Free Printable Calculus Skills Assessment

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?

a)

3 ft/sec

b)

-7.2 ft/sec

c)

7.2 ft/sec

d)

12 ft/sec

2.

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

a)

0

b)

1

c)

Infinity

d)

Does not exist

3.

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

a)

$x^3 + C$

b)

$9x^2 + C$

c)

$x^3 + 3x + C$

d)

$x^3/3 + C$

4.

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

a)

$1/x$

b)

$x/\ln(x)$

c)

$\ln(x+1)$

d)

$e^x$

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.

a)

$5/4\pi cm/sec$

b)

$1/4\pi cm/sec$

c)

$5/8\pi cm/sec$

d)

$1/8\pi cm/sec$

6.

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

a)

9

b)

0

c)

-3

d)

3

7.

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

a)

$2e^{2x}$

b)

$e^{2x}$

c)

$2x e^{2x}$

d)

$e^{x^2}$

8.

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

a)

1/2

b)

1/3

c)

1/4

d)

1/5

9.

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

a)

0

b)

1

c)

e

d)

Does not exist

10.

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

a)

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

b)

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

c)

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

d)

$12t^3 - 6t + 7$

11.

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

a)

$-\sin(x)$

b)

$\sin(x)$

c)

$-\cos(x)$

d)

$\cos(x)$

12.

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

a)

$x^2 + 3x + C$

b)

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

c)

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

d)

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

13.

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

a)

4

b)

12

c)

8

d)

0

14.

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

a)

$4x + 6$

b)

$4x^2 + 12x + 9$

c)

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

d)

$2x + 3$

15.

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

a)

$\ln|x| + C$

b)

$x \ln|x| + C$

c)

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

d)

$\ln(x^2) + C$

16.

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

a)

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

b)

$\sqrt{x}$

c)

$2\sqrt{x}$

d)

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

17.

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

a)

$2x \cos(x^2)$

b)

$\cos(x^2)$

c)

$2 \sin(x^2)$

d)

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

18.

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

a)

21

b)

18

c)

24

d)

15

19.

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

a)

0

b)

1

c)

Infinity

d)

Does not exist

20.

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

a)

$x = 0, x = 1$

b)

$x = 1, x = 2$

c)

$x = 0, x = 2$

d)

$x = -1, x = 2$

21.

Find the derivative of $g(x) = \tan(x)$ .

a)

$\sec^2(x)$

b)

$\sin^2(x)$

c)

$\cos^2(x)$

d)

$\tan^2(x)$

22.

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

a)

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

b)

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

c)

$2e^x + C$

d)

$2e^{-x} + C$

23.

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

a)

0

b)

1/2

c)

1

d)

Does not exist

24.

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

a)

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

b)

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

c)

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

d)

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

25.

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

a)

9

b)

10

c)

11

d)

12

26.

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

a)

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

b)

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

c)

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

d)

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

27.

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

a)

0

b)

1

c)

2

d)

$\pi$

28.

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

a)

5

b)

0

c)

10

d)

Does not exist

Calculus Skills Assessment

Total questions: 28

Worksheet time: 2hrs 20mins

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