Exam 3: Differentiation Rules

The graph of the first derivative $f ^ { \prime } ( x )$ of a function $f$ is shown below. At what values of $x$ does $f$ have a local maximum or minimum? $The graph of the first derivative f ^ { \prime } ( x ) of a function f is shown below. At what values of x does f have a local maximum or minimum?$

(Short Answer)

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Question 141

Find the maximum or minimum point(s) of the function. $F ( x ) = \left( 1 - x ^ { 2 } \right) ^ { 2 } + 6 x ^ { 2 }$

(Short Answer)

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Question 142

$\text { Sketch the graph of the function } y = \cos ^ { 2 } 8 x \text { on } 0 \leq x \leq \frac { \pi } { 4 } \text { using the curve-sketching guidelines. }$

(Short Answer)

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Question 143

For what values of $c$ does the curve have maximum and minimum points for the given function $f ( x ) = c x ^ { 4 } - 5 x ^ { 2 } + 1 ?$ $\text { Select the correct answer. }$

(Essay)

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Question 144

Find the absolute minimum value(s) of $y = 2 x ^ { 2 } - 20 x + 9 \text { on the interval } [ 0,6 ]$

(Short Answer)

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Question 145

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?

(Short Answer)

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Question 146

How many points of inflection are on the graph of the function? $f ( x ) = 18 x ^ { 3 } + 5 x ^ { 2 } - 12 x - 20$ Select the correct answer.

(Multiple Choice)

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Question 147

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

(Essay)

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Question 148

Sketch the graph of the function $f ( x ) = x ^ { 3 } - 3 x$ using the curve-sketching guidelines.

(Multiple Choice)

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Question 149

How many real roots does the equation $x ^ { 5 } - 7 x + c = 0$ have in the interval $[ - 1,1 ]$ ?

(Multiple Choice)

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Question 150

An apple orchard has an average yield of 32 bushels of apples per tree if tree density is 30 trees per acre. For each unit increase in tree density, the yield decreases by 2 bushels per tree. How many trees per acre should be planted to maximize yield?

(Short Answer)

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Question 151

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.

(Short Answer)

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Question 152

Find the maximum and minimum points of the function. $F ( x ) = \frac { 2 x } { 1 + 4 x ^ { 2 } }$

(Short Answer)

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Question 153

Given $f ( x ) = x ^ { 2 } + 6 x$ . (a) Find the intervals on which $f$ is increasing or decreasing. (b) Find the relative maxima and relative minima of $f$ .

(Multiple Choice)

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Question 154

For what values of $c$ does the curve have maximum and minimum points for the given function $f ( x ) = c x ^ { 4 } - 5 x ^ { 2 } + 1 ?$ Select the correct answer.

(Multiple Choice)

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Question 155

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

(Short Answer)

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Question 156

$\text { What can you say about point of inflation for } f ( x ) = x e ^ { - 3 x } \text { ? }$ Select the correct answer.

(Multiple Choice)

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Question 157

Find the absolute maximum and absolute minimum values, if any, of the function $f ( x ) = 8 + 4 \sin 2 x$ on $\left[ 0 , \frac { \pi } { 2 } \right]$ .

(Short Answer)

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Question 158

Find the critical number(s), if any, of the function $f ( x ) = x ^ { 3 } - 21 x + 4$ .

(Multiple Choice)

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Question 159

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.

(Short Answer)

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Question 160

Showing 141 - 160 of 160

Find the maximum or minimum point(s) of the function. $F ( x ) = \left( 1 - x ^ { 2 } \right) ^ { 2 } + 6 x ^ { 2 }$

$\text { Sketch the graph of the function } y = \cos ^ { 2 } 8 x \text { on } 0 \leq x \leq \frac { \pi } { 4 } \text { using the curve-sketching guidelines. }$

For what values of $c$ does the curve have maximum and minimum points for the given function $f ( x ) = c x ^ { 4 } - 5 x ^ { 2 } + 1 ?$ $\text { Select the correct answer. }$

Find the absolute minimum value(s) of $y = 2 x ^ { 2 } - 20 x + 9 \text { on the interval } [ 0,6 ]$

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?

How many points of inflection are on the graph of the function? $f ( x ) = 18 x ^ { 3 } + 5 x ^ { 2 } - 12 x - 20$ Select the correct answer.

Sketch the graph of the function $f ( x ) = x ^ { 3 } - 3 x$ using the curve-sketching guidelines.

How many real roots does the equation $x ^ { 5 } - 7 x + c = 0$ have in the interval $[ - 1,1 ]$ ?

An apple orchard has an average yield of 32 bushels of apples per tree if tree density is 30 trees per acre. For each unit increase in tree density, the yield decreases by 2 bushels per tree. How many trees per acre should be planted to maximize yield?

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.

Find the maximum and minimum points of the function. $F ( x ) = \frac { 2 x } { 1 + 4 x ^ { 2 } }$

Given $f ( x ) = x ^ { 2 } + 6 x$ . (a) Find the intervals on which $f$ is increasing or decreasing. (b) Find the relative maxima and relative minima of $f$ .

For what values of $c$ does the curve have maximum and minimum points for the given function $f ( x ) = c x ^ { 4 } - 5 x ^ { 2 } + 1 ?$ Select the correct answer.

$\text { What can you say about point of inflation for } f ( x ) = x e ^ { - 3 x } \text { ? }$ Select the correct answer.

Find the absolute maximum and absolute minimum values, if any, of the function $f ( x ) = 8 + 4 \sin 2 x$ on $\left[ 0 , \frac { \pi } { 2 } \right]$ .

Find the critical number(s), if any, of the function $f ( x ) = x ^ { 3 } - 21 x + 4$ .

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.

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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Exam 3: Differentiation Rules

The graph of the first derivative f′(x)f ^ { \prime } ( x )f′(x) of a function fff is shown below. At what values of xxx does fff have a local maximum or minimum?

Find the maximum or minimum point(s) of the function. F(x)=(1−x2)2+6x2F ( x ) = \left( 1 - x ^ { 2 } \right) ^ { 2 } + 6 x ^ { 2 }F(x)=(1−x2)2+6x2

For what values of ccc does the curve have maximum and minimum points for the given function f(x)=cx4−5x2+1?f ( x ) = c x ^ { 4 } - 5 x ^ { 2 } + 1 ?f(x)=cx4−5x2+1? Select the correct answer. \text { Select the correct answer. } Select the correct answer.

Find the absolute minimum value(s) of y=2x2−20x+9 on the interval [0,6]y = 2 x ^ { 2 } - 20 x + 9 \text { on the interval } [ 0,6 ]y=2x2−20x+9 on the interval [0,6]

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?

How many points of inflection are on the graph of the function? f(x)=18x3+5x2−12x−20f ( x ) = 18 x ^ { 3 } + 5 x ^ { 2 } - 12 x - 20f(x)=18x3+5x2−12x−20 Select the correct answer.

Sketch the graph of the function f(x)=x3−3xf ( x ) = x ^ { 3 } - 3 xf(x)=x3−3x using the curve-sketching guidelines.

How many real roots does the equation x5−7x+c=0x ^ { 5 } - 7 x + c = 0x5−7x+c=0 have in the interval [−1,1][ - 1,1 ][−1,1] ?

An apple orchard has an average yield of 32 bushels of apples per tree if tree density is 30 trees per acre. For each unit increase in tree density, the yield decreases by 2 bushels per tree. How many trees per acre should be planted to maximize yield?

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.

Find the maximum and minimum points of the function. F(x)=2x1+4x2F ( x ) = \frac { 2 x } { 1 + 4 x ^ { 2 } }F(x)=1+4x22x​

Given f(x)=x2+6xf ( x ) = x ^ { 2 } + 6 xf(x)=x2+6x . (a) Find the intervals on which fff is increasing or decreasing. (b) Find the relative maxima and relative minima of fff .

For what values of ccc does the curve have maximum and minimum points for the given function f(x)=cx4−5x2+1?f ( x ) = c x ^ { 4 } - 5 x ^ { 2 } + 1 ?f(x)=cx4−5x2+1? Select the correct answer.

A steel pipe is being carried down a hallway 14ft14 \mathrm { ft }14ft wide. At the end of the hall there is a right-angled turn into a narrower hallway 7ft7 \mathrm { ft }7ft wide. What is the length of the longest pipe that can be carried horizontally around the corner?

What can you say about point of inflation for f(x)=xe−3x ? \text { What can you say about point of inflation for } f ( x ) = x e ^ { - 3 x } \text { ? } What can you say about point of inflation for f(x)=xe−3x ? Select the correct answer.

Find the absolute maximum and absolute minimum values, if any, of the function f(x)=8+4sin⁡2xf ( x ) = 8 + 4 \sin 2 xf(x)=8+4sin2x on [0,π2]\left[ 0 , \frac { \pi } { 2 } \right][0,2π​] .

Find the critical number(s), if any, of the function f(x)=x3−21x+4f ( x ) = x ^ { 3 } - 21 x + 4f(x)=x3−21x+4 .

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.

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 the maximum or minimum point(s) of the function. $F ( x ) = \left( 1 - x ^ { 2 } \right) ^ { 2 } + 6 x ^ { 2 }$

For what values of $c$ does the curve have maximum and minimum points for the given function $f ( x ) = c x ^ { 4 } - 5 x ^ { 2 } + 1 ?$ $\text { Select the correct answer. }$

Find the absolute minimum value(s) of $y = 2 x ^ { 2 } - 20 x + 9 \text { on the interval } [ 0,6 ]$

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?

How many points of inflection are on the graph of the function? $f ( x ) = 18 x ^ { 3 } + 5 x ^ { 2 } - 12 x - 20$ Select the correct answer.

Sketch the graph of the function $f ( x ) = x ^ { 3 } - 3 x$ using the curve-sketching guidelines.

How many real roots does the equation $x ^ { 5 } - 7 x + c = 0$ have in the interval $[ - 1,1 ]$ ?

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.

Find the maximum and minimum points of the function. $F ( x ) = \frac { 2 x } { 1 + 4 x ^ { 2 } }$

Given $f ( x ) = x ^ { 2 } + 6 x$ . (a) Find the intervals on which $f$ is increasing or decreasing. (b) Find the relative maxima and relative minima of $f$ .

For what values of $c$ does the curve have maximum and minimum points for the given function $f ( x ) = c x ^ { 4 } - 5 x ^ { 2 } + 1 ?$ Select the correct answer.

$\text { What can you say about point of inflation for } f ( x ) = x e ^ { - 3 x } \text { ? }$ Select the correct answer.

Find the absolute maximum and absolute minimum values, if any, of the function $f ( x ) = 8 + 4 \sin 2 x$ on $\left[ 0 , \frac { \pi } { 2 } \right]$ .

Find the critical number(s), if any, of the function $f ( x ) = x ^ { 3 } - 21 x + 4$ .

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.