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Calculus 1 : How to find rate of change

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Example Questions

Example Question #3701 : Calculus

A cube is growing in size. What is the rate of growth of the cube's surface area if its sides have a length of 8 and a rate of growth of 33?

Possible Answers:

69696

1264

2112

528

3168

Correct answer:

3168

Explanation:

Begin by writing the equations for a cube's dimensions. Namely its surface area in terms of the length of its sides:

A=6s^2

The rates of change of the surface area can be found by taking the derivative of each side of the equation with respect to time:

$\frac{dA}{dt}=12s\frac{ds}{dt}$

Now, with the rate equation known, we can solve for the rate of change of the surface area with what we know about the cube, namely that its sides have a length of 8 and a rate of growth of 33:

$\frac{dA}{dt}=12(8)(33)=3168$

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Example Question #792 : Rate Of Change

A cube is growing in size. What is the rate of growth of the cube's surface area if its sides have a length of 7 and a rate of growth of 34?

Possible Answers:

2856

679728

56644

97104

1666

Correct answer:

2856

Explanation:

Begin by writing the equations for a cube's dimensions. Namely its surface area in terms of the length of its sides:

A=6s^2

The rate of change of the surface area can be found by taking the derivative of each side of the equation with respect to time:

$\frac{dA}{dt}=12s\frac{ds}{dt}$

Now that we have a relationship between the surface area and the side parameters, we can use what we were told about the cube, in particular that its sides have a length of 7 and a rate of growth of 34:

$\frac{dA}{dt}=12(7)(34)=2856$

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Example Question #793 : Rate Of Change

A cube is growing in size. What is the rate of growth of the cube's surface area if its sides have a length of 6 and a rate of growth of 35?

Possible Answers:

22050

1260

44100

7350

2520

Correct answer:

2520

Explanation:

Begin by writing the equations for a cube's dimensions. Namely its surface area in terms of the length of its sides:

A=6s^2

The rate of change of the surface area can be found by taking the derivative of each side of the equation with respect to time:

$\frac{dA}{dt}=12s\frac{ds}{dt}$

Now that we have a relationship between the surface area and the side parameters, we can use what we were told about the cube, in particular that its sides have a length of 6 and a rate of growth of 35:

$\frac{dA}{dt}=12(6)(35)=2520$

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Example Question #794 : Rate Of Change

A cube is growing in size. What is the rate of growth of the cube's surface area if its sides have a length of 5 and a rate of growth of 36?

Possible Answers:

10800

5400

2160

180

900

Correct answer:

2160

Explanation:

Begin by writing the equations for a cube's dimensions. Namely its surface area in terms of the length of its sides:

A=6s^2

The rate of change of the surface area can be found by taking the derivative of each side of the equation with respect to time:

$\frac{dA}{dt}=12s\frac{ds}{dt}$

Now that we have a relationship between the surface area and the side parameters, we can use what we were told about the cube, in particular that its sides have a length of 5 and a rate of growth of 36:

$\frac{dA}{dt}=12(5)(36)=2160$

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Example Question #792 : Rate Of Change

A cube is growing in size. What is the rate of growth of the cube's surface area if its sides have a length of 4 and a rate of growth of 37?

Possible Answers:

1184

296

1776

592

14208

Correct answer:

1776

Explanation:

Begin by writing the equations for a cube's dimensions. Namely its surface area in terms of the length of its sides:

A=6s^2

The rate of change of the surface area can be found by taking the derivative of each side of the equation with respect to time:

$\frac{dA}{dt}=12s\frac{ds}{dt}$

Now that we have a relationship between the surface area and the side parameters, we can use what we were told about the cube, in particular that its sides have a length of 4 and a rate of growth of 37:

$\frac{dA}{dt}=12(4)(37)=1776$

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Example Question #796 : Rate Of Change

A cube is growing in size. What is the rate of growth of the cube's surface area if its sides have a length of 3 and a rate of growth of 38?

Possible Answers:

684

1368

4104

342

2052

Correct answer:

1368

Explanation:

Begin by writing the equations for a cube's dimensions. Namely its surface area in terms of the length of its sides:

A=6s^2

The rate of change of the surface area can be found by taking the derivative of each side of the equation with respect to time:

$\frac{dA}{dt}=12s\frac{ds}{dt}$

Now that we have a relationship between the surface area and the side parameters, we can use what we were told about the cube, in particular that its sides have a length of 3 and a rate of growth of 38:

$\frac{dA}{dt}=12(3)(38)=1368$

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Example Question #797 : Rate Of Change

A cube is growing in size. What is the rate of growth of the cube's surface area if its sides have a length of 2 and a rate of growth of 39?

Possible Answers:

936

312

Correct answer:

936

Explanation:

Begin by writing the equations for a cube's dimensions. Namely its surface area in terms of the length of its sides:

A=6s^2

The rate of change of the surface area can be found by taking the derivative of each side of the equation with respect to time:

$\frac{dA}{dt}=12s\frac{ds}{dt}$

Now that we have a relationship between the surface area and the side parameters, we can use what we were told about the cube, in particular that its sides have a length of 2 and a rate of growth of 39:

$\frac{dA}{dt}=12(2)(39)=936$

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Example Question #2681 : Functions

A cube is growing in size. What is the rate of growth of the cube's surface area if its sides have a length of 1 and a rate of growth of 40?

Possible Answers:

960

240

480

Correct answer:

480

Explanation:

Begin by writing the equations for a cube's dimensions. Namely its surface area in terms of the length of its sides:

A=6s^2

The rate of change of the surface area can be found by taking the derivative of each side of the equation with respect to time:

$\frac{dA}{dt}=12s\frac{ds}{dt}$

Now that we have a relationship between the surface area and the side parameters, we can use what we were told about the cube, in particular that its sides have a length of 1 and a rate of growth of 40:

$\frac{dA}{dt}=12(1)(40)=480$

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Example Question #791 : Rate Of Change

Find the rate of change of y=sec(2x) from $x:(0, \pi)$

Possible Answers:

$\frac{2}{\pi}$

$\frac{-2}{\pi}$

$\frac{1}{\pi}$

Correct answer:

Explanation:

To do rate of change , remember it is equivalent to finding slope.

$m=\frac{f(\pi)-f(0)}{\pi-0}=\frac{sec(2\pi)-sec(0)}{\pi-0} =\frac{1-1}{\pi-0} = 0$

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Example Question #2683 : Functions

The position of a car is defined by the equation $r(t)=5t^{2}-7t+6$ . What is the average velocity of the car between t=2 and t=4 ?

Possible Answers:

$v_{average}=29$

$v_{average}=20$

$v_{average}=0$

$v_{average}=58$

Correct answer:

$v_{average}=29$

Explanation:

The average velocity of an object between t=a and t=b is given by the equation

$v_{average}=\frac{\Delta y}{\Delta x}=\frac{r(b)-r(a)}{b-a}$

In this problem,

$r(4)=5*(4)^{2}-7(4)+6=80-28+6=58$

$r(2)=5*(2)^{2}-7(2)+6=20-14+6=0$

$v_{average}=\frac{r(4)-r(2)}{4-2}=\frac{58-0}{4-2}=29$

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Calculus 1 : How to find rate of change

Study concepts, example questions & explanations for Calculus 1

All Calculus 1 Resources

Example Questions

Example Question #3701 : Calculus

Example Question #792 : Rate Of Change

Example Question #793 : Rate Of Change

Example Question #794 : Rate Of Change

Example Question #792 : Rate Of Change

Example Question #796 : Rate Of Change

Example Question #797 : Rate Of Change

Example Question #2681 : Functions

Example Question #791 : Rate Of Change

Example Question #2683 : Functions

All Calculus 1 Resources

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