Question 1

Find $f$.
$$f ^ { \prime } ( x ) = 3 \cos ( x ) + 4 \sin ( x ) , f ( 0 ) = 7$$

Accepted Answer

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

Question 2

Find the dimensions of the rectangle of largest area that can be inscribed in an equilateral triangle of side $L = 9 \mathrm {~cm}$ if one side of the rectangle lies on the base of the triangle.
Round your answer to the nearest tenth.

Accepted Answer

The answer of Find the dimensions of the rectangle of...

Question 3

Find the dimensions of a rectangle of area $400 \mathrm { ft } ^ { 2 }$ that has the smallest possible perimeter.

Accepted Answer

A)  $4 \mathrm { ft }$ by $100 \mathrm { ft }$ 
B)  $1 \mathrm { ft }$ by $400 \mathrm { ft }$ 
C)  $2 \mathrm { ft }$ by $200 \mathrm { ft }$ 
D)  $20 \mathrm { ft }$ by $20 \mathrm { ft }$ 
A)  $4 \mathrm { ft }$ by $100 \mathrm { ft }$ 
B)  $1 \mathrm { ft }$ by $400 \mathrm { ft }$ 
C)  $2 \mathrm { ft }$ by $200 \mathrm { ft }$ 
D)  $20 \mathrm { ft }$ by $20 \mathrm { ft }$

Question 4

Find $f$. $f ^ { \prime } ( t ) = 2 t - 4 \sin t , \quad f ( 0 ) = 5$

Accepted Answer

A)  $f ( t ) = t ^ { 2 } - 3 \cos t - 5$ 
B)  $f ( t ) = t ^ { 2 } + 4 \cos t + 1$ 
C)  $f ( t ) = 2 t - 3 \sin t + 5$ 
D)  $f ( t ) = t ^ { 2 } + 3 \cos t$ 
E)  None of these 
A)  $f ( t ) = t ^ { 2 } - 3 \cos t - 5$ 
B)  $f ( t ) = t ^ { 2 } + 4 \cos t + 1$ 
C)  $f ( t ) = 2 t - 3 \sin t + 5$ 
D)  $f ( t ) = t ^ { 2 } + 3 \cos t$ 
E)  None of these

Question 5

$$\text { Find the intervals where } f ( x ) = \frac { \log 9 x } { x } \text { is increasing and where it is decreasing. }$$

Accepted Answer

The answer of \[\text { Find the intervals where }...

Question 6

Given that the graph of $f$ passes through the point $( 4,69 )$ and that the slope of its tangent line at $( x , f ( x ) )$ is $11 x - 3$, find $f ( 1 )$. Select the correct answer.

Accepted Answer

A)  $- 4.5$ 
B)  8 
C)  12 
D)  11 
E)  1 
A)  $- 4.5$ 
B)  8 
C)  12 
D)  11 
E)  1

Question 7

$$\text { Find the relative extrema, if any, of } f ( t ) = e ^ { t } - 7 t - 3 \text {. Use the Second Derivative Test, if possible. }$$

Accepted Answer

The answer of \[\text { Find the relative extrema, if...

Question 8

Given $f ( x ) = x + \frac { 36 } { x }$.
(a) Find the intervals on which $f$ is increasing or decreasing.
(b) Find the relative maxima and relative minima of $f$.

Accepted Answer

A)  (a) Increasing on $( - 6,6 )$, decreasing on $( - \infty , - 6 )$ and $( 6 , \infty )$
(b) Rel. $\max . f ( 6 ) = 12$,
rel. min. $f ( - 6 ) = - 12$ 
B)  (a) Increasing on $( - 6,0 )$ and $( 0,6 )$, decreasing on $( - \infty , - 6 )$ and $( 6 , \infty )$
(b) Rel. $\max . f ( 6 ) = 12$,
rel. min. $f ( - 6 ) = - 12$ 
C)  (a) Increasing on $( - \infty , - 6 )$ and $( 6 , \infty )$, decreasing on $( - 6,0 )$ and $( 0,6 )$
(b) Rel. max. $f ( - 6 ) = - 12$,
rel. min. $f ( 6 ) = 12$ 
D)  (a) Increasing on $( - \infty , - 6 )$ and $( 6 , \infty )$, decreasing on $( - 6,6 )$
(b) Rel. max. $f ( - 6 ) = - 12$,
rel. min. $f ( 6 ) = 12$ 
A)  (a) Increasing on $( - 6,6 )$, decreasing on $( - \infty , - 6 )$ and $( 6 , \infty )$
(b) Rel. $\max . f ( 6 ) = 12$,
rel. min. $f ( - 6 ) = - 12$ 
B)  (a) Increasing on $( - 6,0 )$ and $( 0,6 )$, decreasing on $( - \infty , - 6 )$ and $( 6 , \infty )$
(b) Rel. $\max . f ( 6 ) = 12$,
rel. min. $f ( - 6 ) = - 12$ 
C)  (a) Increasing on $( - \infty , - 6 )$ and $( 6 , \infty )$, decreasing on $( - 6,0 )$ and $( 0,6 )$
(b) Rel. max. $f ( - 6 ) = - 12$,
rel. min. $f ( 6 ) = 12$ 
D)  (a) Increasing on $( - \infty , - 6 )$ and $( 6 , \infty )$, decreasing on $( - 6,6 )$
(b) Rel. max. $f ( - 6 ) = - 12$,
rel. min. $f ( 6 ) = 12$

Question 9

Find the critical number(s), if any of the function $f ( t ) = 3 t ^ { \frac { 3 } { 4 } } - 3 t ^ { \frac { 1 } { 4 } }$.

Accepted Answer

A)  $\frac { 1 } { 3 }$ 
B)  $\frac { 1 } { 9 }$ 
C)  $0 , \frac { 1 } { 3 }$ 
D)  $0 , \frac { 1 } { 9 }$ 
A)  $\frac { 1 } { 3 }$ 
B)  $\frac { 1 } { 9 }$ 
C)  $0 , \frac { 1 } { 3 }$ 
D)  $0 , \frac { 1 } { 9 }$

Question 10

Find two positive numbers whose product is 196 and whose sum is a minimum.

Accepted Answer

A)  14,14 
B)  3,48 
C)  6,24 
D)  4,36 
E)  2,72 
A)  14,14 
B)  3,48 
C)  6,24 
D)  4,36 
E)  2,72

Question 11

$$\text { Sketch the graph of the function } g ( x ) = \frac { x - 2 } { x - 1 } \text { using the curve-sketching guidelines. }$$

Accepted Answer

The answer of \[\text { Sketch the graph of the...

Question 12

Determine where the graph of $f ( x ) = \ln | x - 4 |$ is concave upward and where it is concave downward. Also, find all inflection points of the function.

Accepted Answer

The answer of Determine where the graph of \(f (...

Question 13

What is the minimum vertical distance between the parabolas $$y = x ^ { 2 } + 4 \text { and } y = x - x ^ { 2 } ?$$

Accepted Answer

The answer of What is the minimum vertical distance between...

Question 14

At 4:00 P.M. a car's speedometer reads $29 \mathrm { mi } ^ { 2 }$. At 4:15 it reads $72 \mathrm { mi } / \mathrm { h } ^ { 2 }$. At some time between $4 : 00$ and $4 : 15$ the acceleration is exactly $x \mathrm { mi } ^ { 2 } \mathrm {~h} ^ { 2 }$. Find $x$.

Accepted Answer

The answer of At 4:00 P.M. a car's speedometer reads...

Question 15

$$\text { Find the point on the line } y = 4 x + 8 \text { that is closest to the origin. }$$

Accepted Answer

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

Question 16

A woman at a point $A$ on the shore of a circular lake with radius $4 \mathrm { mi }$ wants to arrive at the point $C$ diametrically opposite on the other side of the lake in the shortest possible time. She can walk at the rate of $6 \mathrm { mi } / \mathrm { h }$ and row a boat at $2 \mathrm { mi } / \mathrm { h }$. How should she proceed? (Find $\theta$ ). Round the result, if necessary, to the nearest hundredth.

Accepted Answer

A)  $\quad 0.62$ radians 
B)  $0.44$ radians 
C)  $0.46$ radians 
D)  She should walk around the lake from point A to point C. 
E)  She should row from point A to point $C$ radians 
A)  $\quad 0.62$ radians 
B)  $0.44$ radians 
C)  $0.46$ radians 
D)  She should walk around the lake from point A to point C. 
E)  She should row from point A to point $C$ radians

Question 17

A manufacturer has been selling 1,200 television sets a week at $\$ 400$ each. A market survey indicates that for each $\$ 30$ rebate offered to the buyer, the number of sets sold will increase by 60 per week. Find the demand function.

Accepted Answer

The answer of A manufacturer has been selling 1,200 television...

Question 18

The graph of the derivative $f ^ { \prime } ( x )$ of a continuous function $\mathrm { f }$ is shown. On what intervals is $f$ decreasing?
 
 .

Accepted Answer

The answer of The graph of the derivative \(f ^...

Question 19

Find $f$.
$$f ^ { \prime } ( t ) = 2 t - 4 \sin t , \quad f ( 0 ) = 5$$ Select the correct answer.

Accepted Answer

A)  $f ( t ) = t ^ { 2 } - 3 \cos t - 5$ 
B)  $f ( t ) = t ^ { 2 } + 4 \cos t + 1$ 
C)  $f ( t ) = 2 t - 3 \sin t + 5$ 
D)  $f ( t ) = t ^ { 2 } + 3 \cos t$ 
E)  None of these 
A)  $f ( t ) = t ^ { 2 } - 3 \cos t - 5$ 
B)  $f ( t ) = t ^ { 2 } + 4 \cos t + 1$ 
C)  $f ( t ) = 2 t - 3 \sin t + 5$ 
D)  $f ( t ) = t ^ { 2 } + 3 \cos t$ 
E)  None of these

Question 20

The average cost of producing $x$ units of a commodity is given by the equation
$$C ( x ) = 20.4 - 0.0007 x$$
Find the marginal cost at a production level of 1,255 units.

Accepted Answer

The answer of The average cost of producing $x$ units...

Find $f$ . $f ^ { \prime } ( x ) = 3 \cos ( x ) + 4 \sin ( x ) , f ( 0 ) = 7$

Find the dimensions of the rectangle of largest area that can be inscribed in an equilateral triangle of side $L = 9 \mathrm {~cm}$ if one side of the rectangle lies on the base of the triangle. Round your answer to the nearest tenth.

Find the dimensions of a rectangle of area $400 \mathrm { ft } ^ { 2 }$ that has the smallest possible perimeter.

Find $f$ . $f ^ { \prime } ( t ) = 2 t - 4 \sin t , \quad f ( 0 ) = 5$

$\text { Find the intervals where } f ( x ) = \frac { \log 9 x } { x } \text { is increasing and where it is decreasing. }$

Given that the graph of $f$ passes through the point $( 4,69 )$ and that the slope of its tangent line at $( x , f ( x ) )$ is $11 x - 3$ , find $f ( 1 )$ . Select the correct answer.

$\text { Find the relative extrema, if any, of } f ( t ) = e ^ { t } - 7 t - 3 \text {. Use the Second Derivative Test, if possible. }$

Given $f ( x ) = x + \frac { 36 } { x }$ . (a) Find the intervals on which $f$ is increasing or decreasing. (b) Find the relative maxima and relative minima of $f$ .

Find the critical number(s), if any of the function $f ( t ) = 3 t ^ { \frac { 3 } { 4 } } - 3 t ^ { \frac { 1 } { 4 } }$ .

Find two positive numbers whose product is 196 and whose sum is a minimum.

$\text { Sketch the graph of the function } g ( x ) = \frac { x - 2 } { x - 1 } \text { using the curve-sketching guidelines. }$

Determine where the graph of $f ( x ) = \ln | x - 4 |$ is concave upward and where it is concave downward. Also, find all inflection points of the function.

What is the minimum vertical distance between the parabolas $y = x ^ { 2 } + 4 \text { and } y = x - x ^ { 2 } ?$

At 4:00 P.M. a car's speedometer reads $29 \mathrm { mi } ^ { 2 }$ . At 4:15 it reads $72 \mathrm { mi } / \mathrm { h } ^ { 2 }$ . At some time between $4 : 00$ and $4 : 15$ the acceleration is exactly $x \mathrm { mi } ^ { 2 } \mathrm {~h} ^ { 2 }$ . Find $x$ .

$\text { Find the point on the line } y = 4 x + 8 \text { that is closest to the origin. }$

A manufacturer has been selling 1,200 television sets a week at $\$ 400$ each. A market survey indicates that for each $\$ 30$ rebate offered to the buyer, the number of sets sold will increase by 60 per week. Find the demand function.

The graph of the derivative $f ^ { \prime } ( x )$ of a continuous function $\mathrm { f }$ is shown. On what intervals is $f$ decreasing? $The graph of the derivative f ^ { \prime } ( x ) of a continuous function \mathrm { f } is shown. On what intervals is f decreasing? .$ .

Find $f$ . $f ^ { \prime } ( t ) = 2 t - 4 \sin t , \quad f ( 0 ) = 5$ Select the correct answer.

The average cost of producing $x$ units of a commodity is given by the equation $C ( x ) = 20.4 - 0.0007 x$ Find the marginal cost at a production level of 1,255 units.

Functions and Models

Limits and Derivatives

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

Exam 3: Differentiation Rules

Find fff . f′(x)=3cos⁡(x)+4sin⁡(x),f(0)=7f ^ { \prime } ( x ) = 3 \cos ( x ) + 4 \sin ( x ) , f ( 0 ) = 7f′(x)=3cos(x)+4sin(x),f(0)=7

Find the dimensions of the rectangle of largest area that can be inscribed in an equilateral triangle of side L=9 cmL = 9 \mathrm {~cm}L=9 cm if one side of the rectangle lies on the base of the triangle. Round your answer to the nearest tenth.

Find the dimensions of a rectangle of area 400ft2400 \mathrm { ft } ^ { 2 }400ft2 that has the smallest possible perimeter.

Find fff . f′(t)=2t−4sin⁡t,f(0)=5f ^ { \prime } ( t ) = 2 t - 4 \sin t , \quad f ( 0 ) = 5f′(t)=2t−4sint,f(0)=5

Find the intervals where f(x)=log⁡9xx is increasing and where it is decreasing. \text { Find the intervals where } f ( x ) = \frac { \log 9 x } { x } \text { is increasing and where it is decreasing. } Find the intervals where f(x)=xlog9x​ is increasing and where it is decreasing.

Given that the graph of fff passes through the point (4,69)( 4,69 )(4,69) and that the slope of its tangent line at (x,f(x))( x , f ( x ) )(x,f(x)) is 11x−311 x - 311x−3 , find f(1)f ( 1 )f(1) . Select the correct answer.

Given f(x)=x+36xf ( x ) = x + \frac { 36 } { x }f(x)=x+x36​ . (a) Find the intervals on which fff is increasing or decreasing. (b) Find the relative maxima and relative minima of fff .

Find the critical number(s), if any of the function f(t)=3t34−3t14f ( t ) = 3 t ^ { \frac { 3 } { 4 } } - 3 t ^ { \frac { 1 } { 4 } }f(t)=3t43​−3t41​ .

Find two positive numbers whose product is 196 and whose sum is a minimum.

Determine where the graph of f(x)=ln⁡∣x−4∣f ( x ) = \ln | x - 4 |f(x)=ln∣x−4∣ is concave upward and where it is concave downward. Also, find all inflection points of the function.

What is the minimum vertical distance between the parabolas y=x2+4 and y=x−x2?y = x ^ { 2 } + 4 \text { and } y = x - x ^ { 2 } ?y=x2+4 and y=x−x2?

Find the point on the line y=4x+8 that is closest to the origin. \text { Find the point on the line } y = 4 x + 8 \text { that is closest to the origin. } Find the point on the line y=4x+8 that is closest to the origin.

A manufacturer has been selling 1,200 television sets a week at $400\$ 400$400 each. A market survey indicates that for each $30\$ 30$30 rebate offered to the buyer, the number of sets sold will increase by 60 per week. Find the demand function.

The graph of the derivative f′(x)f ^ { \prime } ( x )f′(x) of a continuous function f\mathrm { f }f is shown. On what intervals is fff decreasing? .

Find fff . f′(t)=2t−4sin⁡t,f(0)=5f ^ { \prime } ( t ) = 2 t - 4 \sin t , \quad f ( 0 ) = 5f′(t)=2t−4sint,f(0)=5 Select the correct answer.

The average cost of producing xxx units of a commodity is given by the equation C(x)=20.4−0.0007xC ( x ) = 20.4 - 0.0007 xC(x)=20.4−0.0007x Find the marginal cost at a production level of 1,255 units.

Functions and Models

Limits and Derivatives

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 $f$ . $f ^ { \prime } ( x ) = 3 \cos ( x ) + 4 \sin ( x ) , f ( 0 ) = 7$

Find the dimensions of the rectangle of largest area that can be inscribed in an equilateral triangle of side $L = 9 \mathrm {~cm}$ if one side of the rectangle lies on the base of the triangle. Round your answer to the nearest tenth.

Find the dimensions of a rectangle of area $400 \mathrm { ft } ^ { 2 }$ that has the smallest possible perimeter.

Find $f$ . $f ^ { \prime } ( t ) = 2 t - 4 \sin t , \quad f ( 0 ) = 5$

$\text { Find the intervals where } f ( x ) = \frac { \log 9 x } { x } \text { is increasing and where it is decreasing. }$

Given that the graph of $f$ passes through the point $( 4,69 )$ and that the slope of its tangent line at $( x , f ( x ) )$ is $11 x - 3$ , find $f ( 1 )$ . Select the correct answer.

Given $f ( x ) = x + \frac { 36 } { x }$ . (a) Find the intervals on which $f$ is increasing or decreasing. (b) Find the relative maxima and relative minima of $f$ .

Find the critical number(s), if any of the function $f ( t ) = 3 t ^ { \frac { 3 } { 4 } } - 3 t ^ { \frac { 1 } { 4 } }$ .

Determine where the graph of $f ( x ) = \ln | x - 4 |$ is concave upward and where it is concave downward. Also, find all inflection points of the function.

What is the minimum vertical distance between the parabolas $y = x ^ { 2 } + 4 \text { and } y = x - x ^ { 2 } ?$

$\text { Find the point on the line } y = 4 x + 8 \text { that is closest to the origin. }$

A manufacturer has been selling 1,200 television sets a week at $\$ 400$ each. A market survey indicates that for each $\$ 30$ rebate offered to the buyer, the number of sets sold will increase by 60 per week. Find the demand function.

The graph of the derivative $f ^ { \prime } ( x )$ of a continuous function $\mathrm { f }$ is shown. On what intervals is $f$ decreasing? $The graph of the derivative f ^ { \prime } ( x ) of a continuous function \mathrm { f } is shown. On what intervals is f decreasing? .$ .

Find $f$ . $f ^ { \prime } ( t ) = 2 t - 4 \sin t , \quad f ( 0 ) = 5$ Select the correct answer.

The average cost of producing $x$ units of a commodity is given by the equation $C ( x ) = 20.4 - 0.0007 x$ Find the marginal cost at a production level of 1,255 units.