Exam 3: Differentiation Rules

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

(Short Answer)

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Question 121

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 122

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

(Multiple Choice)

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Question 123

A right circular cylinder is inscribed in a sphere of radius $r = 5 \mathrm {~cm}$ . Find the largest possible surface area of such a cylinder. Round the result to the nearest hundredth.

(Short Answer)

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Question 124

$\text { Find the slant asymptote of the graph of } f ( x ) = \frac { x ^ { 2 } + 4 x + 3 } { 2 x - 6 } \text { using the curve-sketching guidelines }$

(Short Answer)

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Question 125

Find the critical number(s), if any, of the function $f ( x ) = e ^ { x ^ { 2 } - 10 x }$ .

(Multiple Choice)

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Question 126

$\text { What is the minimum vertical distance between the parabolas } y = x ^ { 2 } + 1 \text { and } y = x - x ^ { 2 } \text { ? }$

(Short Answer)

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Question 127

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.

(Short Answer)

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Question 128

Find an equation of the line through the point $( 8,16 )$ that cuts off the least area from the first quadrant.

(Short Answer)

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Question 129

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

(Multiple Choice)

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Question 130

Find the minimum points of the function. $F ( x ) = x ^ { 4 } - 108 x$

(Short Answer)

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Question 131

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. $f ( x ) = \sin 5 \pi x , \left[ - \frac { 2 } { 5 } , \frac { 2 } { 5 } \right]$

(Multiple Choice)

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Question 132

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

(Short Answer)

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Question 133

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 134

Evaluate $f ( x ) = \sin \left( x ^ { 2 } \right)$ , and tell whether its antiderivative $F$ is increasing or decreasing at the point $x = - 4$ radians.

(Short Answer)

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Question 135

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

(Multiple Choice)

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Question 136

$\text { Find any absolute or local maximum and minimum values of } f ( x ) = 10 - 2 x \text { if } x \geq 6 \text {. }$ Select the correct answer.

(Multiple Choice)

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Question 137

Determine where the graph of the function $f ( x ) = 9 x - \sqrt { 4 - x ^ { 2 } }$ is concave upward and where it is concave downward. Also, find all inflection points of the function. Select the correct answer.

(Multiple Choice)

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Question 138

Determine where the graph of the function $f ( x ) = x ^ { 3 } - 6 x$ is concave upward and where it is concave downward. Also, find all inflection points of the function.

(Short Answer)

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Question 139

$\text { Find the absolute maximum value of } y = \sqrt { 9 - x ^ { 2 } } \text { on the interval } [ - 9,9 ] \text {. }$

(Short Answer)

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Question 140

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

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

A right circular cylinder is inscribed in a sphere of radius $r = 5 \mathrm {~cm}$ . Find the largest possible surface area of such a cylinder. Round the result to the nearest hundredth.

$\text { Find the slant asymptote of the graph of } f ( x ) = \frac { x ^ { 2 } + 4 x + 3 } { 2 x - 6 } \text { using the curve-sketching guidelines }$

Find the critical number(s), if any, of the function $f ( x ) = e ^ { x ^ { 2 } - 10 x }$ .

$\text { What is the minimum vertical distance between the parabolas } y = x ^ { 2 } + 1 \text { and } y = x - x ^ { 2 } \text { ? }$

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.

Find an equation of the line through the point $( 8,16 )$ that cuts off the least area from the first quadrant.

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

Find the minimum points of the function. $F ( x ) = x ^ { 4 } - 108 x$

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. $f ( x ) = \sin 5 \pi x , \left[ - \frac { 2 } { 5 } , \frac { 2 } { 5 } \right]$

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

Evaluate $f ( x ) = \sin \left( x ^ { 2 } \right)$ , and tell whether its antiderivative $F$ is increasing or decreasing at the point $x = - 4$ radians.

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

$\text { Find any absolute or local maximum and minimum values of } f ( x ) = 10 - 2 x \text { if } x \geq 6 \text {. }$ Select the correct answer.

Determine where the graph of the function $f ( x ) = 9 x - \sqrt { 4 - x ^ { 2 } }$ is concave upward and where it is concave downward. Also, find all inflection points of the function. Select the correct answer.

Determine where the graph of the function $f ( x ) = x ^ { 3 } - 6 x$ is concave upward and where it is concave downward. Also, find all inflection points of the function.

$\text { Find the absolute maximum value of } y = \sqrt { 9 - x ^ { 2 } } \text { on the interval } [ - 9,9 ] \text {. }$

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

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?

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

A right circular cylinder is inscribed in a sphere of radius r=5 cmr = 5 \mathrm {~cm}r=5 cm . Find the largest possible surface area of such a cylinder. Round the result to the nearest hundredth.

Find the critical number(s), if any, of the function f(x)=ex2−10xf ( x ) = e ^ { x ^ { 2 } - 10 x }f(x)=ex2−10x .

What is the minimum vertical distance between the parabolas y=x2+1 and y=x−x2 ? \text { What is the minimum vertical distance between the parabolas } y = x ^ { 2 } + 1 \text { and } y = x - x ^ { 2 } \text { ? } What is the minimum vertical distance between the parabolas y=x2+1 and y=x−x2 ?

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.

Find an equation of the line through the point (8,16)( 8,16 )(8,16) that cuts off the least area from the first quadrant.

Find the minimum points of the function. F(x)=x4−108xF ( x ) = x ^ { 4 } - 108 xF(x)=x4−108x

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

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?

Evaluate f(x)=sin⁡(x2)f ( x ) = \sin \left( x ^ { 2 } \right)f(x)=sin(x2) , and tell whether its antiderivative FFF is increasing or decreasing at the point x=−4x = - 4x=−4 radians.

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

Determine where the graph of the function f(x)=9x−4−x2f ( x ) = 9 x - \sqrt { 4 - x ^ { 2 } }f(x)=9x−4−x2​ is concave upward and where it is concave downward. Also, find all inflection points of the function. Select the correct answer.

Determine where the graph of the function f(x)=x3−6xf ( x ) = x ^ { 3 } - 6 xf(x)=x3−6x is concave upward and where it is concave downward. Also, find all inflection points of the function.

Find the absolute maximum value of y=9−x2 on the interval [−9,9]. \text { Find the absolute maximum value of } y = \sqrt { 9 - x ^ { 2 } } \text { on the interval } [ - 9,9 ] \text {. } Find the absolute maximum value of y=9−x2​ on the interval [−9,9].

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

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

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

A right circular cylinder is inscribed in a sphere of radius $r = 5 \mathrm {~cm}$ . Find the largest possible surface area of such a cylinder. Round the result to the nearest hundredth.

Find the critical number(s), if any, of the function $f ( x ) = e ^ { x ^ { 2 } - 10 x }$ .

$\text { What is the minimum vertical distance between the parabolas } y = x ^ { 2 } + 1 \text { and } y = x - x ^ { 2 } \text { ? }$

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.

Find an equation of the line through the point $( 8,16 )$ that cuts off the least area from the first quadrant.

Find the minimum points of the function. $F ( x ) = x ^ { 4 } - 108 x$

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

Evaluate $f ( x ) = \sin \left( x ^ { 2 } \right)$ , and tell whether its antiderivative $F$ is increasing or decreasing at the point $x = - 4$ radians.

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

Determine where the graph of the function $f ( x ) = 9 x - \sqrt { 4 - x ^ { 2 } }$ is concave upward and where it is concave downward. Also, find all inflection points of the function. Select the correct answer.

Determine where the graph of the function $f ( x ) = x ^ { 3 } - 6 x$ is concave upward and where it is concave downward. Also, find all inflection points of the function.

$\text { Find the absolute maximum value of } y = \sqrt { 9 - x ^ { 2 } } \text { on the interval } [ - 9,9 ] \text {. }$