Question 1

Differentiate the function.
$$g ( x ) = \frac { \ln x } { x + 4 }$$

Accepted Answer

A)  $\frac { 1 } { x ( x + 4 ) }$ 
B)  $\frac { x ( 1 - \ln x ) + 4 } { x ( x + 4 ) ^ { 2 } }$ 
C)  $\frac { x ( 4 - \ln x ) + 1 } { x ( x + 4 ) ^ { 2 } }$ 
D)  $- \frac { 1 } { x ( x + 4 ) ^ { 2 } }$ 
A)  $\frac { 1 } { x ( x + 4 ) }$ 
B)  $\frac { x ( 1 - \ln x ) + 4 } { x ( x + 4 ) ^ { 2 } }$ 
C)  $\frac { x ( 4 - \ln x ) + 1 } { x ( x + 4 ) ^ { 2 } }$ 
D)  $- \frac { 1 } { x ( x + 4 ) ^ { 2 } }$

Question 2

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

Accepted Answer

The answer of \[\text { Find an equation of the...

Question 3

Find the solution of the equation correct to four decimal places.
$$e ^ { 1 + 2 x } = 190$$

Accepted Answer

The answer of Find the solution of the equation correct...

Question 4

Evaluate the integral.
$$\int _ { 0 } ^ { 4 } x e ^ { - 2 x ^ { 2 } } d t$$ al.

Accepted Answer

The answer of Evaluate the integral.
\[\int _ { 0 }...

Question 5

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 6

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

Accepted Answer

The answer of Find \(g ^ { t } (...

Question 7

A painting in an art gallery has height $h$ and is hung so that lower edge is a distance $d$ above the eye of an observer (as in the figure). How far from the wall should the observer stand to get the best view? (In other words, where should the observer stand so as to maximize the angle $\theta$ subtended at his eye by the painting?)

Accepted Answer

A)  $x = \sqrt { h ( d + h ) }$ 
B)  $x = h ^ { 2 } ( h + d ) ^ { 2 }$ 
C)  $x = d ^ { 2 } ( h + d ) ^ { 2 }$ 
D)  $x = \sqrt { d ( h + d ) }$ 
E)  $x = h ^ { 2 } ( h - d ) ^ { 2 }$ 
A)  $x = \sqrt { h ( d + h ) }$ 
B)  $x = h ^ { 2 } ( h + d ) ^ { 2 }$ 
C)  $x = d ^ { 2 } ( h + d ) ^ { 2 }$ 
D)  $x = \sqrt { d ( h + d ) }$ 
E)  $x = h ^ { 2 } ( h - d ) ^ { 2 }$

Question 8

Find the inverse function.
$$y = \frac { 2 + e ^ { x } } { 9 - e ^ { x } }$$

Accepted Answer

The answer of Find the inverse function.
\[y = \frac {...

Question 9

Find the solution of the equation correct to four decimal places.
$$e ^ { 1 + 2 x } = 190$$

Accepted Answer

The answer of Find the solution of the equation correct...

Question 10

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 11

A painting in an art gallery has height $h$ and is hung so that lower edge is a distance $d$ above the eye of an observer (as in the figure). How far from the wall should the observer stand to get the best view? (In other words, where should the observer stand so as to maximize the angle $\theta$ subtended at his eye by the painting?)

Accepted Answer

A)  $x = \sqrt { h ( d + h ) }$ 
B)  $x = h ^ { 2 } ( h + d ) ^ { 2 }$ 
C)  $x = d ^ { 2 } ( h + d ) ^ { 2 }$ 
D)  $x = \sqrt { d ( h + d ) }$ 
E)  $x = h ^ { 2 } ( h - d ) ^ { 2 }$ 
A)  $x = \sqrt { h ( d + h ) }$ 
B)  $x = h ^ { 2 } ( h + d ) ^ { 2 }$ 
C)  $x = d ^ { 2 } ( h + d ) ^ { 2 }$ 
D)  $x = \sqrt { d ( h + d ) }$ 
E)  $x = h ^ { 2 } ( h - d ) ^ { 2 }$

Question 12

Find the inverse of the function.
$$f ( x ) = \frac { 1 + 8 x } { 8 - 5 x }$$

Accepted Answer

A) 
$f ^ { - 1 } ( x ) = \frac { 8 x - 1 } { x + 8 }$ 
B) 
$f ^ { - 1 } ( x ) = \frac { 8 x - 1 } { 5 x + 8 }$ 
C) 
$f ^ { - 1 } ( x ) = \frac { x - 1 } { 5 x + 8 }$ 
D) 
$f ^ { - 1 } ( x ) = \frac { 8 x - 1 } { 8 x + 5 }$ 
E) 
$f ^ { - 1 } ( x ) = \frac { 5 x - 1 } { 5 x + 8 }$ 
A) 
$f ^ { - 1 } ( x ) = \frac { 8 x - 1 } { x + 8 }$ 
B) 
$f ^ { - 1 } ( x ) = \frac { 8 x - 1 } { 5 x + 8 }$ 
C) 
$f ^ { - 1 } ( x ) = \frac { x - 1 } { 5 x + 8 }$ 
D) 
$f ^ { - 1 } ( x ) = \frac { 8 x - 1 } { 8 x + 5 }$ 
E) 
$f ^ { - 1 } ( x ) = \frac { 5 x - 1 } { 5 x + 8 }$

Question 13

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

Accepted Answer

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

Question 14

Find the given integral.
$$\int \frac { 2 \tan ^ { - 1 } 2 x } { 1 + 4 x ^ { 2 } } d x$$

Accepted Answer

The answer of Find the given integral.
\[\int \frac { 2...

Question 15

Find the volume of the solid obtained by rotating about the $y$ axis the region bounded by the curves.
$y = e ^ { - x ^ { 2 } } , y = 0 , x = 0$ and $x = 9$

Accepted Answer

A)  $V = \pi \left( \frac { 1 } { e ^ { 81 } } \right)$ 
B) 
$V = \pi \left( 1 - \frac { 1 } { e ^ { 81 } } \right)$ 
C)  $V = \pi e ^ { 81 }$ 
D)  $ \pi \left( \frac { 1 } { e ^ { 81 } } - 1 \right)$ 
E)  $V = \pi \left( e ^ { 81 } - 1 \right)$ 
A)  $V = \pi \left( \frac { 1 } { e ^ { 81 } } \right)$ 
B) 
$V = \pi \left( 1 - \frac { 1 } { e ^ { 81 } } \right)$ 
C)  $V = \pi e ^ { 81 }$ 
D)  $ \pi \left( \frac { 1 } { e ^ { 81 } } - 1 \right)$ 
E)  $V = \pi \left( e ^ { 81 } - 1 \right)$

Question 16

Differentiate the function. Select the correct answer.
$$y = e ^ { \cos 3 x }$$

Accepted Answer

A)  $- e ^ { \cos 3 x } \sin 3 x$ 
B)  $- 3 e ^ { \cos 3 x } \sin 3 x$ 
C)  $- 3 e ^ { \cos 3 x } \sin x$ 
D)  $e ^ { \cos 3 x } \sin 3 x$ 
A)  $- e ^ { \cos 3 x } \sin 3 x$ 
B)  $- 3 e ^ { \cos 3 x } \sin 3 x$ 
C)  $- 3 e ^ { \cos 3 x } \sin x$ 
D)  $e ^ { \cos 3 x } \sin 3 x$

Question 17

The geologist C. F. Richter defined the magnitude of an earthquake to be $\log _ { 10 } \left( \frac { I } { S } \right)$ where $I$ is the intensity of the quake (measured by the amplitude of a seismograph $100 \mathrm {~km}$ from the epicenter) and $S$ is the intensity of a "standard" earthquake (where the amplitude is only $1 \mathrm { micron } = 10 ^ { - 4 }$ $\mathrm { cm }$. The 1989 Loma Prieta earthquake that shook San Francisco had a magnitude of $7.9$ on the Richter scale. The 1906 San Francisco earthquake was 12 times as intense. What was its magnitude on the Richter scale?

Accepted Answer

The answer of The geologist C. F. Richter defined the...

Question 18

Use the laws of logarithms to expand the expression.
$\ln \left( \frac { x + 5 } { x - 6 } \right) ^ { 1 / 2 }$

Accepted Answer

A)  $\frac { 1 } { 2 } \ln \left( \frac { x + 5 } { x - 6 } \right)$ 
B)  $\frac { 1 } { 2 } \ln [ ( x + 5 ) ( x - 6 ) ]$ 
C)  $\frac { 1 } { 2 } \ln \left( x - \frac { 5 } { 6 } \right)$ 
D)  $\frac { 1 } { 2 } \ln ( x + 5 ) - \frac { 1 } { 2 } \ln ( x - 6 )$ 
A)  $\frac { 1 } { 2 } \ln \left( \frac { x + 5 } { x - 6 } \right)$ 
B)  $\frac { 1 } { 2 } \ln [ ( x + 5 ) ( x - 6 ) ]$ 
C)  $\frac { 1 } { 2 } \ln \left( x - \frac { 5 } { 6 } \right)$ 
D)  $\frac { 1 } { 2 } \ln ( x + 5 ) - \frac { 1 } { 2 } \ln ( x - 6 )$

Question 19

Determine $f ( x )$ from the table.
The values from the table are given to the nearest ten thousandth. Select the correct answer.
$\begin{array}{|r|c|}
\hline x & f(x) \
\hline-1 & 20.0855 \
\hline 0 & 1 \
\hline 1 & 0.0498 \
\hline 2 & 0.0025 \
\hline
\end{array}$

Accepted Answer

A)  $f ( x ) = e ^ { 3 x }$ 
B)  $f ( x ) = e ^ { - 3 x }$ 
C)  $f ( x ) = 3 x ^ { - 3 }$ 
D)  $f ( x ) = x e ^ { - 3 }$ 
E)  $f ( x ) = x ^ { - 3 }$ 
A)  $f ( x ) = e ^ { 3 x }$ 
B)  $f ( x ) = e ^ { - 3 x }$ 
C)  $f ( x ) = 3 x ^ { - 3 }$ 
D)  $f ( x ) = x e ^ { - 3 }$ 
E)  $f ( x ) = x ^ { - 3 }$

Question 20

Find the domain of the function.
$$f ( x ) = \ln ( 8 x + 7 )$$

Accepted Answer

The answer of Find the domain of the function.
\[f (...

Differentiate the function. $g ( x ) = \frac { \ln x } { x + 4 }$

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

Find the solution of the equation correct to four decimal places. $e ^ { 1 + 2 x } = 190$

Evaluate the integral. $\int _ { 0 } ^ { 4 } x e ^ { - 2 x ^ { 2 } } d t$ al.

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

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

Find the inverse function. $y = \frac { 2 + e ^ { x } } { 9 - e ^ { x } }$

Find the solution of the equation correct to four decimal places. $e ^ { 1 + 2 x } = 190$

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

Find the inverse of the function. $f ( x ) = \frac { 1 + 8 x } { 8 - 5 x }$

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

Find the given integral. $\int \frac { 2 \tan ^ { - 1 } 2 x } { 1 + 4 x ^ { 2 } } d x$

Find the volume of the solid obtained by rotating about the $y$ axis the region bounded by the curves. $y = e ^ { - x ^ { 2 } } , y = 0 , x = 0$ and $x = 9$

Differentiate the function. Select the correct answer. $y = e ^ { \cos 3 x }$

Use the laws of logarithms to expand the expression. $\ln \left( \frac { x + 5 } { x - 6 } \right) ^ { 1 / 2 }$

Determine $f ( x )$ from the table. The values from the table are given to the nearest ten thousandth. Select the correct answer. x f(x) -1 20.0855 0 1 1 0.0498 2 0.0025

Find the domain of the function. $f ( x ) = \ln ( 8 x + 7 )$

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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Exam 6: Applications of Integration

Differentiate the function. g(x)=ln⁡xx+4g ( x ) = \frac { \ln x } { x + 4 }g(x)=x+4lnx​

Find the solution of the equation correct to four decimal places. e1+2x=190e ^ { 1 + 2 x } = 190e1+2x=190

Evaluate the integral. ∫04xe−2x2dt\int _ { 0 } ^ { 4 } x e ^ { - 2 x ^ { 2 } } d t∫04​xe−2x2dt al.

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

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

Find the inverse function. y=2+ex9−exy = \frac { 2 + e ^ { x } } { 9 - e ^ { x } }y=9−ex2+ex​

Find the solution of the equation correct to four decimal places. e1+2x=190e ^ { 1 + 2 x } = 190e1+2x=190

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

Find the inverse of the function. f(x)=1+8x8−5xf ( x ) = \frac { 1 + 8 x } { 8 - 5 x }f(x)=8−5x1+8x​

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

Find the given integral. ∫2tan⁡−12x1+4x2dx\int \frac { 2 \tan ^ { - 1 } 2 x } { 1 + 4 x ^ { 2 } } d x∫1+4x22tan−12x​dx

Find the volume of the solid obtained by rotating about the yyy axis the region bounded by the curves. y=e−x2,y=0,x=0y = e ^ { - x ^ { 2 } } , y = 0 , x = 0y=e−x2,y=0,x=0 and x=9x = 9x=9

Differentiate the function. Select the correct answer. y=ecos⁡3xy = e ^ { \cos 3 x }y=ecos3x

Use the laws of logarithms to expand the expression. ln⁡(x+5x−6)1/2\ln \left( \frac { x + 5 } { x - 6 } \right) ^ { 1 / 2 }ln(x−6x+5​)1/2

Determine f(x)f ( x )f(x) from the table. The values from the table are given to the nearest ten thousandth. Select the correct answer. x f(x) -1 20.0855 0 1 1 0.0498 2 0.0025

Find the domain of the function. f(x)=ln⁡(8x+7)f ( x ) = \ln ( 8 x + 7 )f(x)=ln(8x+7)

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

Differentiate the function. $g ( x ) = \frac { \ln x } { x + 4 }$

Find the solution of the equation correct to four decimal places. $e ^ { 1 + 2 x } = 190$

Evaluate the integral. $\int _ { 0 } ^ { 4 } x e ^ { - 2 x ^ { 2 } } d t$ al.

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

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

Find the inverse function. $y = \frac { 2 + e ^ { x } } { 9 - e ^ { x } }$

Find the solution of the equation correct to four decimal places. $e ^ { 1 + 2 x } = 190$

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

Find the inverse of the function. $f ( x ) = \frac { 1 + 8 x } { 8 - 5 x }$

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

Find the given integral. $\int \frac { 2 \tan ^ { - 1 } 2 x } { 1 + 4 x ^ { 2 } } d x$

Find the volume of the solid obtained by rotating about the $y$ axis the region bounded by the curves. $y = e ^ { - x ^ { 2 } } , y = 0 , x = 0$ and $x = 9$

Differentiate the function. Select the correct answer. $y = e ^ { \cos 3 x }$

Use the laws of logarithms to expand the expression. $\ln \left( \frac { x + 5 } { x - 6 } \right) ^ { 1 / 2 }$

Determine $f ( x )$ from the table. The values from the table are given to the nearest ten thousandth. Select the correct answer. x f(x) -1 20.0855 0 1 1 0.0498 2 0.0025

Find the domain of the function. $f ( x ) = \ln ( 8 x + 7 )$