Exam 3: Differentiation Rules

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?

(Short Answer)

4.7/5

(68)

Question 41

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)

4.9/5

(75)

Question 42

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.

(Short Answer)

4.8/5

(74)

Question 43

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)

5.0/5

(72)

Question 44

$\text { Find the critical number(s) of the function. }$ $y = x ^ { 2 } + 10 x$

(Multiple Choice)

4.8/5

(71)

Question 45

Find the local and absolute extreme values of the function on the given interval. $f ( x ) = x ^ { 3 } - 6 x ^ { 2 } + 9 x + 2$

(Short Answer)

4.9/5

(85)

Question 46

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

(Multiple Choice)

4.8/5

(72)

Question 47

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

(Short Answer)

4.8/5

(89)

Question 48

For what values of $a$ and $b$ is $( 2,2.5 )$ is an inflection point of the curve $x ^ { 2 } y + a x + b y = 0 ?$ What additional inflection points does the curve have?

(Short Answer)

4.7/5

(78)

Question 49

Find the point(s) of inflection of the graph of $f ( x ) = \cos ^ { - 1 } 7 x$ .

(Multiple Choice)

4.9/5

(89)

Question 50

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.

(Multiple Choice)

4.8/5

(84)

Question 51

$\text { For what values of } c \text { does the curve have maximum and minimum points? }$ $F ( x ) = 4 x ^ { 3 } + c x ^ { 2 } + 10 x$

(Multiple Choice)

5.0/5

(76)

Question 52

Estimate the value of $\sqrt [ 3 ] { 5 }$ by using three iterations of Newton's method to solve the equation $x ^ { 3 } - 5 = 0$ with initial estimate $x _ { 0 } = 2$ . Round your final estimate to four decimal places.

(Multiple Choice)

4.9/5

(90)

Question 53

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

(Short Answer)

4.9/5

(75)

Question 54

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

(Short Answer)

4.9/5

(72)

Question 55

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

(Multiple Choice)

4.8/5

(92)

Question 56

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)

5.0/5

(85)

Question 57

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

(Multiple Choice)

4.9/5

(73)

Question 58

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

(Short Answer)

4.8/5

(68)

Question 59

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)

4.8/5

(79)

Question 60

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?

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 $f$ .

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.

$\text { Find the critical number(s) of the function. }$ $y = x ^ { 2 } + 10 x$

Find the local and absolute extreme values of the function on the given interval. $f ( x ) = x ^ { 3 } - 6 x ^ { 2 } + 9 x + 2$

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

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

For what values of $a$ and $b$ is $( 2,2.5 )$ is an inflection point of the curve $x ^ { 2 } y + a x + b y = 0 ?$ What additional inflection points does the curve have?

Find the point(s) of inflection of the graph of $f ( x ) = \cos ^ { - 1 } 7 x$ .

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.

$\text { For what values of } c \text { does the curve have maximum and minimum points? }$ $F ( x ) = 4 x ^ { 3 } + c x ^ { 2 } + 10 x$

Estimate the value of $\sqrt [ 3 ] { 5 }$ by using three iterations of Newton's method to solve the equation $x ^ { 3 } - 5 = 0$ with initial estimate $x _ { 0 } = 2$ . Round your final estimate to four decimal places.

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

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

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

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

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

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

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.

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

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?

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 .

The average cost of producing x 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.

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 critical number(s) of the function. \text { Find the critical number(s) of the function. } Find the critical number(s) of the function. y=x2+10xy = x ^ { 2 } + 10 xy=x2+10x

Find the local and absolute extreme values of the function on the given interval. f(x)=x3−6x2+9x+2f ( x ) = x ^ { 3 } - 6 x ^ { 2 } + 9 x + 2f(x)=x3−6x2+9x+2

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 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].

For what values of aaa and bbb is (2,2.5)( 2,2.5 )(2,2.5) is an inflection point of the curve x2y+ax+by=0?x ^ { 2 } y + a x + b y = 0 ?x2y+ax+by=0? What additional inflection points does the curve have?

Find the point(s) of inflection of the graph of f(x)=cos⁡−17xf ( x ) = \cos ^ { - 1 } 7 xf(x)=cos−17x .

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.

Estimate the value of 53\sqrt [ 3 ] { 5 }35​ by using three iterations of Newton's method to solve the equation x3−5=0x ^ { 3 } - 5 = 0x3−5=0 with initial estimate x0=2x _ { 0 } = 2x0​=2 . Round your final estimate to four decimal places.

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 ?

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​

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

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.

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 .

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].

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.

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

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 $f$ .

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.

$\text { Find the critical number(s) of the function. }$ $y = x ^ { 2 } + 10 x$

Find the local and absolute extreme values of the function on the given interval. $f ( x ) = x ^ { 3 } - 6 x ^ { 2 } + 9 x + 2$

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

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

For what values of $a$ and $b$ is $( 2,2.5 )$ is an inflection point of the curve $x ^ { 2 } y + a x + b y = 0 ?$ What additional inflection points does the curve have?

Find the point(s) of inflection of the graph of $f ( x ) = \cos ^ { - 1 } 7 x$ .

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.

Estimate the value of $\sqrt [ 3 ] { 5 }$ by using three iterations of Newton's method to solve the equation $x ^ { 3 } - 5 = 0$ with initial estimate $x _ { 0 } = 2$ . Round your final estimate to four decimal places.

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

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

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

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

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

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

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.