Question 1

Evaluate the integral.
$\int 5 ^ { \sin \theta } \cos \theta d \theta$

Accepted Answer

A)  $\frac { 5 ^ { \cos \theta } } { \ln ( 5 ) } + C$ 
B) 
$\frac { 5 ^ { \sin \theta } } { \ln ( \sin \theta ) } + C$ 
C)  $\frac { 5 ^ { \sin \theta } } { \ln ( 5 ) } + C$ 
D)  $\frac { 5 ^ { \cos \theta } } { \ln ( \sin \theta ) }$ 
E)  none of these 
A)  $\frac { 5 ^ { \cos \theta } } { \ln ( 5 ) } + C$ 
B) 
$\frac { 5 ^ { \sin \theta } } { \ln ( \sin \theta ) } + C$ 
C)  $\frac { 5 ^ { \sin \theta } } { \ln ( 5 ) } + C$ 
D)  $\frac { 5 ^ { \cos \theta } } { \ln ( \sin \theta ) }$ 
E)  none of these

Question 2

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

Accepted Answer

A)  $e ^ { ( 7 \sin x ) }$ 
B)  $e ^ { ( \sin x \text { (In') } }$ 
C)  $7 e ^ { ( \sin x ) }$ 
D)  $e ^ { ( \sin 7 x ) }$ 
A)  $e ^ { ( 7 \sin x ) }$ 
B)  $e ^ { ( \sin x \text { (In') } }$ 
C)  $7 e ^ { ( \sin x ) }$ 
D)  $e ^ { ( \sin 7 x ) }$

Question 3

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

Accepted Answer

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

Question 4

Find $\left( f ^ { - 1 } \right) ^ { t } ( a )$
$f ( x ) = 8 \sqrt { x ^ { 3 } + x ^ { 2 } + x + 6 } , a = 24$

Accepted Answer

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

Question 5

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

Accepted Answer

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

Question 6

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...

Question 7

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

Accepted Answer

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

Question 8

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

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 9

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 10

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 11

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 12

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 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

$\text { Find the values of } \lambda \text { for which } y = e ^ { \lambda x } \text { satisfies the equation } 
7 y + 7 y ^ {t } = y ^ {tt } \text {. }$

Accepted Answer

The answer of \(\text { Find the values of }...

Question 15

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 16

Solve the equation for x. $$11 e ^ { x } - e ^ { 2 x } = 30$$

Accepted Answer

The answer of Solve the equation for x. \[11 e...

Question 17

Find the absolute extrema of the function on the indicated interval. $$f ( x ) = x e ^ { - 3 x } ; \quad [ - 1,2 ]$$

Accepted Answer

Absolute m

Question 18

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 19

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

Accepted Answer

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

Question 20

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) 
$\frac { d ^ { 4 } } { d x ^ { 4 } } \left( x ^ { 3 } \ln x \right) = \frac { 6 } { x ^ { 4 } }$

Evaluate the integral. $\int 5 ^ { \sin \theta } \cos \theta d \theta$

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

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

Find $\left( f ^ { - 1 } \right) ^ { t } ( a )$ $f ( x ) = 8 \sqrt { x ^ { 3 } + x ^ { 2 } + x + 6 } , a = 24$

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

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

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

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

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

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

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

$\text { Find the values of } \lambda \text { for which } y = e ^ { \lambda x } \text { satisfies the equation } 7 y + 7 y ^ {t } = y ^ {tt } \text {. }$

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 $G ^ {t } ( 3 )$ .

Solve the equation for x. $11 e ^ { x } - e ^ { 2 x } = 30$

Find the absolute extrema of the function on the indicated interval. $f ( x ) = x e ^ { - 3 x } ; \quad [ - 1,2 ]$

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

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

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

Evaluate the integral. ∫5sin⁡θcos⁡θdθ\int 5 ^ { \sin \theta } \cos \theta d \theta∫5sinθcosθdθ

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

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

Find (f−1)t(a)\left( f ^ { - 1 } \right) ^ { t } ( a )(f−1)t(a) f(x)=8x3+x2+x+6,a=24f ( x ) = 8 \sqrt { x ^ { 3 } + x ^ { 2 } + x + 6 } , a = 24f(x)=8x3+x2+x+6​,a=24

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

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

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

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

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

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

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

Find the values of λ for which y=eλx satisfies the equation 7y+7yt=ytt. \text { Find the values of } \lambda \text { for which } y = e ^ { \lambda x } \text { satisfies the equation } 7 y + 7 y ^ {t } = y ^ {tt } \text {. } Find the values of λ for which y=eλx satisfies the equation 7y+7yt=ytt.

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) .

Solve the equation for x. 11ex−e2x=3011 e ^ { x } - e ^ { 2 x } = 3011ex−e2x=30

Find the absolute extrema of the function on the indicated interval. f(x)=xe−3x;[−1,2]f ( x ) = x e ^ { - 3 x } ; \quad [ - 1,2 ]f(x)=xe−3x;[−1,2]

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

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

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

Evaluate the integral. $\int 5 ^ { \sin \theta } \cos \theta d \theta$

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

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

Find $\left( f ^ { - 1 } \right) ^ { t } ( a )$ $f ( x ) = 8 \sqrt { x ^ { 3 } + x ^ { 2 } + x + 6 } , a = 24$

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

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

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

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

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

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

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

$\text { Find the values of } \lambda \text { for which } y = e ^ { \lambda x } \text { satisfies the equation } 7 y + 7 y ^ {t } = y ^ {tt } \text {. }$

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 $G ^ {t } ( 3 )$ .

Solve the equation for x. $11 e ^ { x } - e ^ { 2 x } = 30$

Find the absolute extrema of the function on the indicated interval. $f ( x ) = x e ^ { - 3 x } ; \quad [ - 1,2 ]$

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

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