Quiz about Calculus Skills Assessment

Question 1

Rachel is standing atop a 13 ft ladder. The ladder is leaning against a vertical wall. The ladder starts sliding away from the wall at a rate of 3 ft/sec. How fast is the ladder sliding down the wall when the tip of the ladder is 5 ft high?

Accepted Answer

-7.2 ft/sec

Answer

3 ft/sec

Answer

7.2 ft/sec

Answer

12 ft/sec

Question 2

Evaluate the limit: $$\lim_{x o 0} \frac{\sin(x)}{x}$$ .

Accepted Answer

1

Answer

0

Answer

Infinity

Answer

Does not exist

Question 3

What is the integral of $$\int 3x^2 dx$$ ?

Accepted Answer

$$x^3 + C$$

Answer

$$9x^2 + C$$

Answer

$$x^3 + 3x + C$$

Answer

$$x^3/3 + C$$

Question 4

Find the derivative of $$f(x) = \ln(x)$$ .

Accepted Answer

$$1/x$$

Answer

$$x/\ln(x)$$

Answer

$$\ln(x+1)$$

Answer

$$e^x$$

Question 5

If a balloon is being filled with air such that its volume increases at a rate of $$5 cm^3/sec$$ , how fast is the radius of the balloon increasing when the radius is 2 cm? Assume the balloon is a perfect sphere.

Accepted Answer

$$5/8\pi cm/sec$$

Answer

$$5/4\pi cm/sec$$

Answer

$$1/4\pi cm/sec$$

Answer

$$1/8\pi cm/sec$$

Question 6

What is the maximum value of the function $$f(x) = 3x^2 - 12x + 9$$ ?

Accepted Answer

-3

Answer

9

Answer

0

Answer

3

Question 7

Find the derivative of $$g(x) = e^{2x}$$ .

Accepted Answer

$$2e^{2x}$$

Answer

$$e^{2x}$$

Answer

$$2x e^{2x}$$

Answer

$$e^{x^2}$$

Question 8

Evaluate the integral $$\int_0^1 (x^4 - 2x^2 + 1)dx$$ .

Accepted Answer

1/3

Answer

1/2

Answer

1/4

Answer

1/5

Question 9

What is the limit of $$\frac{e^x - 1}{x}$$ as $$x$$ approaches 0?

Accepted Answer

1

Answer

0

Answer

e

Answer

Does not exist

Question 10

If $$y = 3t^4 - 2t^3 + 7t - 5$$ , find $$\frac{dy}{dt}$$ .

Accepted Answer

$$12t^3 - 6t^2 + 7$$

Answer

$$12t^3 - 6t^2 + 7t$$

Answer

$$12t^4 - 6t^3 + 7$$

Answer

$$12t^3 - 6t + 7$$

Question 11

What is the derivative of $$f(x) = \cos(x)$$ ?

Accepted Answer

$$-\sin(x)$$

Answer

$$\sin(x)$$

Answer

$$-\cos(x)$$

Answer

$$\cos(x)$$

Question 12

Evaluate the integral $$\int (2x + 3)dx$$ .

Accepted Answer

$$x^2 + 3x + C$$

Answer

$$x^2 + 3x^2 + C$$

Answer

$$2x^2/2 + 3x + C$$

Answer

$$x^2 + 3x^2/2 + C$$

Question 13

What is the limit of $$\frac{x^3 - 8}{x - 2}$$ as $$x$$ approaches 2?

Accepted Answer

12

Answer

4

Answer

8

Answer

0

Question 14

Given $$f(x) = x^2$$ and $$g(x) = 2x + 3$$ , find the derivative of the function $$h(x) = f(g(x))$$ .

Accepted Answer

$$2(2x + 3)(2) = 8x + 12$$

Answer

$$4x + 6$$

Answer

$$4x^2 + 12x + 9$$

Answer

$$2x + 3$$

Question 15

What is the integral of $$\int \frac{1}{x} dx$$ ?

Accepted Answer

$$\ln|x| + C$$

Answer

$$x \ln|x| + C$$

Answer

$$\frac{1}{x^2} + C$$

Answer

$$\ln(x^2) + C$$

Question 16

Find the derivative of $$f(x) = \sqrt{x}$$ .

Accepted Answer

$$\frac{1}{2\sqrt{x}}$$

Answer

$$\sqrt{x}$$

Answer

$$2\sqrt{x}$$

Answer

$$\frac{1}{\sqrt{x}}$$

Question 17

If $$y = \sin(x^2)$$ , find $$\frac{dy}{dx}$$ .

Accepted Answer

$$2x \cos(x^2)$$

Answer

$$\cos(x^2)$$

Answer

$$2 \sin(x^2)$$

Answer

$$x^2 \cos(x^2)$$

Question 18

Evaluate the integral $$\int_1^4 (3x + 2)dx$$ .

Accepted Answer

21

Answer

18

Answer

24

Answer

15

Question 19

What is the limit of $$\frac{ an(x)}{x}$$ as $$x$$ approaches 0?

Accepted Answer

1

Answer

0

Answer

Infinity

Answer

Does not exist

Question 20

If $$f(x) = x^3 - 3x^2 + 2x$$ , find the critical points.

Accepted Answer

$$x = 1, x = 2$$

Answer

$$x = 0, x = 1$$

Answer

$$x = 0, x = 2$$

Answer

$$x = -1, x = 2$$

Question 21

Find the derivative of $$g(x) = an(x)$$ .

Accepted Answer

$$\sec^2(x)$$

Answer

$$\sin^2(x)$$

Answer

$$\cos^2(x)$$

Answer

$$ an^2(x)$$

Question 22

Evaluate the integral $$\int (e^x - e^{-x})dx$$ .

Accepted Answer

$$e^x - e^{-x} + C$$

Answer

$$e^x + e^{-x} + C$$

Answer

$$2e^x + C$$

Answer

$$2e^{-x} + C$$

Question 23

What is the limit of $$\frac{1 - \cos(x)}{x^2}$$ as $$x$$ approaches 0?

Accepted Answer

1/2

Answer

0

Answer

1

Answer

Does not exist

Question 24

If $$y = e^{3x} + e^{-3x}$$ , find $$\frac{dy}{dx}$$ .

Accepted Answer

$$3e^{3x} + 3e^{-3x}$$

Answer

$$3e^{3x} - 3e^{-3x}$$

Answer

$$e^{3x} - e^{-3x}$$

Answer

$$e^{3x} + e^{-3x}$$

Question 25

What is the maximum value of the function $$f(x) = -x^2 + 4x + 5$$ ?

Accepted Answer

9

Answer

10

Answer

11

Answer

12

Question 26

Find the derivative of $$g(x) = \frac{1}{x^2}$$ .

Accepted Answer

$$-\frac{2}{x^3}$$

Answer

$$\frac{2}{x^3}$$

Answer

$$-\frac{1}{2x^2}$$

Answer

$$\frac{1}{2x^3}$$

Question 27

Evaluate the integral $$\int_0^{\pi} \sin(x) dx$$ .

Accepted Answer

2

Answer

0

Answer

1

Answer

$$\pi$$

Question 28

What is the limit of $$\frac{3x^2 + 2x - 5}{x^2 - 1}$$ as $$x$$ approaches 1?

Accepted Answer

5

Answer

0

Answer

10

Answer

Does not exist

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