Question 1

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

Accepted Answer

A) 
$g ( x ) = 3 + \frac { 1 } { x } , g ^ { t } ( 2 ) = - \frac { 1 } { 4 }$, domain $= ( 0 , \infty )$, range $= ( 3 , \infty )$ 
B) 
$g ( x ) = 4 + \frac { 1 } { x } , g ^ { t } ( 2 ) = - \frac { 1 } { 4 }$, domain $= ( 0 , \infty )$, range $= ( 4 , \infty )$ 
C) 
$g ( x ) = 2 + \frac { 1 } { x } , g ^ { t } ( 2 ) = - \frac { 1 } { 2 }$, domain $= ( 0 , \infty )$, range $= ( 3 , \infty )$ 
D) 
$g ( x ) = 4 + \frac { 1 } { 2 x } , g ^ {te } ( 2 ) = - \frac { 1 } { 8 }$, domain $= ( 0 , \infty )$, range $= ( 3 , \infty )$ 
E) 
$g ( x ) = 1 + \frac { 3 } { x } , g ^ { t } ( 2 ) = - \frac { 3 } { 4 }$, domain $= ( 0 , \infty )$, range $= ( 1 , \infty )$ 
A) 
$g ( x ) = 3 + \frac { 1 } { x } , g ^ { t } ( 2 ) = - \frac { 1 } { 4 }$, domain $= ( 0 , \infty )$, range $= ( 3 , \infty )$ 
B) 
$g ( x ) = 4 + \frac { 1 } { x } , g ^ { t } ( 2 ) = - \frac { 1 } { 4 }$, domain $= ( 0 , \infty )$, range $= ( 4 , \infty )$ 
C) 
$g ( x ) = 2 + \frac { 1 } { x } , g ^ { t } ( 2 ) = - \frac { 1 } { 2 }$, domain $= ( 0 , \infty )$, range $= ( 3 , \infty )$ 
D) 
$g ( x ) = 4 + \frac { 1 } { 2 x } , g ^ {te } ( 2 ) = - \frac { 1 } { 8 }$, domain $= ( 0 , \infty )$, range $= ( 3 , \infty )$ 
E) 
$g ( x ) = 1 + \frac { 3 } { x } , g ^ { t } ( 2 ) = - \frac { 3 } { 4 }$, domain $= ( 0 , \infty )$, range $= ( 1 , \infty )$

Question 2

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

Accepted Answer

The answer of \[\text { Suppose } g \text {...

Question 3

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

A)  $f^{-1}(x)=\sqrt{4+x^{2}}, x \geq 0$
  
B) $f ^ { - 1 } ( x ) = - \sqrt { 16 - x ^ { 2 } } , x \geq 0$
  
C) $f^{-1}(x)=\sqrt{16-x^{2}}, x \geq 0$ 
  
D)  $f ^ { - 1 } ( x ) = - \sqrt { 4 + x ^ { 2 } } , x \geq 0$
  
A)  $f^{-1}(x)=\sqrt{4+x^{2}}, x \geq 0$
  
B) $f ^ { - 1 } ( x ) = - \sqrt { 16 - x ^ { 2 } } , x \geq 0$
  
C) $f^{-1}(x)=\sqrt{16-x^{2}}, x \geq 0$ 
  
D)  $f ^ { - 1 } ( x ) = - \sqrt { 4 + x ^ { 2 } } , x \geq 0$

Question 4

Use Newton's method to find the roots of the equation correct to five decimal places. Select the correct answer.
$x \ln x - 1 = 0$

Accepted Answer

A)  $0.88161$ 
B)  $1.76322$ 
C)  $2.26322$ 
D)  $1.26322$ 
A)  $0.88161$ 
B)  $1.76322$ 
C)  $2.26322$ 
D)  $1.26322$

Question 5

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

Accepted Answer

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

Question 6

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

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 7

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

Accepted Answer

The answer of Find the limit.
\[\lim _ { x \rightarrow...

Question 8

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 $P$ ? Select the correct answer.

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 9

Find the points of intersection of the graphs of the functions. Express your answers accurate to five decimal places.
$$f ( x ) = 0.2 x ^ { 2 } - 1.8 x - 3.8 ; g ( x ) = - 0.1 x ^ { 2 } + 0.8 x + 6.7$$

Accepted Answer

The answer of Find the points of intersection of the...

Question 10

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

Accepted Answer

The answer of Find the volume of the solid obtained...

Question 11

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

The answer of Find the volume of the solid obtained...

Question 12

Solve each equation for $x$.
(a) $\ln x = 4$
(b) $e ^ { e ^ { x } } = 7$

Accepted Answer

The answer of Solve each equation for $x$.
(a) \(\ln x...

Question 13

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

Accepted Answer

A)  $0.732$ 
B)  $0.194$ 
C)  $0.094$ 
D)  $0.533$ 
E)  $0.183$ 
A)  $0.732$ 
B)  $0.194$ 
C)  $0.094$ 
D)  $0.533$ 
E)  $0.183$

Question 14

Write the expression in algebraic form.
$$\sec \left( \sin ^ { - 1 } 8 x \right)$$

Accepted Answer

The answer of Write the expression in algebraic form.
\[\sec \left(...

Question 15

Use the laws of logarithms to write the expression as the logarithm of a single quantity.
$4 \ln 3 - \frac { 3 } { 4 } \ln ( x + 3 )$

Accepted Answer

A)  $\ln \frac { 81 } { ( x + 3 ) ^ { 3 / 4 } }$ 
B)  $\frac { 3 \ln 81 } { 4 ( x + 3 ) }$ 
C)  $\ln 3 ^ { 4 } - \ln ( x + 3 ) ^ { 3 / 4 }$ 
D)  $\frac { 4 \ln 3 } { \frac { 3 } { 4 } \ln ( x + 3 ) }$ 
A)  $\ln \frac { 81 } { ( x + 3 ) ^ { 3 / 4 } }$ 
B)  $\frac { 3 \ln 81 } { 4 ( x + 3 ) }$ 
C)  $\ln 3 ^ { 4 } - \ln ( x + 3 ) ^ { 3 / 4 }$ 
D)  $\frac { 4 \ln 3 } { \frac { 3 } { 4 } \ln ( x + 3 ) }$

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

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

Accepted Answer

The answer of Write the expression as an exponent with...

Question 18

Evaluate the integral.
$$\int _ { 0 } ^ { 2 } \frac { e ^ { x } } { 1 + e ^ { 2 x } } d x$$

Accepted Answer

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

Question 19

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

The answer of Suppose that the graph of \(y =...

Question 20

$$\text { Find the exponential function } f ( x ) = b a ^ { x } \text { whose graph is given. }$$

Accepted Answer

The answer of \[\text { Find the exponential function }...

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

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

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

Use Newton's method to find the roots of the equation correct to five decimal places. Select the correct answer. $x \ln x - 1 = 0$

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

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

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

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 $P$ ? Select the correct answer.

Find the points of intersection of the graphs of the functions. Express your answers accurate to five decimal places. $f ( x ) = 0.2 x ^ { 2 } - 1.8 x - 3.8 ; g ( x ) = - 0.1 x ^ { 2 } + 0.8 x + 6.7$

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

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

Solve each equation for $x$ . (a) $\ln x = 4$ (b) $e ^ { e ^ { x } } = 7$

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

Write the expression in algebraic form. $\sec \left( \sin ^ { - 1 } 8 x \right)$

Use the laws of logarithms to write the expression as the logarithm of a single quantity. $4 \ln 3 - \frac { 3 } { 4 } \ln ( x + 3 )$

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

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

Evaluate the integral. $\int _ { 0 } ^ { 2 } \frac { e ^ { x } } { 1 + e ^ { 2 x } } d x$

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 $2 \mathrm { ft }$ .

$\text { Find the exponential function } f ( x ) = b a ^ { x } \text { whose graph is given. }$ $\text { Find the exponential function } f ( x ) = b a ^ { x } \text { whose graph is given. }$

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

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Exam 6: Applications of Integration

Calculate g(x)g ( x )g(x) , where g=f−1g = f ^ { - 1 }g=f−1 . State the domain and range of ggg . Calculate gt(a)g ^ { t } ( a )gt(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

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

Use Newton's method to find the roots of the equation correct to five decimal places. Select the correct answer. xln⁡x−1=0x \ln x - 1 = 0xlnx−1=0

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

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

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

Find the points of intersection of the graphs of the functions. Express your answers accurate to five decimal places. f(x)=0.2x2−1.8x−3.8;g(x)=−0.1x2+0.8x+6.7f ( x ) = 0.2 x ^ { 2 } - 1.8 x - 3.8 ; g ( x ) = - 0.1 x ^ { 2 } + 0.8 x + 6.7f(x)=0.2x2−1.8x−3.8;g(x)=−0.1x2+0.8x+6.7

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 y = \ln x , y = 1 , y = 5 , x = 0 ; \text { about the } y - \text { axis }y=lnx,y=1,y=5,x=0; about the y− axis

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

Solve each equation for xxx . (a) ln⁡x=4\ln x = 4lnx=4 (b) eex=7e ^ { e ^ { x } } = 7eex=7

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

Write the expression in algebraic form. sec⁡(sin⁡−18x)\sec \left( \sin ^ { - 1 } 8 x \right)sec(sin−18x)

Use the laws of logarithms to write the expression as the logarithm of a single quantity. 4ln⁡3−34ln⁡(x+3)4 \ln 3 - \frac { 3 } { 4 } \ln ( x + 3 )4ln3−43​ln(x+3)

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

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

Evaluate the integral. ∫02ex1+e2xdx\int _ { 0 } ^ { 2 } \frac { e ^ { x } } { 1 + e ^ { 2 x } } d x∫02​1+e2xex​dx

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 .

Find the exponential function f(x)=bax whose graph is given. \text { Find the exponential function } f ( x ) = b a ^ { x } \text { whose graph is given. } Find the exponential function f(x)=bax whose graph is given.

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

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

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

Use Newton's method to find the roots of the equation correct to five decimal places. Select the correct answer. $x \ln x - 1 = 0$

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

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

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

Find the points of intersection of the graphs of the functions. Express your answers accurate to five decimal places. $f ( x ) = 0.2 x ^ { 2 } - 1.8 x - 3.8 ; g ( x ) = - 0.1 x ^ { 2 } + 0.8 x + 6.7$

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

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

Solve each equation for $x$ . (a) $\ln x = 4$ (b) $e ^ { e ^ { x } } = 7$

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

Write the expression in algebraic form. $\sec \left( \sin ^ { - 1 } 8 x \right)$

Use the laws of logarithms to write the expression as the logarithm of a single quantity. $4 \ln 3 - \frac { 3 } { 4 } \ln ( x + 3 )$

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

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

Evaluate the integral. $\int _ { 0 } ^ { 2 } \frac { e ^ { x } } { 1 + e ^ { 2 x } } d x$

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 $2 \mathrm { ft }$ .

$\text { Find the exponential function } f ( x ) = b a ^ { x } \text { whose graph is given. }$ $\text { Find the exponential function } f ( x ) = b a ^ { x } \text { whose graph is given. }$