Question 1

$$\text { If } f ( t ) = \sqrt { 9 t + 1 } \text {, find } f ^ { \prime \prime } ( 4 )$$

Accepted Answer

The answer of \[\text { If } f ( t...

Question 2

Find $\frac { d ^ { 2 } y } { d x ^ { 2 } }$ in terms of $x$ and $y$.
$$x ^ { 7 } - y ^ { 7 } = 1$$

Accepted Answer

The answer of Find \(\frac { d ^ { 2...

Question 3

$$\text { Find the instantaneous rate of change of the function } f ( x ) = \sqrt { 3 x } \text { when } x = 3 \text {. }$$

Accepted Answer

The answer of \[\text { Find the instantaneous rate of...

Question 4

$$\text { If } y = 2 x ^ { 2 } + 7 x \text { and } \frac { d x } { d t } = 6 , \text { find } \frac { d y } { d t } \text { when } x = 4$$

Accepted Answer

The answer of \[\text { If } y = 2...

Question 5

$$\text { Find the points on the curve } y = 2 x ^ { 3 } + 3 x ^ { 2 } - 36 x + 19 \text { where the tangent is horizontal. }$$

Accepted Answer

The answer of \[\text { Find the points on the...

Question 6

Find $f ^ { \prime }$ in terms of $g ^ { \prime }$.
$$f ( x ) = [ g ( x ) ] ^ { 4 }$$

Accepted Answer

The answer of Find $f ^ { \prime }$ in...

Question 7

Find the derivative of the function.
$$f ( x ) = \frac { 2 \sqrt { x } } { x ^ { 2 } + 9 }$$

Accepted Answer

A) 
$\frac { - 3 x ^ { 2 } + 9 } { \sqrt { x } \left( x ^ { 2 } + 9 \right) ^ { 2 } }$ 
B) 
$\frac { 1 } { 2 x \sqrt { x } \left( x ^ { 2 } + 9 \right) }$ 
C)  $\frac { 1 } { 2 x \sqrt { x } }$ 
D)  $\frac { - 3 x ^ { 2 } + 9 } { \sqrt { x } \left( x ^ { 2 } + 9 \right) }$ 
A) 
$\frac { - 3 x ^ { 2 } + 9 } { \sqrt { x } \left( x ^ { 2 } + 9 \right) ^ { 2 } }$ 
B) 
$\frac { 1 } { 2 x \sqrt { x } \left( x ^ { 2 } + 9 \right) }$ 
C)  $\frac { 1 } { 2 x \sqrt { x } }$ 
D)  $\frac { - 3 x ^ { 2 } + 9 } { \sqrt { x } \left( x ^ { 2 } + 9 \right) }$

Question 8

Find $f ^ { \prime \prime } ( x )$
$$f ( x ) = ( 2 x ) ^ { 5 } - ( 7 x ) ^ { 2 } + 5$$

Accepted Answer

The answer of Find \(f ^ { \prime \prime }...

Question 9

Find the derivative of the function.
$$y = 3 \cos ^ { - 1 } \left( \sin ^ { - 1 } t \right)$$

Accepted Answer

A) 
$$y ^ { \prime } = - \frac { 3 } { \sqrt { \left( 1 - t ^ { 2 } \right) \left( 1 - \left( \sin ^ { - 1 } ( t ) \right) ^ { 2 } \right) } }$$ 
B)  
$y ^ { t } = - \frac { 3 } { \sqrt { \left( 1 - t ^ { 2 } \right) \left( 1 - \sin ^ { - 1 } ( t ) \right) } }$ 
C) 
 $y ^ { t } = - \frac { 3 } { \sqrt { \left( 1 - t ^ { 2 } \right) } }$ 
D) 
$$y ^ { \prime } = - \frac { 3 } { \sqrt { 1 - \left( \sin ^ { - 1 } ( t ) \right) ^ { 2 } } }$$ 
E) 
$$y ^ { \prime } = - \frac { 3 } { \sqrt { \left( 1 + t ^ { 2 } \right) \left( 1 + \left( \sin ^ { - 1 } ( t ) \right) ^ { 2 } \right) } }$$ 
A) 
$$y ^ { \prime } = - \frac { 3 } { \sqrt { \left( 1 - t ^ { 2 } \right) \left( 1 - \left( \sin ^ { - 1 } ( t ) \right) ^ { 2 } \right) } }$$ 
B)  
$y ^ { t } = - \frac { 3 } { \sqrt { \left( 1 - t ^ { 2 } \right) \left( 1 - \sin ^ { - 1 } ( t ) \right) } }$ 
C) 
 $y ^ { t } = - \frac { 3 } { \sqrt { \left( 1 - t ^ { 2 } \right) } }$ 
D) 
$$y ^ { \prime } = - \frac { 3 } { \sqrt { 1 - \left( \sin ^ { - 1 } ( t ) \right) ^ { 2 } } }$$ 
E) 
$$y ^ { \prime } = - \frac { 3 } { \sqrt { \left( 1 + t ^ { 2 } \right) \left( 1 + \left( \sin ^ { - 1 } ( t ) \right) ^ { 2 } \right) } }$$

Question 10

Gravel is being dumped from a conveyor belt at a rate of $34 \mathrm { ft } / \mathrm { min }$ and its coarseness is such that it forms a pile in the shape of a cone whose base diameter and height are always equal. How fast is the height of the pile increasing when the pile is $13 \mathrm { ft }$ high? Round the result to the nearest hundredth.

Accepted Answer

A)  $\quad 0.6 \mathrm { ft } / \mathrm { min }$ 
B)  $0.26 \mathrm { ft } / \mathrm { min }$ 
C)  $0.14 \mathrm { ft } / \mathrm { min }$ 
D)  $\quad 0.27 \mathrm { ft } / \mathrm { min }$ 
E)  $1.24 \mathrm { ft } / \mathrm { min }$ 
A)  $\quad 0.6 \mathrm { ft } / \mathrm { min }$ 
B)  $0.26 \mathrm { ft } / \mathrm { min }$ 
C)  $0.14 \mathrm { ft } / \mathrm { min }$ 
D)  $\quad 0.27 \mathrm { ft } / \mathrm { min }$ 
E)  $1.24 \mathrm { ft } / \mathrm { min }$

Question 11

Find the derivative of the function.
$$g ( v ) = \sin v - 8 v \csc v$$

Accepted Answer

The answer of Find the derivative of the function.
\[g (...

Question 12

The top of a ladder slides down a vertical wall at a rate of $0.15 \mathrm {~m} / \mathrm { s }$. At the moment when the bottom of the ladder is $1.5 \mathrm {~m}$ from the wall, it slides away from the wall at a rate of $0.3 \mathrm {~m} / \mathrm { s }$. How long is the ladder?

Accepted Answer

The answer of The top of a ladder slides down...

Question 13

Calculate $y ^ { \prime }$.
$x y ^ { 3 } + x ^ { 3 } y = x + 3 y$

Accepted Answer

A)

$y ^ { \prime } = \frac { 1 - y ^ { 3 } - 3 x ^ { 2 } y } { 3 x y ^ { 2 } + x ^ { 3 } - 3 }$ 
B)

$y ^ { \prime } = \frac { 1 - y ^ { 3 } - 2 x ^ { 3 } } { 3 x y ^ { 2 } + x ^ { 2 } - 3 }$ 
C)

$$y ^ { t } = \frac { - y ^ { 4 } - 3 x y } { 4 x y ^ { 3 } + x ^ { 2 } }$$ 
D) 
$$y ^ { t } = \frac { x y ^ { 2 } + 2 x - 3 } { x ^ { 2 } y ^ { 2 } ( 3 x - 1 ) }$$ 
E)  none of these 
A)

$y ^ { \prime } = \frac { 1 - y ^ { 3 } - 3 x ^ { 2 } y } { 3 x y ^ { 2 } + x ^ { 3 } - 3 }$ 
B)

$y ^ { \prime } = \frac { 1 - y ^ { 3 } - 2 x ^ { 3 } } { 3 x y ^ { 2 } + x ^ { 2 } - 3 }$ 
C)

$$y ^ { t } = \frac { - y ^ { 4 } - 3 x y } { 4 x y ^ { 3 } + x ^ { 2 } }$$ 
D) 
$$y ^ { t } = \frac { x y ^ { 2 } + 2 x - 3 } { x ^ { 2 } y ^ { 2 } ( 3 x - 1 ) }$$ 
E)  none of these

Question 14

$s ( t )$ is the position of a body moving along a coordinate line; $s ( t )$ is measured in feet and $t$ in seconds, where $t \geq 0$. Find the position, velocity, and speed of the body at the indicated time.
$$s ( t ) = t ^ { 10 } e ^ { - t } ; \quad t = 1$$

Accepted Answer

The answer of $s ( t )$ is the position...

Question 15

Find $f ^ { \prime }$ in terms of $g ^ { \prime }$.
$$f ( x ) = [ g ( x ) ] ^ { 4 }$$

Accepted Answer

The answer of Find $f ^ { \prime }$ in...

Question 16

Use differentials to estimate the amount of paint needed to apply a coat of paint $0.0017 \mathrm {~cm}$ thick to a hemispherical dome with diameter $70 \mathrm {~m}$.
Select the correct answer.

Accepted Answer

A)  $4.165 \pi$ 
B)  $2.52 \pi$ 
C)  $4.11 \pi$ 
D)  $3.82 \pi$ 
E)  $2.28 \pi$ 
A)  $4.165 \pi$ 
B)  $2.52 \pi$ 
C)  $4.11 \pi$ 
D)  $3.82 \pi$ 
E)  $2.28 \pi$

Question 17

Calculate $y ^ { \prime }$.
$$x y ^ { 3 } + x ^ { 3 } y = x + 3 y$$

Accepted Answer

The answer of Calculate $y ^ { \prime }$.
\[x y...

Question 18

$$\text { Find the tangent line to the ellipse } \frac { x ^ { 2 } } { 40 } + \frac { y ^ { 2 } } { 10 } = 1 \text { at the point } ( 2 , - \sqrt { 3 } ) \text {. }$$

Accepted Answer

The answer of \[\text { Find the tangent line to...

Question 19

If $f$ is a differentiable function, find an expression for the derivative of $y = x ^ { 3 } f ( x )$.

Accepted Answer

A)  $\frac { d } { d x } \left( x ^ { 3 } f ( x ) \right) = 3 x ^ { 2 } f ( x ) + x ^ { 3 } f ^ { \prime } ( x )$ 
B)  $\frac { d } { d x } \left( x ^ { 3 } f ( x ) \right) = 3 x ^ { 3 } f ( x ) + x ^ { 2 } f ^ { \prime } ( x )$ 
C) 
$\frac { d } { d x } \left( x ^ { 3 } f ( x ) \right) = 2 x ^ { 2 } f ( x ) - x ^ { 3 } f ^ { \prime } ( x )$ 
D)  $\frac { d } { d x } \left( x ^ { 3 } f ( x ) \right) = 3 x ^ { 2 } f ( x ) - x ^ { 3 } f ^ { \prime } ( x )$ 
E) 
$\frac { d } { d x } \left( x ^ { 3 } f ( x ) \right) = 3 x ^ { 3 } f ( x ) - x ^ { 2 } f ^ { \prime } ( x )$ 
A)  $\frac { d } { d x } \left( x ^ { 3 } f ( x ) \right) = 3 x ^ { 2 } f ( x ) + x ^ { 3 } f ^ { \prime } ( x )$ 
B)  $\frac { d } { d x } \left( x ^ { 3 } f ( x ) \right) = 3 x ^ { 3 } f ( x ) + x ^ { 2 } f ^ { \prime } ( x )$ 
C) 
$\frac { d } { d x } \left( x ^ { 3 } f ( x ) \right) = 2 x ^ { 2 } f ( x ) - x ^ { 3 } f ^ { \prime } ( x )$ 
D)  $\frac { d } { d x } \left( x ^ { 3 } f ( x ) \right) = 3 x ^ { 2 } f ( x ) - x ^ { 3 } f ^ { \prime } ( x )$ 
E) 
$\frac { d } { d x } \left( x ^ { 3 } f ( x ) \right) = 3 x ^ { 3 } f ( x ) - x ^ { 2 } f ^ { \prime } ( x )$

$\text { If } f ( t ) = \sqrt { 9 t + 1 } \text {, find } f ^ { \prime \prime } ( 4 )$

Find $\frac { d ^ { 2 } y } { d x ^ { 2 } }$ in terms of $x$ and $y$ . $x ^ { 7 } - y ^ { 7 } = 1$

$\text { Find the instantaneous rate of change of the function } f ( x ) = \sqrt { 3 x } \text { when } x = 3 \text {. }$

$\text { If } y = 2 x ^ { 2 } + 7 x \text { and } \frac { d x } { d t } = 6 , \text { find } \frac { d y } { d t } \text { when } x = 4$

$\text { Find the points on the curve } y = 2 x ^ { 3 } + 3 x ^ { 2 } - 36 x + 19 \text { where the tangent is horizontal. }$

Find $f ^ { \prime }$ in terms of $g ^ { \prime }$ . $f ( x ) = [ g ( x ) ] ^ { 4 }$

Find the derivative of the function. $f ( x ) = \frac { 2 \sqrt { x } } { x ^ { 2 } + 9 }$

Find $f ^ { \prime \prime } ( x )$ $f ( x ) = ( 2 x ) ^ { 5 } - ( 7 x ) ^ { 2 } + 5$

Find the derivative of the function. $y = 3 \cos ^ { - 1 } \left( \sin ^ { - 1 } t \right)$

Find the derivative of the function. $g ( v ) = \sin v - 8 v \csc v$

The top of a ladder slides down a vertical wall at a rate of $0.15 \mathrm {~m} / \mathrm { s }$ . At the moment when the bottom of the ladder is $1.5 \mathrm {~m}$ from the wall, it slides away from the wall at a rate of $0.3 \mathrm {~m} / \mathrm { s }$ . How long is the ladder?

Calculate $y ^ { \prime }$ . $x y ^ { 3 } + x ^ { 3 } y = x + 3 y$

$s ( t )$ is the position of a body moving along a coordinate line; $s ( t )$ is measured in feet and $t$ in seconds, where $t \geq 0$ . Find the position, velocity, and speed of the body at the indicated time. $s ( t ) = t ^ { 10 } e ^ { - t } ; \quad t = 1$

Find $f ^ { \prime }$ in terms of $g ^ { \prime }$ . $f ( x ) = [ g ( x ) ] ^ { 4 }$

Use differentials to estimate the amount of paint needed to apply a coat of paint $0.0017 \mathrm {~cm}$ thick to a hemispherical dome with diameter $70 \mathrm {~m}$ . Select the correct answer.

Calculate $y ^ { \prime }$ . $x y ^ { 3 } + x ^ { 3 } y = x + 3 y$

$\text { Find the tangent line to the ellipse } \frac { x ^ { 2 } } { 40 } + \frac { y ^ { 2 } } { 10 } = 1 \text { at the point } ( 2 , - \sqrt { 3 } ) \text {. }$

If $f$ is a differentiable function, find an expression for the derivative of $y = x ^ { 3 } f ( x )$ .

Functions and Models

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Techniques of Integration

Further Applications of Integration

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Exam 2: Limits and Derivatives

If f(t)=9t+1, find f′′(4)\text { If } f ( t ) = \sqrt { 9 t + 1 } \text {, find } f ^ { \prime \prime } ( 4 ) If f(t)=9t+1​, find f′′(4)

Find d2ydx2\frac { d ^ { 2 } y } { d x ^ { 2 } }dx2d2y​ in terms of xxx and yyy . x7−y7=1x ^ { 7 } - y ^ { 7 } = 1x7−y7=1

Find the instantaneous rate of change of the function f(x)=3x when x=3. \text { Find the instantaneous rate of change of the function } f ( x ) = \sqrt { 3 x } \text { when } x = 3 \text {. } Find the instantaneous rate of change of the function f(x)=3x​ when x=3.

If y=2x2+7x and dxdt=6, find dydt when x=4\text { If } y = 2 x ^ { 2 } + 7 x \text { and } \frac { d x } { d t } = 6 , \text { find } \frac { d y } { d t } \text { when } x = 4 If y=2x2+7x and dtdx​=6, find dtdy​ when x=4

Find the points on the curve y=2x3+3x2−36x+19 where the tangent is horizontal. \text { Find the points on the curve } y = 2 x ^ { 3 } + 3 x ^ { 2 } - 36 x + 19 \text { where the tangent is horizontal. } Find the points on the curve y=2x3+3x2−36x+19 where the tangent is horizontal.

Find f′f ^ { \prime }f′ in terms of g′g ^ { \prime }g′ . f(x)=[g(x)]4f ( x ) = [ g ( x ) ] ^ { 4 }f(x)=[g(x)]4

Find the derivative of the function. f(x)=2xx2+9f ( x ) = \frac { 2 \sqrt { x } } { x ^ { 2 } + 9 }f(x)=x2+92x​​

Find f′′(x)f ^ { \prime \prime } ( x )f′′(x) f(x)=(2x)5−(7x)2+5f ( x ) = ( 2 x ) ^ { 5 } - ( 7 x ) ^ { 2 } + 5f(x)=(2x)5−(7x)2+5

Find the derivative of the function. y=3cos⁡−1(sin⁡−1t)y = 3 \cos ^ { - 1 } \left( \sin ^ { - 1 } t \right)y=3cos−1(sin−1t)

Find the derivative of the function. g(v)=sin⁡v−8vcsc⁡vg ( v ) = \sin v - 8 v \csc vg(v)=sinv−8vcscv

Calculate y′y ^ { \prime }y′ . xy3+x3y=x+3yx y ^ { 3 } + x ^ { 3 } y = x + 3 yxy3+x3y=x+3y

Find f′f ^ { \prime }f′ in terms of g′g ^ { \prime }g′ . f(x)=[g(x)]4f ( x ) = [ g ( x ) ] ^ { 4 }f(x)=[g(x)]4

Use differentials to estimate the amount of paint needed to apply a coat of paint 0.0017 cm0.0017 \mathrm {~cm}0.0017 cm thick to a hemispherical dome with diameter 70 m70 \mathrm {~m}70 m . Select the correct answer.

Calculate y′y ^ { \prime }y′ . xy3+x3y=x+3yx y ^ { 3 } + x ^ { 3 } y = x + 3 yxy3+x3y=x+3y

If fff is a differentiable function, find an expression for the derivative of y=x3f(x)y = x ^ { 3 } f ( x )y=x3f(x) .

Functions and Models

Differentiation Rules

Applications of Differentiation

Integrals

Applications of Integration

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

$\text { If } f ( t ) = \sqrt { 9 t + 1 } \text {, find } f ^ { \prime \prime } ( 4 )$

Find $\frac { d ^ { 2 } y } { d x ^ { 2 } }$ in terms of $x$ and $y$ . $x ^ { 7 } - y ^ { 7 } = 1$

$\text { Find the instantaneous rate of change of the function } f ( x ) = \sqrt { 3 x } \text { when } x = 3 \text {. }$

$\text { If } y = 2 x ^ { 2 } + 7 x \text { and } \frac { d x } { d t } = 6 , \text { find } \frac { d y } { d t } \text { when } x = 4$

$\text { Find the points on the curve } y = 2 x ^ { 3 } + 3 x ^ { 2 } - 36 x + 19 \text { where the tangent is horizontal. }$

Find $f ^ { \prime }$ in terms of $g ^ { \prime }$ . $f ( x ) = [ g ( x ) ] ^ { 4 }$

Find the derivative of the function. $f ( x ) = \frac { 2 \sqrt { x } } { x ^ { 2 } + 9 }$

Find $f ^ { \prime \prime } ( x )$ $f ( x ) = ( 2 x ) ^ { 5 } - ( 7 x ) ^ { 2 } + 5$

Find the derivative of the function. $y = 3 \cos ^ { - 1 } \left( \sin ^ { - 1 } t \right)$

Find the derivative of the function. $g ( v ) = \sin v - 8 v \csc v$

Calculate $y ^ { \prime }$ . $x y ^ { 3 } + x ^ { 3 } y = x + 3 y$

Find $f ^ { \prime }$ in terms of $g ^ { \prime }$ . $f ( x ) = [ g ( x ) ] ^ { 4 }$

Use differentials to estimate the amount of paint needed to apply a coat of paint $0.0017 \mathrm {~cm}$ thick to a hemispherical dome with diameter $70 \mathrm {~m}$ . Select the correct answer.

Calculate $y ^ { \prime }$ . $x y ^ { 3 } + x ^ { 3 } y = x + 3 y$

If $f$ is a differentiable function, find an expression for the derivative of $y = x ^ { 3 } f ( x )$ .