Question 1

Find the limit. $$\lim _ { t \rightarrow \infty } \left( \frac { 8 t ^ { 2 } + 7 } { 9 t ^ { 2 } - 7 } \right) e ^ { - 03 t }$$

Accepted Answer

A)  0 
B)  $\frac { 9 } { 8 }$ 
C)  $\frac { 8 } { 9 }$ 
D)  $- \frac { 7 } { 8 }$ 
A)  0 
B)  $\frac { 9 } { 8 }$ 
C)  $\frac { 8 } { 9 }$ 
D)  $- \frac { 7 } { 8 }$

Question 2

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 3

Find an equation of the line tangent to the graph of $y = 7 ^ { x } + 2$ at the point $( 0,3 )$.

Accepted Answer

A)  $y = ( \ln 7 ) x + 3$ 
B)  $y = 7 x + 3$ 
C)  $y = 7 x + 2$ 
D)  $y = ( \ln 7 ) x + 2$ 
A)  $y = ( \ln 7 ) x + 3$ 
B)  $y = 7 x + 3$ 
C)  $y = 7 x + 2$ 
D)  $y = ( \ln 7 ) x + 2$

Question 4

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

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 5

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...

Question 6

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 7

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 8

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 9

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 10

Use the graph of y = ln x as an aid to sketch the graph of the function. $$g ( x ) = \ln ( x + 1 )$$

Accepted Answer

The answer of Use the graph of y = ln...

Question 11

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 12

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

Accepted Answer

A)  3 
B)  $\frac { 1 } { 2 }$ 
C) 4 
D)  $\frac { 2 } { 3 }$ 
A)  3 
B)  $\frac { 1 } { 2 }$ 
C) 4 
D)  $\frac { 2 } { 3 }$

Question 13

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

The answer of Suppose $g$ is the inverse function of...

Question 14

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 15

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 16

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

Accepted Answer

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

Question 17

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 18

Use transformations to sketch the graph of the function. $$y = 3 \ln ( x - 5 )$$

Accepted Answer

The answer of Use transformations to sketch the graph of...

Question 19

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

Accepted Answer

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

Question 20

Find the integral. $$\int ( x + 1 ) 2 ^ { x ^ { 2 } + 2 x } d x$$

Accepted Answer

The answer of Find the integral. \[\int ( x +...

Find the limit. $\lim _ { t \rightarrow \infty } \left( \frac { 8 t ^ { 2 } + 7 } { 9 t ^ { 2 } - 7 } \right) e ^ { - 03 t }$

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

Find an equation of the line tangent to the graph of $y = 7 ^ { x } + 2$ at the point $( 0,3 )$ .

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

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

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

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

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

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

Use the graph of y = ln x as an aid to sketch the graph of the function. $g ( x ) = \ln ( x + 1 )$

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

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

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 $G ^ { t } ( 3 )$

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

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

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

Use transformations to sketch the graph of the function. $y = 3 \ln ( x - 5 )$

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

Find the integral. $\int ( x + 1 ) 2 ^ { x ^ { 2 } + 2 x } d x$

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

Find the limit. lim⁡t→∞(8t2+79t2−7)e−03t\lim _ { t \rightarrow \infty } \left( \frac { 8 t ^ { 2 } + 7 } { 9 t ^ { 2 } - 7 } \right) e ^ { - 03 t }limt→∞​(9t2−78t2+7​)e−03t

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 an equation of the line tangent to the graph of y=7x+2y = 7 ^ { x } + 2y=7x+2 at the point (0,3)( 0,3 )(0,3) .

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

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 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

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)

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

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

Use the graph of y = ln x as an aid to sketch the graph of the function. g(x)=ln⁡(x+1)g ( x ) = \ln ( x + 1 )g(x)=ln(x+1)

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)

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

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)

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

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

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) .

Use transformations to sketch the graph of the function. y=3ln⁡(x−5)y = 3 \ln ( x - 5 )y=3ln(x−5)

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

Find the integral. ∫(x+1)2x2+2xdx\int ( x + 1 ) 2 ^ { x ^ { 2 } + 2 x } d x∫(x+1)2x2+2xdx

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 the limit. $\lim _ { t \rightarrow \infty } \left( \frac { 8 t ^ { 2 } + 7 } { 9 t ^ { 2 } - 7 } \right) e ^ { - 03 t }$

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

Find an equation of the line tangent to the graph of $y = 7 ^ { x } + 2$ at the point $( 0,3 )$ .

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

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

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

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

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

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

Use the graph of y = ln x as an aid to sketch the graph of the function. $g ( x ) = \ln ( x + 1 )$

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

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

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 $G ^ { t } ( 3 )$

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

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

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

Use transformations to sketch the graph of the function. $y = 3 \ln ( x - 5 )$

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

Find the integral. $\int ( x + 1 ) 2 ^ { x ^ { 2 } + 2 x } d x$