Question 1

What can you say about point of inflation for $f ( x ) = x e ^ { - 3 x }$ ?

Accepted Answer

A)  point of inflation gets closer to $\pm 6$ 
B)  point of inflation goes away from the origin 
C)  point of inflation gets closer to the origin 
D)  point of inflation goes away from $\pm 3$ 
A)  point of inflation gets closer to $\pm 6$ 
B)  point of inflation goes away from the origin 
C)  point of inflation gets closer to the origin 
D)  point of inflation goes away from $\pm 3$

Question 2

A fence $10 \mathrm { ft }$ tall runs parallel to a tall building at a distance of $5 \mathrm { ft }$ from the building. What is the length of the shortest ladder that will reach from the ground over the fence to the wall of the building? Round the result to the nearest hundredth.

Accepted Answer

The answer of A fence $10 \mathrm { ft }$...

Question 3

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

Accepted Answer

The answer of \(\text { What is the minimum vertical...

Question 4

The function $f ( t ) = \frac { t } { t - 8 }$ satisfies the hypotheses of the Mean Value Theorem on the interval $[ - 2,0 ]$. Find all values of $c$ that satisfy the conclusion of the theorem.

Accepted Answer

The answer of The function \(f ( t ) =...

Question 5

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

Accepted Answer

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

Question 6

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

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 7

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

Accepted Answer

A)  increasing: $( 9 e , \infty )$, decreasing: $( 0,9 e )$ 
B)  increasing: $\left( 0 , \frac { e } { 9 } \right)$, decreasing: $\left( \frac { e } { 9 } , \infty \right)$ 
C)  increasing: $( 90 , \infty )$, decreasing: $( 0,90 )$ 
D)  increasing: $\left( 0 , \frac { 9 } { 10 } \right)$, decreasing: $\left( \frac { 9 } { 10 } , \infty \right)$ 
A)  increasing: $( 9 e , \infty )$, decreasing: $( 0,9 e )$ 
B)  increasing: $\left( 0 , \frac { e } { 9 } \right)$, decreasing: $\left( \frac { e } { 9 } , \infty \right)$ 
C)  increasing: $( 90 , \infty )$, decreasing: $( 0,90 )$ 
D)  increasing: $\left( 0 , \frac { 9 } { 10 } \right)$, decreasing: $\left( \frac { 9 } { 10 } , \infty \right)$

Question 8

Find a cubic function $f ( x ) = a x ^ { 3 } - b x ^ { 2 } + c x - d$ that has a local maximum value of 112 at 1 and a local minimum value of $- 1,184$ at 7 .

Accepted Answer

The answer of Find a cubic function \(f ( x...

Question 9

Find $f$.
$$f ^ { tt } ( x ) = x ^ { 3 } + \sinh x , f ( 0 ) = 6 , f ( 2 ) = \frac { 13 } { 5 }$$

Accepted Answer

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

Question 10

Verify that the function satisfies the three hypotheses of Rolle's Theorem on the given interval. Then find all numbers c that satisfy the conclusion of Rolle's Theorem. $$f ( x ) = \sin 5 \pi x , \left[ - \frac { 2 } { 5 } , \frac { 2 } { 5 } \right]$$

Accepted Answer

A)  $c _ { 1 } = \pm \frac { 1 } { 10 } , c _ { 2 } = \pm \frac { 3 } { 10 }$ 
B)  $c _ { 1 } = \pm \frac { 1 } { 10 } , c _ { 2 } = \frac { 3 } { 10 }$ 
C)  $c _ { 1 } = - \frac { 1 } { 10 } , c _ { 2 } = \pm \frac { 1 } { 10 }$ 
D)  $c _ { 1 } = \frac { 1 } { 10 } , c _ { 2 } = \pm \frac { 3 } { 10 }$ 
E)  None of these 
A)  $c _ { 1 } = \pm \frac { 1 } { 10 } , c _ { 2 } = \pm \frac { 3 } { 10 }$ 
B)  $c _ { 1 } = \pm \frac { 1 } { 10 } , c _ { 2 } = \frac { 3 } { 10 }$ 
C)  $c _ { 1 } = - \frac { 1 } { 10 } , c _ { 2 } = \pm \frac { 1 } { 10 }$ 
D)  $c _ { 1 } = \frac { 1 } { 10 } , c _ { 2 } = \pm \frac { 3 } { 10 }$ 
E)  None of these

Question 11

Find the critical numbers of $f ( x ) = x ^ { 4 } ( x - 3 ) ^ { 3 }$.

Accepted Answer

A)  $0,3 , \frac { 12 } { 7 }$ 
B)  $0,4 , \frac { 7 } { 12 }$ 
C)  $0,4 , \frac { 12 } { 7 }$ 
D)  $0,3 , \frac { 12 } { 11 }$ 
E)  $0,3 , \frac { 7 } { 12 }$ 
A)  $0,3 , \frac { 12 } { 7 }$ 
B)  $0,4 , \frac { 7 } { 12 }$ 
C)  $0,4 , \frac { 12 } { 7 }$ 
D)  $0,3 , \frac { 12 } { 11 }$ 
E)  $0,3 , \frac { 7 } { 12 }$

Question 12

Find the absolute maximum value of $y = \sqrt { 16 - x ^ { 2 } }$ on the interval $[ - 6,6 ]$.

Accepted Answer

A)  7 
B)  5 
C)  8 
D)  6 
E)  4 
A)  7 
B)  5 
C)  8 
D)  6 
E)  4

Question 13

Find the critical numbers of $f ( x ) = x ^ { 4 } ( x - 3 ) ^ { 3 }$.

Accepted Answer

A)  $0,3 , \frac { 12 } { 7 }$ 
B)  $0,4 , \frac { 7 } { 12 }$ 
C)  $0,4 , \frac { 12 } { 7 }$ 
D)  $0,3 , \frac { 12 } { 11 }$ 
E)  $0,3 , \frac { 7 } { 12 }$ 
A)  $0,3 , \frac { 12 } { 7 }$ 
B)  $0,4 , \frac { 7 } { 12 }$ 
C)  $0,4 , \frac { 12 } { 7 }$ 
D)  $0,3 , \frac { 12 } { 11 }$ 
E)  $0,3 , \frac { 7 } { 12 }$

Question 14

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

Accepted Answer

The answer of \[\text { What is the minimum vertical...

Question 15

A steel pipe is being carried down a hallway 14 ft wide.At the end of the hall there is a right-angled turn into a narrower hallway 7 ft wide.What is the length of the longest pipe that can be carried horizontally around the corner?

Accepted Answer

The answer of A steel pipe is being carried down...

Question 16

Find the critical number(s), if any, of the function $g ( t ) = t ^ { 2 } \ln 9 t$.

Accepted Answer

A)  $\frac { 1 } { 9 } e ^ { \frac { - 1 } { 2 } }$ 
B)  $\frac { 1 } { 9 } e ^ { \frac { 1 } { 9 } }$ 
C)  $e ^ { \frac { - 1 } { 9 } }$ 
D)  $e ^ { \frac { 1 } { 9 } }$ 
A)  $\frac { 1 } { 9 } e ^ { \frac { - 1 } { 2 } }$ 
B)  $\frac { 1 } { 9 } e ^ { \frac { 1 } { 9 } }$ 
C)  $e ^ { \frac { - 1 } { 9 } }$ 
D)  $e ^ { \frac { 1 } { 9 } }$

Question 17

What constant acceleration is required to increase the speed of a car from $20 \mathrm { mi } / \mathrm { h }$ to $45 \mathrm { mi } / \mathrm { h }$ in 5 s?

Accepted Answer

The answer of What constant acceleration is required to increase...

Question 18

A production editor decided that a promotional flyer should have a 1-in. margin at the top and the bottom, and a $\frac { 1 } { 2 }$-in. margin on each side. The editor further stipulated that the flyer should have an area of 72 in. $^ { 2 }$. Determine the dimensions of the flyer that will result in the maximum printed area on the flyer.

Accepted Answer

A)  6 in. $\times 12$ in. 
B)  $3 \mathrm { in. } \times 24 \mathrm { in }$. 
C)  $6 \sqrt { 2 }$ in. $\times 6 \sqrt { 2 }$ in. 
D)  $3 \sqrt { 2 }$ in. $\times 3 \sqrt { 2 }$ in. 
A)  6 in. $\times 12$ in. 
B)  $3 \mathrm { in. } \times 24 \mathrm { in }$. 
C)  $6 \sqrt { 2 }$ in. $\times 6 \sqrt { 2 }$ in. 
D)  $3 \sqrt { 2 }$ in. $\times 3 \sqrt { 2 }$ in.

Question 19

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

Accepted Answer

A)  relative minimum: none, relative maximum: $f ( \ln 3 ) = - 7 \ln 7$ 
B)  relative minimum: $f ( \ln 7 ) = 4 - 7 \ln 7$, relative maximum: none 
C)  relative minimum: $f ( \ln 3 ) = - 7 \ln 7$, relative maximum: none 
D)  relative minimum: none, relative maximum: $f ( \ln 7 ) = 4 - 7 \ln 7$ 
A)  relative minimum: none, relative maximum: $f ( \ln 3 ) = - 7 \ln 7$ 
B)  relative minimum: $f ( \ln 7 ) = 4 - 7 \ln 7$, relative maximum: none 
C)  relative minimum: $f ( \ln 3 ) = - 7 \ln 7$, relative maximum: none 
D)  relative minimum: none, relative maximum: $f ( \ln 7 ) = 4 - 7 \ln 7$

Question 20

$$\text { Find two positive numbers whose product is } 196 \text { and whose sum is a minimum. }$$

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

What can you say about point of inflation for $f ( x ) = x e ^ { - 3 x }$ ?

A fence $10 \mathrm { ft }$ tall runs parallel to a tall building at a distance of $5 \mathrm { ft }$ from the building. What is the length of the shortest ladder that will reach from the ground over the fence to the wall of the building? Round the result to the nearest hundredth.

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

The function $f ( t ) = \frac { t } { t - 8 }$ satisfies the hypotheses of the Mean Value Theorem on the interval $[ - 2,0 ]$ . Find all values of $c$ that satisfy the conclusion of the theorem.

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

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

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

Find a cubic function $f ( x ) = a x ^ { 3 } - b x ^ { 2 } + c x - d$ that has a local maximum value of 112 at 1 and a local minimum value of $- 1,184$ at 7 .

Find $f$ . $f ^ { tt } ( x ) = x ^ { 3 } + \sinh x , f ( 0 ) = 6 , f ( 2 ) = \frac { 13 } { 5 }$

Verify that the function satisfies the three hypotheses of Rolle's Theorem on the given interval. Then find all numbers c that satisfy the conclusion of Rolle's Theorem. $f ( x ) = \sin 5 \pi x , \left[ - \frac { 2 } { 5 } , \frac { 2 } { 5 } \right]$

Find the critical numbers of $f ( x ) = x ^ { 4 } ( x - 3 ) ^ { 3 }$ .

Find the absolute maximum value of $y = \sqrt { 16 - x ^ { 2 } }$ on the interval $[ - 6,6 ]$ .

Find the critical numbers of $f ( x ) = x ^ { 4 } ( x - 3 ) ^ { 3 }$ .

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

A steel pipe is being carried down a hallway 14 ft wide.At the end of the hall there is a right-angled turn into a narrower hallway 7 ft wide.What is the length of the longest pipe that can be carried horizontally around the corner?

Find the critical number(s), if any, of the function $g ( t ) = t ^ { 2 } \ln 9 t$ .

What constant acceleration is required to increase the speed of a car from $20 \mathrm { mi } / \mathrm { h }$ to $45 \mathrm { mi } / \mathrm { h }$ in 5 s?

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

$\text { Find two positive numbers whose product is } 196 \text { and whose sum is a minimum. }$

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

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Partial Derivatives

Multiple Integrals

Vector Calculus

Second-Order Differential Equations

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Exam 3: Differentiation Rules

What can you say about point of inflation for f(x)=xe−3xf ( x ) = x e ^ { - 3 x }f(x)=xe−3x ?

A fence 10ft10 \mathrm { ft }10ft tall runs parallel to a tall building at a distance of 5ft5 \mathrm { ft }5ft from the building. What is the length of the shortest ladder that will reach from the ground over the fence to the wall of the building? Round the result to the nearest hundredth.

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

The function f(t)=tt−8f ( t ) = \frac { t } { t - 8 }f(t)=t−8t​ satisfies the hypotheses of the Mean Value Theorem on the interval [−2,0][ - 2,0 ][−2,0] . Find all values of ccc that satisfy the conclusion of the theorem.

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

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

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

Find a cubic function f(x)=ax3−bx2+cx−df ( x ) = a x ^ { 3 } - b x ^ { 2 } + c x - df(x)=ax3−bx2+cx−d that has a local maximum value of 112 at 1 and a local minimum value of −1,184- 1,184−1,184 at 7 .

Find fff . ftt(x)=x3+sinh⁡x,f(0)=6,f(2)=135f ^ { tt } ( x ) = x ^ { 3 } + \sinh x , f ( 0 ) = 6 , f ( 2 ) = \frac { 13 } { 5 }ftt(x)=x3+sinhx,f(0)=6,f(2)=513​

Find the critical numbers of f(x)=x4(x−3)3f ( x ) = x ^ { 4 } ( x - 3 ) ^ { 3 }f(x)=x4(x−3)3 .

Find the absolute maximum value of y=16−x2y = \sqrt { 16 - x ^ { 2 } }y=16−x2​ on the interval [−6,6][ - 6,6 ][−6,6] .

Find the critical numbers of f(x)=x4(x−3)3f ( x ) = x ^ { 4 } ( x - 3 ) ^ { 3 }f(x)=x4(x−3)3 .

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

A steel pipe is being carried down a hallway 14 ft wide.At the end of the hall there is a right-angled turn into a narrower hallway 7 ft wide.What is the length of the longest pipe that can be carried horizontally around the corner?

Find the critical number(s), if any, of the function g(t)=t2ln⁡9tg ( t ) = t ^ { 2 } \ln 9 tg(t)=t2ln9t .

What constant acceleration is required to increase the speed of a car from 20mi/h20 \mathrm { mi } / \mathrm { h }20mi/h to 45mi/h45 \mathrm { mi } / \mathrm { h }45mi/h in 5 s?

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

Find two positive numbers whose product is 196 and whose sum is a minimum. \text { Find two positive numbers whose product is } 196 \text { and whose sum is a minimum. } Find two positive numbers whose product is 196 and whose sum is a minimum.

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

What can you say about point of inflation for $f ( x ) = x e ^ { - 3 x }$ ?

A fence $10 \mathrm { ft }$ tall runs parallel to a tall building at a distance of $5 \mathrm { ft }$ from the building. What is the length of the shortest ladder that will reach from the ground over the fence to the wall of the building? Round the result to the nearest hundredth.

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

The function $f ( t ) = \frac { t } { t - 8 }$ satisfies the hypotheses of the Mean Value Theorem on the interval $[ - 2,0 ]$ . Find all values of $c$ that satisfy the conclusion of the theorem.

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

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

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

Find a cubic function $f ( x ) = a x ^ { 3 } - b x ^ { 2 } + c x - d$ that has a local maximum value of 112 at 1 and a local minimum value of $- 1,184$ at 7 .

Find $f$ . $f ^ { tt } ( x ) = x ^ { 3 } + \sinh x , f ( 0 ) = 6 , f ( 2 ) = \frac { 13 } { 5 }$

Find the critical numbers of $f ( x ) = x ^ { 4 } ( x - 3 ) ^ { 3 }$ .

Find the absolute maximum value of $y = \sqrt { 16 - x ^ { 2 } }$ on the interval $[ - 6,6 ]$ .

Find the critical numbers of $f ( x ) = x ^ { 4 } ( x - 3 ) ^ { 3 }$ .

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

Find the critical number(s), if any, of the function $g ( t ) = t ^ { 2 } \ln 9 t$ .

What constant acceleration is required to increase the speed of a car from $20 \mathrm { mi } / \mathrm { h }$ to $45 \mathrm { mi } / \mathrm { h }$ in 5 s?

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

$\text { Find two positive numbers whose product is } 196 \text { and whose sum is a minimum. }$