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Calculus 1 : Calculus

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Example Questions

Example Question #3461 : Calculus

A cube is diminishing in size. What is the length of the sides of the cube at the time that the rate of shrinkage of the cube's surface area is equal to $\frac{26}{3}$ times the rate of shrinkage of its diagonal?

Possible Answers:

$\frac{13\sqrt{3}}{6}$

$\frac{18\sqrt{3}}{13}$

$\frac{13\sqrt{3}}{9}$

$\frac{13\sqrt{3}}{18}$

$\frac{26\sqrt{3}}{13}$

Correct answer:

$\frac{13\sqrt{3}}{18}$

Explanation:

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

A=6s^2

$d=s\sqrt{3}$

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

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

$\frac{dd}{dt}=\sqrt{3}\frac{ds}{dt}$

Now, solve for the length of the side of the cube to satisfy the problem condition, the rate of shrinkage of the cube's surface area is equal to $\frac{26}{3}$ times the rate of shrinkage of its diagonal:

$12s\frac{ds}{dt}=(\frac{26}{3})\sqrt{3}\frac{ds}{dt}$

$s=(\frac{26}{3})\frac{\sqrt{3}}{12}=\frac{13\sqrt{3}}{18}$

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Example Question #3462 : Calculus

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

$f(x) = 2x^4 - \cos x$

Possible Answers:

8.8

2.4

6.3

5.9

Answer not listed

Correct answer:

8.8

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: $f'(x) = 8x^{3} + \sin x$

Then plug x=1 into the derivative function: $f'(1) = 8({\color{Blue} 1})^{3} + \sin ({\color{Blue} 1})$

Therefore, the slope is: 8.8

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Example Question #3463 : Calculus

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

f(x) = 23+ e

Possible Answers:

Answer not listed

$\infty$

Correct answer:

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) = 0

Then plug x=1 into the derivative function: f'(1) = 0

Therefore, the slope is:

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Example Question #3461 : Calculus

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

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

Possible Answers:

Answer not listed

$\frac{7}{16}$

$\frac{3}{8}$

$\frac{2}{5}$

$\frac{9}{11}$

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 #3462 : Calculus

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

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

Possible Answers:

Answer not listed

565.2

10.3

2.5

12,488

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 #3463 : Calculus

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

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

Possible Answers:

448

Answer not listed

561

789

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 #3464 : Calculus

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

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

Possible Answers:

-0.84

8.5

Answer not listed

3.2

-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 #3465 : 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:

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

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

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

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

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

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 #3466 : 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 74 times the rate of growth of the surface area?

Possible Answers:

148

296

185

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 #2441 : Functions

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:

2048

512

1024

256

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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Calculus 1 : Calculus

Study concepts, example questions & explanations for Calculus 1

All Calculus 1 Resources

Example Questions

Example Question #3461 : Calculus

Example Question #3462 : Calculus

Example Question #3463 : Calculus

Example Question #3461 : Calculus

Example Question #3462 : Calculus

Example Question #3463 : Calculus

Example Question #3464 : Calculus

Example Question #3465 : Calculus

Example Question #3466 : Calculus

Example Question #2441 : Functions

All Calculus 1 Resources

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