Question 1

Differentiate the function.
$$y = \ln \left( x ^ { 3 } \sin ^ { 2 } x \right)$$

Accepted Answer

A) 
$y ^ { \prime } = \frac { 3 \cos x + 2 x \sin x } { x \cos x }$ 
B) 
$y ^ { \prime } = \frac { 3 x ^ { 2 } + \cos x \sin x } { \cos x }$ 
C) 
$y ^ { \prime } = \frac { 3 \sin x + x \cos x } { x ^ { 3 } \sin ^ { 2 } x }$ 
D) 
$y ^ { \prime } = \frac { 3 \sin x - 2 x } { x \sin x }$ 
E) 
$y ^ { t } = \frac { 3 \sin x + 2 x \cos x } { x \sin x }$ 
A) 
$y ^ { \prime } = \frac { 3 \cos x + 2 x \sin x } { x \cos x }$ 
B) 
$y ^ { \prime } = \frac { 3 x ^ { 2 } + \cos x \sin x } { \cos x }$ 
C) 
$y ^ { \prime } = \frac { 3 \sin x + x \cos x } { x ^ { 3 } \sin ^ { 2 } x }$ 
D) 
$y ^ { \prime } = \frac { 3 \sin x - 2 x } { x \sin x }$ 
E) 
$y ^ { t } = \frac { 3 \sin x + 2 x \cos x } { x \sin x }$

Question 2

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

Accepted Answer

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

Question 3

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?) Select the correct answer.

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 4

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 5

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

Accepted Answer

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

Question 6

$$\text { Suppose that } g \text { is the inverse of a function } f \text {. If } f ( 4 ) = 3 \text { and } f ^ { \prime } ( 4 ) = 2 \text {, find } g ^ { \prime } ( 3 ) \text {. }$$

Accepted Answer

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

Question 7

Find the limit.
$$\lim _ { x \rightarrow \infty } \frac { 6 x } { \ln \left( 2 + 5 e ^ { x } \right) }$$

Accepted Answer

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

Question 8

Evaluate the given integral. Select the correct answer.
$$\int _ { 0 } ^ { 1 / 7 } \frac { 1 } { 1 + 49 x ^ { 2 } } d x$$

Accepted Answer

A)  $\frac { \pi } { 35 }$ 
B)  $\frac { \pi } { 21 }$ 
C)  $\frac { \pi } { 14 }$ 
D)  $\frac { \pi } { 28 }$ 
A)  $\frac { \pi } { 35 }$ 
B)  $\frac { \pi } { 21 }$ 
C)  $\frac { \pi } { 14 }$ 
D)  $\frac { \pi } { 28 }$

Question 9

Solve the equation.
$$2 e ^ { x + 5 } = 4$$

Accepted Answer

The answer of Solve the equation.
\[2 e ^ { x...

Question 10

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 11

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

Accepted Answer

The answer of Use Newton's method to find the roots...

Question 12

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

Accepted Answer

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

Question 13

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 14

$$\text { The graph of } f \text { is given. Sketch the graph of } f ^ { - 1 } \text { on the same set of axes. }$$

Accepted Answer

The answer of \[\text { The graph of } f...

Question 15

Find $f ^ { t } ( x )$
$f ( x ) = 8 e ^ { x } - 3 x ^ { 2 } \arctan ( x )$

Accepted Answer

A) 
$f ^ { t } ( x ) = 8 e ^ { x } - \frac { 3 x ^ { 2 } } { 1 + x ^ { 2 } } - 2 x ( \arctan ( x ) )$ 
B) 
$f ^ {t } ( x ) = 8 e ^ { x } - \frac { 3 x ^ { 2 } } { 1 + x ^ { 2 } } + 6 x ( \arctan ( x ) )$ 
C) 
$f ^ { t } ( x ) = 8 e ^ { x } - \frac { 3 x ^ { 2 } } { 1 + x ^ { 2 } } - 6 x ( \arcsin ( x ) )$ 
D) 
$f ^ { t } ( x ) = 8 e ^ { x } - \frac { 3 x ^ { 2 } } { 1 + x ^ { 2 } } - 6 x ( \arctan ( x ) )$ 
E)  none of these 
A) 
$f ^ { t } ( x ) = 8 e ^ { x } - \frac { 3 x ^ { 2 } } { 1 + x ^ { 2 } } - 2 x ( \arctan ( x ) )$ 
B) 
$f ^ {t } ( x ) = 8 e ^ { x } - \frac { 3 x ^ { 2 } } { 1 + x ^ { 2 } } + 6 x ( \arctan ( x ) )$ 
C) 
$f ^ { t } ( x ) = 8 e ^ { x } - \frac { 3 x ^ { 2 } } { 1 + x ^ { 2 } } - 6 x ( \arcsin ( x ) )$ 
D) 
$f ^ { t } ( x ) = 8 e ^ { x } - \frac { 3 x ^ { 2 } } { 1 + x ^ { 2 } } - 6 x ( \arctan ( x ) )$ 
E)  none of these

Question 16

Find the absolute minimum value of the function.
$$g ( x ) = \frac { e ^ { x } } { x ^ { 4 } } , x > 0$$

Accepted Answer

The answer of Find the absolute minimum value of the...

Question 17

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

Accepted Answer

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

Question 18

Find an equation of the tangent line to the curve at the given point.
$$y = 6 e ^ { 2 x } \cos \pi x , ( 0,6 )$$

Accepted Answer

The answer of Find an equation of the tangent line...

Question 19

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 20

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

Accepted Answer

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

Differentiate the function. $y = \ln \left( x ^ { 3 } \sin ^ { 2 } x \right)$

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

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

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

$\text { Suppose that } g \text { is the inverse of a function } f \text {. If } f ( 4 ) = 3 \text { and } f ^ { \prime } ( 4 ) = 2 \text {, find } g ^ { \prime } ( 3 ) \text {. }$

Find the limit. $\lim _ { x \rightarrow \infty } \frac { 6 x } { \ln \left( 2 + 5 e ^ { x } \right) }$

Evaluate the given integral. Select the correct answer. $\int _ { 0 } ^ { 1 / 7 } \frac { 1 } { 1 + 49 x ^ { 2 } } d x$

Solve the equation. $2 e ^ { x + 5 } = 4$

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

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

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

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

$\text { The graph of } f \text { is given. Sketch the graph of } f ^ { - 1 } \text { on the same set of axes. }$ $\text { The graph of } f \text { is given. Sketch the graph of } f ^ { - 1 } \text { on the same set of axes. }$

Find $f ^ { t } ( x )$ $f ( x ) = 8 e ^ { x } - 3 x ^ { 2 } \arctan ( x )$

Find the absolute minimum value of the function. $g ( x ) = \frac { e ^ { x } } { x ^ { 4 } } , x > 0$

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

Find an equation of the tangent line to the curve at the given point. $y = 6 e ^ { 2 x } \cos \pi x , ( 0,6 )$

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

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

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

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Exam 6: Applications of Integration

Differentiate the function. y=ln⁡(x3sin⁡2x)y = \ln \left( x ^ { 3 } \sin ^ { 2 } x \right)y=ln(x3sin2x)

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=0 and x=9y = e ^ { - x ^ { 2 } } , y = 0 , x = 0 \text { and } x = 9y=e−x2,y=0,x=0 and x=9

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

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

Find the limit. lim⁡x→∞6xln⁡(2+5ex)\lim _ { x \rightarrow \infty } \frac { 6 x } { \ln \left( 2 + 5 e ^ { x } \right) }limx→∞​ln(2+5ex)6x​

Evaluate the given integral. Select the correct answer. ∫01/711+49x2dx\int _ { 0 } ^ { 1 / 7 } \frac { 1 } { 1 + 49 x ^ { 2 } } d x∫01/7​1+49x21​dx

Solve the equation. 2ex+5=42 e ^ { x + 5 } = 42ex+5=4

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

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

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

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)​

The graph of f is given. Sketch the graph of f−1 on the same set of axes. \text { The graph of } f \text { is given. Sketch the graph of } f ^ { - 1 } \text { on the same set of axes. } The graph of f is given. Sketch the graph of f−1 on the same set of axes.

Find ft(x)f ^ { t } ( x )ft(x) f(x)=8ex−3x2arctan⁡(x)f ( x ) = 8 e ^ { x } - 3 x ^ { 2 } \arctan ( x )f(x)=8ex−3x2arctan(x)

Find the absolute minimum value of the function. g(x)=exx4,x>0g ( x ) = \frac { e ^ { x } } { x ^ { 4 } } , x > 0g(x)=x4ex​,x>0

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

Find an equation of the tangent line to the curve at the given point. y=6e2xcos⁡πx,(0,6)y = 6 e ^ { 2 x } \cos \pi x , ( 0,6 )y=6e2xcosπx,(0,6)

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