Question 1

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

Accepted Answer

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

Question 2

Find the solution of the equation correct to four decimal places.
$$\ln \left( e ^ { x } - 3 \right) = 2$$

Accepted Answer

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

Question 3

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 4

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

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 5

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

Accepted Answer

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

Question 6

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

Accepted Answer

A)  $\frac { \ln ( a ) } { 11 } + \frac { \ln ( 2 b + 2 c ) } { 11 }$ 
B) 
$\frac { 11 \ln ( a ) } { 2 } + \frac { 11 \ln \left( b ^ { 2 } + c ^ { 2 } \right) } { 2 }$ 
C)  $11 \ln ( a ) - 11 \ln \left( b ^ { 2 } + c ^ { 2 } \right)$ 
D) 
$\frac { \ln ( a ) } { 11 } - \frac { \ln \left( b ^ { 2 } + c ^ { 2 } \right) } { 11 }$ 
E)  $11 \ln ( a ) + 11 \ln \left( b ^ { 2 } + c ^ { 2 } \right)$ 
A)  $\frac { \ln ( a ) } { 11 } + \frac { \ln ( 2 b + 2 c ) } { 11 }$ 
B) 
$\frac { 11 \ln ( a ) } { 2 } + \frac { 11 \ln \left( b ^ { 2 } + c ^ { 2 } \right) } { 2 }$ 
C)  $11 \ln ( a ) - 11 \ln \left( b ^ { 2 } + c ^ { 2 } \right)$ 
D) 
$\frac { \ln ( a ) } { 11 } - \frac { \ln \left( b ^ { 2 } + c ^ { 2 } \right) } { 11 }$ 
E)  $11 \ln ( a ) + 11 \ln \left( b ^ { 2 } + c ^ { 2 } \right)$

Question 7

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

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 8

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

Accepted Answer

A)  $\frac { 2 } { 5 }$ 
B)  $\infty$ 
C)  $- \infty$ 
D)  6 
E)  0 
A)  $\frac { 2 } { 5 }$ 
B)  $\infty$ 
C)  $- \infty$ 
D)  6 
E)  0

Question 9

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

Accepted Answer

The answer of \[\text { Find } \frac { d...

Question 10

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

The answer of Use the laws of logarithms to write...

Question 11

Use logarithmic differentiation to find the derivative of the function.
$$y = x ^ { 6 x }$$

Accepted Answer

The answer of Use logarithmic differentiation to find the derivative...

Question 12

If $f ( x ) = 8 x + \ln x$, find $f ^ { - 1 } ( 8 )$

Accepted Answer

A)  $f ^ { - 1 } ( 8 ) = 1$ 
B)  $f ^ { - 1 } ( 8 ) = 0$ 
C)  $f ^ { - 1 } ( 8 ) = 8$ 
D)  $f ^ { - 1 } ( 8 ) = - 1$ 
E)  $f ^ { - 1 } ( 8 ) = - 8$ 
A)  $f ^ { - 1 } ( 8 ) = 1$ 
B)  $f ^ { - 1 } ( 8 ) = 0$ 
C)  $f ^ { - 1 } ( 8 ) = 8$ 
D)  $f ^ { - 1 } ( 8 ) = - 1$ 
E)  $f ^ { - 1 } ( 8 ) = - 8$

Question 13

Find the limit.
$$\lim _ { x \rightarrow \infty } [ \ln ( 9 + x ) - \ln ( 4 + x ) ]$$

Accepted Answer

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

Question 14

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

Accepted Answer

The answer of Determine $f ( x )$ from the...

Question 15

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 16

Find the exact value of the given expression.
$$\sin \left( \sin ^ { - 1 } \left( - \frac { \sqrt { 3 } } { 2 } \right) \right)$$

Accepted Answer

The answer of Find the exact value of the given...

Question 17

Find the exact value of the given expression.
$$\sin \left( \sin ^ { - 1 } \left( - \frac { \sqrt { 3 } } { 2 } \right) \right)$$

Accepted Answer

The answer of Find the exact value of the given...

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

Find the solution of the equation correct to four decimal places. $\ln \left( e ^ { x } - 3 \right) = 2$

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

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

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

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

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

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

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

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

Use logarithmic differentiation to find the derivative of the function. $y = x ^ { 6 x }$

If $f ( x ) = 8 x + \ln x$ , find $f ^ { - 1 } ( 8 )$

Find the limit. $\lim _ { x \rightarrow \infty } [ \ln ( 9 + x ) - \ln ( 4 + x ) ]$

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

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

Find the exact value of the given expression. $\sin \left( \sin ^ { - 1 } \left( - \frac { \sqrt { 3 } } { 2 } \right) \right)$

Find the exact value of the given expression. $\sin \left( \sin ^ { - 1 } \left( - \frac { \sqrt { 3 } } { 2 } \right) \right)$

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

Filters

Exam 6: Applications of Integration

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

Find the solution of the equation correct to four decimal places. ln⁡(ex−3)=2\ln \left( e ^ { x } - 3 \right) = 2ln(ex−3)=2

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)​

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

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)​

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

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

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

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)

Use logarithmic differentiation to find the derivative of the function. y=x6xy = x ^ { 6 x }y=x6x

If f(x)=8x+ln⁡xf ( x ) = 8 x + \ln xf(x)=8x+lnx , find f−1(8)f ^ { - 1 } ( 8 )f−1(8)

Find the limit. lim⁡x→∞[ln⁡(9+x)−ln⁡(4+x)]\lim _ { x \rightarrow \infty } [ \ln ( 9 + x ) - \ln ( 4 + x ) ]limx→∞​[ln(9+x)−ln(4+x)]

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

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

Find the exact value of the given expression. sin⁡(sin⁡−1(−32))\sin \left( \sin ^ { - 1 } \left( - \frac { \sqrt { 3 } } { 2 } \right) \right)sin(sin−1(−23​​))

Find the exact value of the given expression. sin⁡(sin⁡−1(−32))\sin \left( \sin ^ { - 1 } \left( - \frac { \sqrt { 3 } } { 2 } \right) \right)sin(sin−1(−23​​))

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

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

Find the solution of the equation correct to four decimal places. $\ln \left( e ^ { x } - 3 \right) = 2$

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

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

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

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

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

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

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

Use logarithmic differentiation to find the derivative of the function. $y = x ^ { 6 x }$

If $f ( x ) = 8 x + \ln x$ , find $f ^ { - 1 } ( 8 )$

Find the limit. $\lim _ { x \rightarrow \infty } [ \ln ( 9 + x ) - \ln ( 4 + x ) ]$

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

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

Find the exact value of the given expression. $\sin \left( \sin ^ { - 1 } \left( - \frac { \sqrt { 3 } } { 2 } \right) \right)$

Find the exact value of the given expression. $\sin \left( \sin ^ { - 1 } \left( - \frac { \sqrt { 3 } } { 2 } \right) \right)$