Question 1

Find the derivative of the function. $f ( x ) = - x ^ { 2 } + x + 2$

Accepted Answer

The answer of Find the derivative of the function. \(f...

Question 2

Find the linearization $L ( x )$ of the function at $a$.
$f ( x ) = x ^ { 2 / 3 } ; \quad a = 64$

Accepted Answer

A)  $\frac { 8 } { 3 } x + \frac { 464 } { 3 }$ 
B)  $\frac { 8 } { 3 } x + \frac { 512 } { 3 }$ 
C)  $\frac { 8 } { 3 } x - \frac { 512 } { 3 }$ 
D)  $\frac { 8 } { 3 } x - \frac { 464 } { 3 }$ 
A)  $\frac { 8 } { 3 } x + \frac { 464 } { 3 }$ 
B)  $\frac { 8 } { 3 } x + \frac { 512 } { 3 }$ 
C)  $\frac { 8 } { 3 } x - \frac { 512 } { 3 }$ 
D)  $\frac { 8 } { 3 } x - \frac { 464 } { 3 }$

Question 3

Find the point(s) on the graph of $f$ where the tangent line is horizontal.
$f ( x ) = x ^ { 2 } e ^ { - x }$

Accepted Answer

A)  $( 0,0 ) , \left( 2 , \frac { 2 ^ { 2 } } { e ^ { 2 } } \right)$ 
B)  $\left( 1 , \frac { 1 } { e } \right)$ 
C)  $( 0,0 )$ 
D)  $\left( 2 , \frac { 2 ^ { 2 } } { e ^ { 2 } } \right)$ 
A)  $( 0,0 ) , \left( 2 , \frac { 2 ^ { 2 } } { e ^ { 2 } } \right)$ 
B)  $\left( 1 , \frac { 1 } { e } \right)$ 
C)  $( 0,0 )$ 
D)  $\left( 2 , \frac { 2 ^ { 2 } } { e ^ { 2 } } \right)$

Question 4

Two chemicals react to form another chemical.Suppose that the amount of chemical formed in
time t (in hours) is given by $$x ( t ) = \frac { 11 \left[ 1 - \left( \frac { 2 } { 3 } \right) ^ { 3 t } \right] } { 1 - \frac { 1 } { 4 } \left( \frac { 2 } { 3 } \right) ^ { 3 t } }$$
where $x ( t )$ is measured in pounds.
a. Find the rate at which the chemical is formed when $t = 4$. Round to two decimal places.
b. How many pounds of the chemical are formed eventually?

Accepted Answer

The answer of Two chemicals react to form another chemical.Suppose...

Question 5

The level of nitrogen dioxide present on a certain June day in downtown Megapolis is approximated by
$$A ( t ) = 0.03 t ^ { 3 } ( t - 7 ) ^ { 4 } + 64.8 \quad 0 \leq t \leq 7$$
where $A ( t )$ is measured in pollutant standard index and $t$ is measured in hours with $t = 0$ corresponding to 7 a.m. What is the average level of nitrogen dioxide in the atmosphere from 1 a.m. to 2 p.m. on that day? Round to three decimal places.

Accepted Answer

153.037 po

Question 6

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

A)  $3.9 \mathrm {~m}$ 
B)  $2.9 \mathrm {~m}$ 
C)  $4.4 \mathrm {~m}$ 
D)  $3.4 \mathrm {~m}$ 
E)  $2.4 \mathrm {~m}$ 
A)  $3.9 \mathrm {~m}$ 
B)  $2.9 \mathrm {~m}$ 
C)  $4.4 \mathrm {~m}$ 
D)  $3.4 \mathrm {~m}$ 
E)  $2.4 \mathrm {~m}$

Question 7

$s ( t )$ 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.
$$s ( t ) = \frac { 4 t } { t ^ { 2 } + 1 } ; \quad t = 3$$

Accepted Answer

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

Question 8

Two curves are said to be orthogonal if their tangent lines are perpendicular at each point of
intersection of the curves.Show that the curves of the given equations are orthogonal. $$y - \frac { 7 } { 4 } x = \frac { \pi } { 2 } , \quad x = \frac { 7 } { 4 } \cos y$$

Accepted Answer

The curves

Question 9

Find the derivative of the function. $$f ( x ) = 0.2 x ^ { - 13 }$$

Accepted Answer

A)  $- \frac { 0.34 } { x ^ { 0.2 } }$ 
B)  $- \frac { 0.34 } { x ^ { 2.7 } }$ 
C)  $- 0.34 x ^ { 2.7 }$ 
D)  $- 0.34 x ^ { 0.7 }$ 
A)  $- \frac { 0.34 } { x ^ { 0.2 } }$ 
B)  $- \frac { 0.34 } { x ^ { 2.7 } }$ 
C)  $- 0.34 x ^ { 2.7 }$ 
D)  $- 0.34 x ^ { 0.7 }$

Question 10

The volume of a right circular cone of radius $r$ and height $h$ is $V = \frac { \pi } { 3 } r ^ { 2 } h$. Suppose that the radius and height of the cone are changing with respect to time $t$.
a. Find a relationship between $\frac { d V } { d t } , \frac { d r } { d t }$, and $\frac { d h } { d t }$.
b. At a certain instant of time, the radius and height of the cone are $12 \mathrm { in }$. and 13 in. and are increasing at the rate of $0.2 \mathrm { in } . / \mathrm { sec }$ and $0.5 \mathrm { in } . / \mathrm { sec }$, respectively. How fast is the volume of the cone increasing?

Accepted Answer

The answer of The volume of a right circular cone...

Question 11

The curve with the equation $x ^ { 2 / 3 } + y ^ { 2 / 3 } = 25$ is called an asteroid. Find an equation of the tangent to the curve at the point $( 48 \sqrt { 6 } , 1 )$.

Accepted Answer

The answer of The curve with the equation \(x ^...

Question 12

If $f$ is the focal length of a convex lens and an object is placed at a distance $v$ from the lens, then its image will be at a distance $u$ from the lens, where $f , v$, and $u$ are related by the lens equation $\frac { 1 } { f } = \frac { 1 } { v } + \frac { 1 } { u } .$
Find the rate of change of $v$ with respect to $u$.

Accepted Answer

A)  $\frac { d v } { d u } = - \frac { f } { ( u - f ) ^ { 2 } }$ 
B)  $\frac { d v } { d u } = - \frac { f ^ { 2 } } { u - f }$ 
C)  $\frac { d v } { d u } = \frac { 2 f ^ { 2 } } { ( u - f ) ^ { 2 } }$ 
D) 
$$\frac { d v } { d u } = \frac { f ^ { 2 } } { ( u - f ) ^ { 2 } }$$ 
E) 
$$\frac { d v } { d u } = - \frac { f ^ { 2 } } { ( u - f ) ^ { 2 } }$$ 
A)  $\frac { d v } { d u } = - \frac { f } { ( u - f ) ^ { 2 } }$ 
B)  $\frac { d v } { d u } = - \frac { f ^ { 2 } } { u - f }$ 
C)  $\frac { d v } { d u } = \frac { 2 f ^ { 2 } } { ( u - f ) ^ { 2 } }$ 
D) 
$$\frac { d v } { d u } = \frac { f ^ { 2 } } { ( u - f ) ^ { 2 } }$$ 
E) 
$$\frac { d v } { d u } = - \frac { f ^ { 2 } } { ( u - f ) ^ { 2 } }$$

Question 13

Find $f ^ { \prime }$ in terms of $g ^ { \prime }$.
$f ( x ) = x ^ { 7 } g ( x )$

Accepted Answer

A)  $f ^ { t } ( x ) = 7 x ^ { 6 } f ^ { \prime } ( x ) + x ^ { 7 } g ^ { \prime } ( x )$ 
B)  $f ^ { t } ( x ) = 7 x ^ { 6 } g ^ { \prime } ( x )$ 
C)  $f ^ { t } ( x ) = 7 x ^ { 6 } g ( x ) + 7 x ^ { 7 } g ^ { \prime } ( x )$ 
D)  $f ^ { t } ( x ) = 7 x ^ { 6 } g ( x ) + x ^ { 7 } g ^ { \prime } ( x )$ 
E)  $f ^ { t } ( x ) = 7 x ^ { 6 } + g ^ { \prime } ( x )$ 
A)  $f ^ { t } ( x ) = 7 x ^ { 6 } f ^ { \prime } ( x ) + x ^ { 7 } g ^ { \prime } ( x )$ 
B)  $f ^ { t } ( x ) = 7 x ^ { 6 } g ^ { \prime } ( x )$ 
C)  $f ^ { t } ( x ) = 7 x ^ { 6 } g ( x ) + 7 x ^ { 7 } g ^ { \prime } ( x )$ 
D)  $f ^ { t } ( x ) = 7 x ^ { 6 } g ( x ) + x ^ { 7 } g ^ { \prime } ( x )$ 
E)  $f ^ { t } ( x ) = 7 x ^ { 6 } + g ^ { \prime } ( x )$

Question 14

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) 
$\frac { d } { d x } \left( x ^ { 3 } f ( x ) \right) = 3 x ^ { 3 } f ( x ) - x ^ { 2 } f ^ { \prime } ( x )$

Question 15

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 $70 \mathrm {~m}$.

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 16

The graph shows the percentage of households in a certain city watching television during a 24-hr period on a weekday ( $t = 0$ corresponds to 6 a.m. ). By computing the slope of the respective tangent line, estimate the rate of change of the percentage of households watching television at a-12 p.m.
Note that $d y = 0.03$

Accepted Answer

The answer of The graph shows the percentage of households...

Question 17

Suppose that $f$ and $g$ are functions that are differentiable at $x = 1$ and that $f ( 1 ) = 1 , f ^ { \prime } ( 1 ) = - 3 , g$ $( 1 ) = 2$, and $g ^ { \prime } ( 1 ) = 5$. Find $h ^ { \prime } ( 1 )$.
$$h ( x ) = \frac { x f ( x ) } { x + g ( x ) }$$

Accepted Answer

The answer of Suppose that $f$ and $g$ are functions...

Question 18

Two sides of a triangle are $2 \mathrm {~m}$ and $3 \mathrm {~m}$ in length and the angle between them is increasing at a rate of $0.06 \mathrm { rad } / \mathrm { s }$. Find the rate at which the area of the triangle is increasing when the angle between the sides of fixed length is $\frac { \pi } { 3 }$.

Accepted Answer

A)  $1.145 \mathrm {~m} ^ { 2 } / \mathrm { s }$ 
B)  $- 0.955 \mathrm {~m} ^ { 2 } / \mathrm { s }$ 
C)  $0.090 \mathrm {~m} ^ { 2 } / \mathrm { s }$ 
D)  $5.045 \mathrm {~m} ^ { 2 } / \mathrm { s }$ 
E)  $- 1.955 \mathrm {~m} ^ { 2 } / \mathrm { s }$ 
A)  $1.145 \mathrm {~m} ^ { 2 } / \mathrm { s }$ 
B)  $- 0.955 \mathrm {~m} ^ { 2 } / \mathrm { s }$ 
C)  $0.090 \mathrm {~m} ^ { 2 } / \mathrm { s }$ 
D)  $5.045 \mathrm {~m} ^ { 2 } / \mathrm { s }$ 
E)  $- 1.955 \mathrm {~m} ^ { 2 } / \mathrm { s }$

Question 19

Find equations of the tangent lines to the curve $y = \frac { x - 8 } { x + 8 }$ that are parallel to the line $x - y = 8 .$
A) $x - y = - 18.5$
B) $x - y = - 4.5$
C) $x - y = - 1.5$
D) $x - y = - 12.5$
E) $x - y = - 15$

Accepted Answer

The answer of Find equations of the tangent lines to...

Question 20

Suppose the total cost in maunufacturing $x$ units of a certain product is $C ( x )$ dollars.
a. What does $C ^ { \prime } ( x )$ measure? Give units.
b. What can you say about the sign of $C ^ { \prime }$ ?
c. Given that $C ^ { \prime } ( 3000 ) = 11$, estimate the additional cost in producing the 3001 st unit of the product.

Accepted Answer

a. @#LAT-DLM&, measured in dollars per u

Find the derivative of the function. $f ( x ) = - x ^ { 2 } + x + 2$

Find the linearization $L ( x )$ of the function at $a$ . $f ( x ) = x ^ { 2 / 3 } ; \quad a = 64$

Find the point(s) on the graph of $f$ where the tangent line is horizontal. $f ( x ) = x ^ { 2 } e ^ { - x }$

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?

$s ( t )$ 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. $s ( t ) = \frac { 4 t } { t ^ { 2 } + 1 } ; \quad t = 3$

Find the derivative of the function. $f ( x ) = 0.2 x ^ { - 13 }$

Find $f ^ { \prime }$ in terms of $g ^ { \prime }$ . $f ( x ) = x ^ { 7 } g ( x )$

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

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 $70 \mathrm {~m}$ .

Suppose that $f$ and $g$ are functions that are differentiable at $x = 1$ and that $f ( 1 ) = 1 , f ^ { \prime } ( 1 ) = - 3 , g$ $( 1 ) = 2$ , and $g ^ { \prime } ( 1 ) = 5$ . Find $h ^ { \prime } ( 1 )$ . $h ( x ) = \frac { x f ( x ) } { x + g ( x ) }$

Find equations of the tangent lines to the curve $y = \frac { x - 8 } { x + 8 }$ that are parallel to the line $x - y = 8 .$ A) $x - y = - 18.5$ B) $x - y = - 4.5$ C) $x - y = - 1.5$ D) $x - y = - 12.5$ E) $x - y = - 15$

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

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Exam 2: Limits and Derivatives

Find the derivative of the function. f(x)=−x2+x+2f ( x ) = - x ^ { 2 } + x + 2f(x)=−x2+x+2

Find the linearization L(x)L ( x )L(x) of the function at aaa . f(x)=x2/3;a=64f ( x ) = x ^ { 2 / 3 } ; \quad a = 64f(x)=x2/3;a=64

Find the point(s) on the graph of fff where the tangent line is horizontal. f(x)=x2e−xf ( x ) = x ^ { 2 } e ^ { - x }f(x)=x2e−x

Find the derivative of the function. f(x)=0.2x−13f ( x ) = 0.2 x ^ { - 13 }f(x)=0.2x−13

The curve with the equation x2/3+y2/3=25x ^ { 2 / 3 } + y ^ { 2 / 3 } = 25x2/3+y2/3=25 is called an asteroid. Find an equation of the tangent to the curve at the point (486,1)( 48 \sqrt { 6 } , 1 )(486​,1) .

Find f′f ^ { \prime }f′ in terms of g′g ^ { \prime }g′ . f(x)=x7g(x)f ( x ) = x ^ { 7 } g ( x )f(x)=x7g(x)

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

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 .

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

Find the derivative of the function. $f ( x ) = - x ^ { 2 } + x + 2$

Find the linearization $L ( x )$ of the function at $a$ . $f ( x ) = x ^ { 2 / 3 } ; \quad a = 64$

Find the point(s) on the graph of $f$ where the tangent line is horizontal. $f ( x ) = x ^ { 2 } e ^ { - x }$

Find the derivative of the function. $f ( x ) = 0.2 x ^ { - 13 }$

Find $f ^ { \prime }$ in terms of $g ^ { \prime }$ . $f ( x ) = x ^ { 7 } g ( x )$

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

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 $70 \mathrm {~m}$ .