthouse is located on a small island, $3 \mathrm {~km}$ away from the nearest point $P$ on a straight shoreline, and its light makes four revolutions per minute. How fast is the beam of light moving along the shoreline when it is $1 \mathrm {~km}$ from $P$ ?

Use the properties of logarithms to expand the quantity. $11 \ln \sqrt { a \left( b ^ { 2 } + c ^ { 2 } \right) }$

Find $f ^ { - 1 } ( a )$ for the function $f$ and the real number $a$ . $f ( x ) = x ^ { 3 } + x - 5 ; a = - 5$

Find the derivative of the function. $y = 3 \sin ^ { - 1 } \left( x ^ { 2 } \right)$

Evaluate the expression. $\log _ { 125 } 25$

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Further Applications of Integration

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Exam 6: Applications of Integration

Simplify the expression. Select the correct answer. e3ln⁡2e ^ { 3 \ln 2 }e3ln2

Suppose ggg is the inverse function of a differentiable function fff and G(x)=1g(x)G ( x ) = \frac { 1 } { g ( x ) }G(x)=g(x)1​ . If f(4)=3f ( 4 ) = 3f(4)=3 and ft(4)=116f ^ { t } ( 4 ) = \frac { 1 } { 16 }ft(4)=161​ , find Gt(3)G ^ {t} ( 3 )Gt(3) .

Find gt(x)g ^ { t } ( x )gt(x) g(x)=∫3xessdsg ( x ) = \int _ { 3 } ^ { \sqrt { x } } \frac { e ^ { s } } { s } d sg(x)=∫3x​​ses​ds

If f is a one-to-one function such that f(7)=1, what is f−1(1) ? \text { If } f \text { is a one-to-one function such that } f ( 7 ) = 1 \text {, what is } f ^ { - 1 } ( 1 ) \text { ? } If f is a one-to-one function such that f(7)=1, what is f−1(1) ? Select the correct answer.

Differentiate the function. f(x)=cos⁡(ln⁡(6x))f ( x ) = \cos ( \ln ( 6 x ) )f(x)=cos(ln(6x))

Use logarithmic differentiation to find the derivative of the function. y=(x+2)7/xy = ( x + 2 ) ^ { 7 / x }y=(x+2)7/x

Find the volume of the solid obtained by rotating the region bounded by the given curves about the specified line. y=ln⁡x,y=1,y=5,x=0y = \ln x , y = 1 , y = 5 , x = 0y=lnx,y=1,y=5,x=0 ; about the yyy - axis

Find the integral. ∫1+8exexdx\int \sqrt { 1 + 8 e ^ { x } } e ^ { x } d x∫1+8ex​exdx

Solve the inequality. ln⁡(x2−6x−6)≤0\ln \left( x ^ { 2 } - 6 x - 6 \right) \leq 0ln(x2−6x−6)≤0

Evaluate the integral. ∫5sin⁡θcos⁡θdθ\int 5 ^ { \sin \theta } \cos \theta d \theta∫5sinθcosθdθ

Find the limit. lim⁡x→0+ln⁡5xx\lim _ { x \rightarrow 0 ^ { + } } \frac { \ln 5 x } { x }limx→0+​xln5x​

Calculate g(x)g ( x )g(x) , where g=f−1g = f ^ { - 1 }g=f−1 . State the domain and range of ggg . Calculate g′(a)g ^ { \prime } ( a )g′(a) . f(x)=1x−3,x>3;a=2f ( x ) = \frac { 1 } { x - 3 } , x > 3 ; a = 2f(x)=x−31​,x>3;a=2

Differentiate the function. y=ln⁡(x5sin⁡2x)y = \ln \left( x ^ { 5 } \sin ^ { 2 } x \right)y=ln(x5sin2x)

Find the limit. Select the correct answer. lim⁡x→∞e4−x4\lim _ { x \rightarrow \infty } e ^ { 4 - x ^ { 4 } }limx→∞​e4−x4

Find the absolute extrema of the function on the indicated interval. f(x)=xe−3x;[−1,2]f ( x ) = x e ^ { - 3 x } ; \quad [ - 1,2 ]f(x)=xe−3x;[−1,2]

A lighthouse is located on a small island, 3 km3 \mathrm {~km}3 km away from the nearest point PPP on a straight shoreline, and its light makes four revolutions per minute. How fast is the beam of light moving along the shoreline when it is 1 km1 \mathrm {~km}1 km from PPP ?

Use the properties of logarithms to expand the quantity. 11ln⁡a(b2+c2)11 \ln \sqrt { a \left( b ^ { 2 } + c ^ { 2 } \right) }11lna(b2+c2)​

Find f−1(a)f ^ { - 1 } ( a )f−1(a) for the function fff and the real number aaa . f(x)=x3+x−5;a=−5f ( x ) = x ^ { 3 } + x - 5 ; a = - 5f(x)=x3+x−5;a=−5

Find the derivative of the function. y=3sin⁡−1(x2)y = 3 \sin ^ { - 1 } \left( x ^ { 2 } \right)y=3sin−1(x2)

Evaluate the expression. log⁡12525\log _ { 125 } 25log125​25

Functions and Models

Limits and Derivatives

Differentiation Rules

Applications of Differentiation

Integrals

Techniques of Integration

Further Applications of Integration

Differential Equations

Parametric Equations and Polar Coordinates

Infinite Sequences and Series

Vectors and the Geometry of Space

Vector Functions

Partial Derivatives

Multiple Integrals

Vector Calculus

Second-Order Differential Equations

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Simplify the expression. Select the correct answer. $e ^ { 3 \ln 2 }$

Suppose $g$ is the inverse function of a differentiable function $f$ and $G ( x ) = \frac { 1 } { g ( x ) }$ . If $f ( 4 ) = 3$ and $f ^ { t } ( 4 ) = \frac { 1 } { 16 }$ , find $G ^ {t} ( 3 )$ .

Find $g ^ { t } ( x )$ $g ( x ) = \int _ { 3 } ^ { \sqrt { x } } \frac { e ^ { s } } { s } d s$

$\text { If } f \text { is a one-to-one function such that } f ( 7 ) = 1 \text {, what is } f ^ { - 1 } ( 1 ) \text { ? }$ Select the correct answer.

Differentiate the function. $f ( x ) = \cos ( \ln ( 6 x ) )$

Use logarithmic differentiation to find the derivative of the function. $y = ( x + 2 ) ^ { 7 / x }$

Find the volume of the solid obtained by rotating the region bounded by the given curves about the specified line. $y = \ln x , y = 1 , y = 5 , x = 0$ ; about the $y$ - axis

Find the integral. $\int \sqrt { 1 + 8 e ^ { x } } e ^ { x } d x$

Solve the inequality. $\ln \left( x ^ { 2 } - 6 x - 6 \right) \leq 0$

Evaluate the integral. $\int 5 ^ { \sin \theta } \cos \theta d \theta$

Find the limit. $\lim _ { x \rightarrow 0 ^ { + } } \frac { \ln 5 x } { x }$

Calculate $g ( x )$ , where $g = f ^ { - 1 }$ . State the domain and range of $g$ . Calculate $g ^ { \prime } ( a )$ . $f ( x ) = \frac { 1 } { x - 3 } , x > 3 ; a = 2$

Differentiate the function. $y = \ln \left( x ^ { 5 } \sin ^ { 2 } x \right)$

Find the limit. Select the correct answer. $\lim _ { x \rightarrow \infty } e ^ { 4 - x ^ { 4 } }$

Find the absolute extrema of the function on the indicated interval. $f ( x ) = x e ^ { - 3 x } ; \quad [ - 1,2 ]$

A lighthouse is located on a small island, $3 \mathrm {~km}$ away from the nearest point $P$ on a straight shoreline, and its light makes four revolutions per minute. How fast is the beam of light moving along the shoreline when it is $1 \mathrm {~km}$ from $P$ ?

Use the properties of logarithms to expand the quantity. $11 \ln \sqrt { a \left( b ^ { 2 } + c ^ { 2 } \right) }$

Find $f ^ { - 1 } ( a )$ for the function $f$ and the real number $a$ . $f ( x ) = x ^ { 3 } + x - 5 ; a = - 5$

Find the derivative of the function. $y = 3 \sin ^ { - 1 } \left( x ^ { 2 } \right)$

Evaluate the expression. $\log _ { 125 } 25$