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Calculus 1 : Rate of Change

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Example Questions

Example Question #551 : Rate Of Change

Find the slope at x=2 given the following function:

f(x) = ln(7x +2)

Possible Answers:

$\frac{3}{8}$

Answer not listed

$\frac{9}{11}$

$\frac{7}{16}$

$\frac{2}{5}$

Correct answer:

$\frac{7}{16}$

Explanation:

In order to find the slope of a certain point given a function, you must first find the derivative.

In this case the derivative is: $f'(x) = \frac{7}{7x+2}$

Then plug x=2 into the derivative function: $f'(2) = \frac{7}{7({\color{Blue} 2})+2}$

Therefore, the slope is: $\frac{7}{16}$

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Example Question #552 : Rate Of Change

Find the slope at x=4 given the following function:

$f(x) = e^{8x^2}$

Possible Answers:

10.3

12,488

Answer not listed

565.2

2.5

Correct answer:

2.5

Explanation:

In order to find the slope of a certain point given a function, you must first find the derivative.

In this case the derivative is: $f'(x) = 16xe^{8x^2}$

Then plug x=4 into the derivative function: $f'(4) = 16({\color{Blue} 4})e^{8({\color{Blue} 4})^2}$

Therefore, the slope is: 2.5

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Example Question #3461 : Calculus

Find the slope at x=1 given the following function:

f(x) = (5x^6 + 2x)^2

Possible Answers:

Answer not listed

789

448

561

Correct answer:

448

Explanation:

In order to find the slope of a certain point given a function, you must first find the derivative.

In this case the derivative is: $f'(x) = 2(5x^6 + 2x) (30x^{5} + 2)$

Then plug x=1 into the derivative function: $f'(1) = 2(5({\color{Blue} 1})^6 + 2({\color{Blue} 1})) (30({\color{Blue} 1})^{5} + 2)$

Therefore, the slope is: 448

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Example Question #3462 : Calculus

Find the slope at x=0 given the following function:

$f(x) = \cos(5x^2 - e^x)$

Possible Answers:

Answer not listed

8.5

3.2

-0.84

-4.9

Correct answer:

-0.84

Explanation:

In order to find the slope of a certain point given a function, you must first find the derivative.

In this case the derivative is: $f'(x) = -\sin(5x^2 - e^x) (10x - e^x)$

Then plug x=0 into the derivative function: $f'(0) = -\sin(5({\color{Blue} 0})^2 - e^{\color{Blue} 0}) (10({\color{Blue} 0}) - e^{\color{Blue} 0})$

Therefore, the slope is: -0.84

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Example Question #3463 : Calculus

A spherical balloon is being filled with air. What is the volume of the sphere at the instance the rate of growth of the volume is $\frac{1}{6}$ times the rate of growth of the surface area?

Possible Answers:

$\frac{8}{243}\pi$

$\frac{4}{81}\pi$

$\frac{4}{243}\pi$

$\frac{2}{81}\pi$

$\frac{4}{27}\pi$

Correct answer:

$\frac{4}{81}\pi$

Explanation:

Let's begin by writing the equations for the volume and surface area of a sphere with respect to the sphere's radius:

$V=\frac{4}{3}\pi r^3$

$A=4\pi r^2$

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

$\frac{dV}{dt}=4\pi r^2 \frac{dr}{dt}$

$\frac{dA}{dt}=8\pi r \frac{dr}{dt}$

The rate of change of the radius is going to be the same for the sphere. So given our problem conditions, the rate of growth of the volume is $\frac{1}{6}$ times the rate of growth of the surface area, let's solve for a radius that satisfies it.

$4\pi r^2 \frac{dr}{dt}=(\frac{1}{6})8\pi r \frac{dr}{dt}$

$r =(\frac{1}{6})2$

$r=\frac{1}{3}$

Now to determine the volume at this point in time:

$V=\frac{4}{3}\pi r^3$

$V=\frac{4}{3}\pi (\frac{1}{3})^3=\frac{4}{81}\pi$

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Example Question #555 : Rate Of Change

A spherical balloon is being filled with air. What is the radius of the sphere at the instance the rate of growth of the volume is 74 times the rate of growth of the surface area?

Possible Answers:

185

148

296

Correct answer:

148

Explanation:

Let's begin by writing the equations for the volume and surface area of a sphere with respect to the sphere's radius:

$V=\frac{4}{3}\pi r^3$

$A=4\pi r^2$

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

$\frac{dV}{dt}=4\pi r^2 \frac{dr}{dt}$

$\frac{dA}{dt}=8\pi r \frac{dr}{dt}$

The rate of change of the radius is going to be the same for the sphere. So given our problem conditions, the rate of growth of the volume is 74 times the rate of growth of the surface area, let's solve for a radius that satisfies it.

$4\pi r^2 \frac{dr}{dt}=(74)8\pi r \frac{dr}{dt}$

r =(74)2

r=148

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Example Question #556 : Rate Of Change

A spherical balloon is being filled with air. What is the radius of the sphere at the instance the rate of growth of the volume is 128 times the rate of growth of the surface area?

Possible Answers:

1024

256

512

2048

Correct answer:

256

Explanation:

Let's begin by writing the equations for the volume and surface area of a sphere with respect to the sphere's radius:

$V=\frac{4}{3}\pi r^3$

$A=4\pi r^2$

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

$\frac{dV}{dt}=4\pi r^2 \frac{dr}{dt}$

$\frac{dA}{dt}=8\pi r \frac{dr}{dt}$

The rate of change of the radius is going to be the same for the sphere. So given our problem conditions, the rate of growth of the volume is 128 times the rate of growth of the surface area, let's solve for a radius that satisfies it.

$4\pi r^2 \frac{dr}{dt}=(128)8\pi r \frac{dr}{dt}$

r =(128)2

r=256

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Example Question #3471 : Calculus

A spherical balloon is being filled with air. What is the radius of the sphere at the instance the rate of growth of the volume is 81 times the rate of growth of the surface area?

Possible Answers:

162

324

243

144

Correct answer:

162

Explanation:

Let's begin by writing the equations for the volume and surface area of a sphere with respect to the sphere's radius:

$V=\frac{4}{3}\pi r^3$

$A=4\pi r^2$

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

$\frac{dV}{dt}=4\pi r^2 \frac{dr}{dt}$

$\frac{dA}{dt}=8\pi r \frac{dr}{dt}$

The rate of change of the radius is going to be the same for the sphere. So given our problem conditions, the rate of growth of the volume is 81 times the rate of growth of the surface area, let's solve for a radius that satisfies it.

$4\pi r^2 \frac{dr}{dt}=(81)8\pi r \frac{dr}{dt}$

r =(81)2

r=162

Report an Error

Example Question #3472 : Calculus

A spherical balloon is being filled with air. What is the radius of the sphere at the instance the rate of growth of the volume is 116 times the rate of growth of the surface area?

Possible Answers:

145

232

290

Correct answer:

232

Explanation:

Let's begin by writing the equations for the volume and surface area of a sphere with respect to the sphere's radius:

$V=\frac{4}{3}\pi r^3$

$A=4\pi r^2$

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

$\frac{dV}{dt}=4\pi r^2 \frac{dr}{dt}$

$\frac{dA}{dt}=8\pi r \frac{dr}{dt}$

The rate of change of the radius is going to be the same for the sphere. So given our problem conditions, the rate of growth of the volume is 116 times the rate of growth of the surface area, let's solve for a radius that satisfies it.

$4\pi r^2 \frac{dr}{dt}=(116)8\pi r \frac{dr}{dt}$

r =(116)2

r=232

Report an Error

Example Question #3473 : Calculus

A spherical balloon is being filled with air. What is the diameter of the sphere at the instance the rate of growth of the volume is 54 times the rate of growth of the surface area?

Possible Answers:

108

243

216

Correct answer:

216

Explanation:

Let's begin by writing the equations for the volume and surface area of a sphere with respect to the sphere's radius:

$V=\frac{4}{3}\pi r^3$

$A=4\pi r^2$

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

$\frac{dV}{dt}=4\pi r^2 \frac{dr}{dt}$

$\frac{dA}{dt}=8\pi r \frac{dr}{dt}$

The rate of change of the radius is going to be the same for the sphere. So given our problem conditions, the rate of growth of the volume is 54 times the rate of growth of the surface area, let's solve for a radius that satisfies it.

$4\pi r^2 \frac{dr}{dt}=(54)8\pi r \frac{dr}{dt}$

r =(54)2

r=108

The diameter is then

d=2r

d=216

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Calculus 1 : Rate of Change

Study concepts, example questions & explanations for Calculus 1

All Calculus 1 Resources

Example Questions

Example Question #551 : Rate Of Change

Example Question #552 : Rate Of Change

Example Question #3461 : Calculus

Example Question #3462 : Calculus

Example Question #3463 : Calculus

Example Question #555 : Rate Of Change

Example Question #556 : Rate Of Change

Example Question #3471 : Calculus

Example Question #3472 : Calculus

Example Question #3473 : Calculus

All Calculus 1 Resources

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