Question 1

Find $\frac { d ^ { 4 } } { d x ^ { 4 } } \left( x ^ { 3 } \ln x \right)$

Accepted Answer

A)  $\frac { d ^ { 4 } } { d x ^ { 4 } } \left( x ^ { 3 } \ln x \right) = \frac { 1 } { x }$ 
B)  $\frac { d ^ { 4 } } { d x ^ { 4 } } \left( x ^ { 3 } \ln x \right) = \frac { 3 } { x ^ { 4 } }$ 
C)  $\frac { d ^ { 4 } } { d x ^ { 4 } } \left( x ^ { 3 } \ln x \right) = \frac { 6 } { x }$ 
D) 
$\frac { d ^ { 4 } } { d x ^ { 4 } } \left( x ^ { 3 } \ln x \right) = \frac { 6 } { x ^ { 2 } }$ 
E) 
$\frac { d ^ { 4 } } { d x ^ { 4 } } \left( x ^ { 3 } \ln x \right) = \frac { 6 } { x ^ { 4 } }$ 
A)  $\frac { d ^ { 4 } } { d x ^ { 4 } } \left( x ^ { 3 } \ln x \right) = \frac { 1 } { x }$ 
B)  $\frac { d ^ { 4 } } { d x ^ { 4 } } \left( x ^ { 3 } \ln x \right) = \frac { 3 } { x ^ { 4 } }$ 
C)  $\frac { d ^ { 4 } } { d x ^ { 4 } } \left( x ^ { 3 } \ln x \right) = \frac { 6 } { x }$ 
D) 
$\frac { d ^ { 4 } } { d x ^ { 4 } } \left( x ^ { 3 } \ln x \right) = \frac { 6 } { x ^ { 2 } }$ 
E) 
$\frac { d ^ { 4 } } { d x ^ { 4 } } \left( x ^ { 3 } \ln x \right) = \frac { 6 } { x ^ { 4 } }$

Question 2

Find an equation of the tangent line to the curve $y = 8 \arccos \left( \frac { x } { 2 } \right)$ at the point $( 1 , \pi )$.

Accepted Answer

A)  $y = \frac { \sqrt { 8 } } { 3 } x + 6 \pi - \frac { \sqrt { 8 } } { 3 }$ 
B)  $y = - \frac { \sqrt { 3 } } { 8 } x + \pi + 6 \frac { \sqrt { 3 } } { 8 }$ 
C) 
$y = - \frac { 8 } { \sqrt { 3 } } x + \pi + \frac { 8 } { \sqrt { 3 } }$ 
D) 
$y = \frac { 8 } { \sqrt { 3 } } x + 3 \pi - \frac { 8 } { \sqrt { 3 } }$ 
E) 
$$y = \frac { \sqrt { 3 } } { 8 } x + \pi - 6 \frac { \sqrt { 3 } } { 8 }$$ 
A)  $y = \frac { \sqrt { 8 } } { 3 } x + 6 \pi - \frac { \sqrt { 8 } } { 3 }$ 
B)  $y = - \frac { \sqrt { 3 } } { 8 } x + \pi + 6 \frac { \sqrt { 3 } } { 8 }$ 
C) 
$y = - \frac { 8 } { \sqrt { 3 } } x + \pi + \frac { 8 } { \sqrt { 3 } }$ 
D) 
$y = \frac { 8 } { \sqrt { 3 } } x + 3 \pi - \frac { 8 } { \sqrt { 3 } }$ 
E) 
$$y = \frac { \sqrt { 3 } } { 8 } x + \pi - 6 \frac { \sqrt { 3 } } { 8 }$$

Question 3

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

Accepted Answer

The answer of Evaluate the expression.
\(\log _ { 125 }...

Question 4

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

Accepted Answer

The answer of Find the derivative of the function.
\[y =...

Question 5

Suppose that the graph of $y = \log _ { 2 } x$ is drawn on a coordinate grid where the unit of measurement is an inch. How many miles to the right of the origin do we have to move before the height of the curve reaches $2 \mathrm { ft }$.

Accepted Answer

A)  $1264.79 \mathrm { mi }$ 
B)  $264.79 \mathrm { mi }$ 
C)  $3264.79 \mathrm { mi }$ 
D)  $4264.79 \mathrm { mi }$ 
E)  $2264.79 \mathrm { mi }$ 
A)  $1264.79 \mathrm { mi }$ 
B)  $264.79 \mathrm { mi }$ 
C)  $3264.79 \mathrm { mi }$ 
D)  $4264.79 \mathrm { mi }$ 
E)  $2264.79 \mathrm { mi }$

Question 6

Write the expression as an exponent with base $e$. $7 ^ { \sin x }$

Accepted Answer

A)  $e ^ { ( 7 \sin x ) }$ 
B)  $e ^ { ( \sin x \text { 从in } y ) }$ 
C)  $7 e ^ { ( \sin x ) }$ 
D)  $e ^ { ( \sin 7 x ) }$ 
A)  $e ^ { ( 7 \sin x ) }$ 
B)  $e ^ { ( \sin x \text { 从in } y ) }$ 
C)  $7 e ^ { ( \sin x ) }$ 
D)  $e ^ { ( \sin 7 x ) }$

Question 7

Find $\left( f ^ { - 1 } \right) ^ { t }$$( a )$ $$f ( x ) = 7 + x ^ { 2 } + \tan \left( \frac { \pi x } { 2 } \right) , - 1 < x < 1 , a = 7$$

Accepted Answer

The answer of Find \(\left( f ^ { - 1...

Question 8

Differentiate the function. Select the correct answer.
$$f ( t ) = \frac { 2 + \ln t } { 5 - \ln t }$$

Accepted Answer

A)  $f ^ { t } ( t ) = \frac { 7 } { t ( 1 - \ln t ) ^ { 2 } }$ 
B)  $f ^ { t } ( t ) = \frac { 7 } { t ( 5 - \ln t ) ^ { 2 } }$ 
C)  $f ^ { t } ( t ) = \frac { - 7 } { 5 t ( 2 - \ln t ) ^ { 3 } }$ 
D)  $f ^ { t } ( t ) = \frac { - 3 } { t ( 5 - \ln t ) ^ { 2 } }$ 
E)  $f ^ { t } ( t ) = \frac { 3 + 2 \ln t } { t ( 2 - \ln t ) ^ { 2 } }$ 
A)  $f ^ { t } ( t ) = \frac { 7 } { t ( 1 - \ln t ) ^ { 2 } }$ 
B)  $f ^ { t } ( t ) = \frac { 7 } { t ( 5 - \ln t ) ^ { 2 } }$ 
C)  $f ^ { t } ( t ) = \frac { - 7 } { 5 t ( 2 - \ln t ) ^ { 3 } }$ 
D)  $f ^ { t } ( t ) = \frac { - 3 } { t ( 5 - \ln t ) ^ { 2 } }$ 
E)  $f ^ { t } ( t ) = \frac { 3 + 2 \ln t } { t ( 2 - \ln t ) ^ { 2 } }$

Question 9

Find the inverse of $f$. Then sketch the graphs of $f$ and $f ^ { - 1 }$ on the same set of axes.
$$f ( x ) = \sqrt { 16 - x ^ { 2 } } , x \geq 0$$

Accepted Answer

The answer of Find the inverse of $f$. Then sketch...

Question 10

$$\text { Find } y ^ { \prime } \text { if } \ln x y = 2 y \sin x$$

Accepted Answer

The answer of \[\text { Find } y ^ {...

Question 11

Find the volume of the solid obtained by rotating the region bounded by the given curves about the specified line. Select the correct answer.
$y = \ln x , y = 1 , y = 5 , x = 0$; 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 12

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

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)  $\frac { 1 } { 2 }$

Question 13

Use logarithmic differentiation to find the derivative of the function.
$$y = x ^ { 8 x }$$

Accepted Answer

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

Question 14

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

Accepted Answer

The answer of Differentiate the function.
\[y = x \left( 5...

Question 15

Determine whether the function is one-to-one. Select the correct answer.
$$f ( x ) = \sqrt { 3 - x }$$

Accepted Answer

A)  Yes 
B)  No 
A)  Yes 
B)  No

Question 16

Evaluate the integral to three decimal places.
$\int _ { e } ^ { 8 } \frac { d x } { x \ln x }$

Accepted Answer

The answer of Evaluate the integral to three decimal places.
\(\int...

Question 17

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

Accepted Answer

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

Question 18

Differentiate the function.
$$h ( t ) = t ^ { 3 } - 3 ^ { t }$$

Accepted Answer

The answer of Differentiate the function.
\[h ( t ) =...

Question 19

Differentiate the function.
$$y = x ^ { e ^ { 7 x } }$$

Accepted Answer

The answer of Differentiate the function.
\[y = x ^ {...

Question 20

Find the limit. Select the correct answer. $\lim _ { x \rightarrow \infty } [ \ln ( 9 + x ) - \ln ( 4 + x ) ]$

Accepted Answer

A)  $\frac { 8 } { 5 }$ 
B)  $- 5$ 
C)  9 
D)  0 
E)  5 
A)  $\frac { 8 } { 5 }$ 
B)  $- 5$ 
C)  9 
D)  0 
E)  5

Find $\frac { d ^ { 4 } } { d x ^ { 4 } } \left( x ^ { 3 } \ln x \right)$

Find an equation of the tangent line to the curve $y = 8 \arccos \left( \frac { x } { 2 } \right)$ at the point $( 1 , \pi )$ .

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

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

Suppose that the graph of $y = \log _ { 2 } x$ is drawn on a coordinate grid where the unit of measurement is an inch. How many miles to the right of the origin do we have to move before the height of the curve reaches $2 \mathrm { ft }$ .

Write the expression as an exponent with base $e$ . $7 ^ { \sin x }$

Find $\left( f ^ { - 1 } \right) ^ { t }$ $( a )$ $f ( x ) = 7 + x ^ { 2 } + \tan \left( \frac { \pi x } { 2 } \right) , - 1 < x < 1 , a = 7$

Differentiate the function. Select the correct answer. $f ( t ) = \frac { 2 + \ln t } { 5 - \ln t }$

Find the inverse of $f$ . Then sketch the graphs of $f$ and $f ^ { - 1 }$ on the same set of axes. $f ( x ) = \sqrt { 16 - x ^ { 2 } } , x \geq 0$

$\text { Find } y ^ { \prime } \text { if } \ln x y = 2 y \sin x$

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

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

Use logarithmic differentiation to find the derivative of the function. $y = x ^ { 8 x }$

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

Determine whether the function is one-to-one. Select the correct answer. $f ( x ) = \sqrt { 3 - x }$

Evaluate the integral to three decimal places. $\int _ { e } ^ { 8 } \frac { d x } { x \ln x }$

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

Differentiate the function. $h ( t ) = t ^ { 3 } - 3 ^ { t }$

Differentiate the function. $y = x ^ { e ^ { 7 x } }$

Find the limit. Select the correct answer. $\lim _ { x \rightarrow \infty } [ \ln ( 9 + x ) - \ln ( 4 + x ) ]$

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

Filters

Exam 6: Applications of Integration

Find d4dx4(x3ln⁡x)\frac { d ^ { 4 } } { d x ^ { 4 } } \left( x ^ { 3 } \ln x \right)dx4d4​(x3lnx)

Find an equation of the tangent line to the curve y=8arccos⁡(x2)y = 8 \arccos \left( \frac { x } { 2 } \right)y=8arccos(2x​) at the point (1,π)( 1 , \pi )(1,π) .

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

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

Suppose that the graph of y=log⁡2xy = \log _ { 2 } xy=log2​x is drawn on a coordinate grid where the unit of measurement is an inch. How many miles to the right of the origin do we have to move before the height of the curve reaches 2ft2 \mathrm { ft }2ft .

Write the expression as an exponent with base eee . 7sin⁡x7 ^ { \sin x }7sinx

Find (f−1)t\left( f ^ { - 1 } \right) ^ { t }(f−1)t(a)( a )(a) f(x)=7+x2+tan⁡(πx2),−1<x<1,a=7f ( x ) = 7 + x ^ { 2 } + \tan \left( \frac { \pi x } { 2 } \right) , - 1 < x < 1 , a = 7f(x)=7+x2+tan(2πx​),−1<x<1,a=7

Differentiate the function. Select the correct answer. f(t)=2+ln⁡t5−ln⁡tf ( t ) = \frac { 2 + \ln t } { 5 - \ln t }f(t)=5−lnt2+lnt​

Find the inverse of fff . Then sketch the graphs of fff and f−1f ^ { - 1 }f−1 on the same set of axes. f(x)=16−x2,x≥0f ( x ) = \sqrt { 16 - x ^ { 2 } } , x \geq 0f(x)=16−x2​,x≥0

Find y′ if ln⁡xy=2ysin⁡x\text { Find } y ^ { \prime } \text { if } \ln x y = 2 y \sin x Find y′ if lnxy=2ysinx

Find the volume of the solid obtained by rotating the region bounded by the given curves about the specified line. Select the correct answer. 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 y−y -y− axis

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

Use logarithmic differentiation to find the derivative of the function. y=x8xy = x ^ { 8 x }y=x8x

Differentiate the function. y=x(52x)y = x \left( 5 ^ { 2 x } \right)y=x(52x)

Determine whether the function is one-to-one. Select the correct answer. f(x)=3−xf ( x ) = \sqrt { 3 - x }f(x)=3−x​

Evaluate the integral to three decimal places. ∫e8dxxln⁡x\int _ { e } ^ { 8 } \frac { d x } { x \ln x }∫e8​xlnxdx​

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

Differentiate the function. h(t)=t3−3th ( t ) = t ^ { 3 } - 3 ^ { t }h(t)=t3−3t

Differentiate the function. y=xe7xy = x ^ { e ^ { 7 x } }y=xe7x

Find the limit. Select the correct answer. lim⁡x→∞[ln⁡(9+x)−ln⁡(4+x)]\lim _ { x \rightarrow \infty } [ \ln ( 9 + x ) - \ln ( 4 + x ) ]limx→∞​[ln(9+x)−ln(4+x)]

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

Filters

Find $\frac { d ^ { 4 } } { d x ^ { 4 } } \left( x ^ { 3 } \ln x \right)$

Find an equation of the tangent line to the curve $y = 8 \arccos \left( \frac { x } { 2 } \right)$ at the point $( 1 , \pi )$ .

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

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

Suppose that the graph of $y = \log _ { 2 } x$ is drawn on a coordinate grid where the unit of measurement is an inch. How many miles to the right of the origin do we have to move before the height of the curve reaches $2 \mathrm { ft }$ .

Write the expression as an exponent with base $e$ . $7 ^ { \sin x }$

Find $\left( f ^ { - 1 } \right) ^ { t }$ $( a )$ $f ( x ) = 7 + x ^ { 2 } + \tan \left( \frac { \pi x } { 2 } \right) , - 1 < x < 1 , a = 7$

Differentiate the function. Select the correct answer. $f ( t ) = \frac { 2 + \ln t } { 5 - \ln t }$

Find the inverse of $f$ . Then sketch the graphs of $f$ and $f ^ { - 1 }$ on the same set of axes. $f ( x ) = \sqrt { 16 - x ^ { 2 } } , x \geq 0$

$\text { Find } y ^ { \prime } \text { if } \ln x y = 2 y \sin x$

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

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

Use logarithmic differentiation to find the derivative of the function. $y = x ^ { 8 x }$

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

Determine whether the function is one-to-one. Select the correct answer. $f ( x ) = \sqrt { 3 - x }$

Evaluate the integral to three decimal places. $\int _ { e } ^ { 8 } \frac { d x } { x \ln x }$

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

Differentiate the function. $h ( t ) = t ^ { 3 } - 3 ^ { t }$

Differentiate the function. $y = x ^ { e ^ { 7 x } }$

Find the limit. Select the correct answer. $\lim _ { x \rightarrow \infty } [ \ln ( 9 + x ) - \ln ( 4 + x ) ]$