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Calculus 1 : Calculus

Study concepts, example questions & explanations for Calculus 1

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Example Questions

Example Question #388 : Rate Of Change

A cube is growing in size. What is the ratio of the rate of growth of the cube's surface area to the rate of growth of its sides when its sides have length 3?

Possible Answers:

Correct answer:

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 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, knowing the length of the sides, simply divide to find the ratio between the rate of change of the surface area and diagonal:

$12s\frac{ds}{dt}=(\phi)\frac{ds}{dt}$

$\phi=12s$

$\phi =12(3)=36$

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Example Question #389 : Rate Of Change

A cube is diminishing in size. What is the ratio of the rate of loss of the cube's surface area to the rate of loss of its sides when its sides have length $\frac{1}{24}$ ?

Possible Answers:

$\frac{1}{3}$

$\frac{1}{6}$

$\frac{1}{12}$

$\frac{1}{4}$

$\frac{1}{2}$

Correct answer:

$\frac{1}{2}$

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 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, knowing the length of the sides, simply divide to find the ratio between the rate of change of the surface area and diagonal:

$12s\frac{ds}{dt}=(\phi)\frac{ds}{dt}$

$\phi=12s$

$\phi =12(\frac{1}{24})=\frac{1}{2}$

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Example Question #390 : Rate Of Change

Find the slope of the following function at x=2 .

$f(x)= x^2 + \frac{1}{6}x^6$

Possible Answers:

Correct answer:

Explanation:

In order to find the slope of a function, you must first differentiate that function.

In this case, the derivative of the given function is: f'(x) = 2x + x^5

Then, plug x=2 into the function to get the slope: f'(2) = 2(2) + (2)^5

Therefore, the slope is:

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Example Question #3304 : Calculus

Find the slope of the following function at x=0 .

$f(x)= \sin x + (x-2)^2$

Possible Answers:

Correct answer:

Explanation:

In order to find the slope of a function, you must first differentiate that function.

In this case, the derivative of the given function is: $f'(x)= \cos x + 2(x-2)$

Then, plug x=0 into the function to get the slope: $f'(0)= \cos (0) + 2((0)-2)$

Therefore, the slope is:

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Example Question #3305 : Calculus

Find the slope of the following function at x=1 .

f(x)= (e^x)^2

Possible Answers:

23.6

14.8

3.21

32.8

Correct answer:

14.8

Explanation:

In order to find the slope of a function, you must first differentiate that function.

In this case, the derivative of the given function is: $f'(x)= 2e^{2x}$

Then, plug x=1 into the function to get the slope: $f'(1)= 2e^{2(1)}$

Therefore, the slope is: 14.8

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Example Question #3306 : Calculus

Find the slope of the following function at x=3 .

$f(x)= \cos(x^3)$

Possible Answers:

-27.8

27.8

-25.8

29.8

25.8

Correct answer:

-25.8

Explanation:

In order to find the slope of a function, you must first differentiate that function.

In this case, the derivative of the given function is: $f'(x)= -\sin(x^3) 3x^2$

Then, plug x=3 into the function to get the slope: $f'(3)= -\sin((3)^3) 3(3)^2$

Therefore, the slope is: -25.8

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Example Question #3307 : Calculus

Find the slope of the following function at x=0 .

f(x)= 5x^3 + (2x - ln(x))^2

Possible Answers:

undefined

Correct answer:

undefined

Explanation:

In order to find the slope of a function, you must first differentiate that function.

In this case, the derivative of the given function is: $f'(x)= 15x^2 + 2(2x - ln(x))(2-\frac{1}{x})$

Then, plug x=0 into the function to get the slope: $f'(0)= 15(0)^2 + 2(2(0) - ln(0))(2-\frac{1}{(0)})$

Therefore, the slope is: undefined

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Example Question #3308 : Calculus

Find the slope of the following function at x=-1 .

$f(x)= ln(2x) + \cos (\sin x)$

Possible Answers:

-1.5

0.8

1.2

-0.6

Correct answer:

-0.6

Explanation:

In order to find the slope of a function, you must first differentiate that function.

In this case, the derivative of the given function is: $f'(x)= \frac{1}{x} -\sin (\sin x) (\cos x)$

Then, plug x=-1 into the function to get the slope: $f'(-1)= \frac{1}{(-1)} -\sin (\sin (-1)) (\cos (-1))$

Therefore, the slope is: -0.6

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Example Question #3309 : Calculus

A cube is diminishing in size. What is the ratio of the rate of loss of the cube's volume to the rate of loss of its diagonal when its sides have length $13\sqrt{3}$ ?

Possible Answers:

$351\sqrt{3}$

$3\sqrt5$

$13\sqrt{3}$

$507\sqrt{3}$

$169\sqrt{3}$

Correct answer:

$507\sqrt{3}$

Explanation:

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

V=s^3

$d=s\sqrt{3}$

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

$\frac{dV}{dt}=3s^2\frac{ds}{dt}$

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

Now, knowing the length of the sides, simply divide to find the ratio between the rate of change of the volume and diagonal:

$3s^2\frac{ds}{dt}=(\phi)\sqrt{3}\frac{ds}{dt}$

$\phi=s^2\sqrt{3}$

$\phi =(13\sqrt{3})^2\sqrt{3}=507\sqrt{3}$

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Example Question #3310 : Calculus

What is the slope at x=3 given the following function: $f(x) = 12x^{\frac{1}{2}} + (6x^2 - 5x +2)^2$

Possible Answers:

1,545.5

545.5

3,545.5

2,545.5

5,545.5

Correct answer:

2,545.5

Explanation:

In order to find the slope of a function, you must find its derivative.

In this case, we must find the derivative of the following: $f(x) = 12x^{\frac{1}{2}} + (6x^2 - 5x +2)^2$

That is done by doing the following: $f'(x) = (\frac{1}{2})12x^{(\frac{1}{2}-1)} + (2)(6x^2 - 5x +2)^{(2-1)}((2)6x^{(2-1)}- (1)5x^{(1-1)}+0 )$

So, the derivative is: $f'(x) = 6x^{-\frac{1}{2}} + 2(6x^2 - 5x +2) (12x- 5 )$

Then, you plug x=3 into the derivative: $f'(3) = 6(3)^{-\frac{1}{2}} + 2(6(3)^2 - 5(3) +2) (12(3)- 5 )$

Therefore, the answer is: 2,545.5

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Calculus 1 : Calculus

Study concepts, example questions & explanations for Calculus 1

All Calculus 1 Resources

Example Questions

Example Question #388 : Rate Of Change

Example Question #389 : Rate Of Change

Example Question #390 : Rate Of Change

Example Question #3304 : Calculus

Example Question #3305 : Calculus

Example Question #3306 : Calculus

Example Question #3307 : Calculus

Example Question #3308 : Calculus

Example Question #3309 : Calculus

Example Question #3310 : Calculus

All Calculus 1 Resources

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