Question 1

Given the graph in $$image_1$$, what type of discontinuity is present at $$x = 5$$?

Accepted Answer

Jump discontinuity

Question 2

Evaluate the following limit: $$\lim_{x 	o 0} \frac{\sin 3x}{4x}.$$

Accepted Answer

$$\frac{3}{4}$$

Question 3

For the function $$f(x) = -2x^2$ + 9x - 7$$, what is the equation of the secant line through $$x = 2$$ and $$x = 5$$?

Accepted Answer

$$y = -5x + 13$$

Question 4

Using the limit definition of derivative, what is $$f'(x)$$ for $$f(x) = 1/(x-3)$$?

Accepted Answer

$$-\frac{1}{(x-3)^2$}$$

Question 5

What is the value of $$\lim_{x 	o \infty} \frac{x^2$ + 5x + 6}{x^3$ - 2x^2$ - 15x}$$?

Accepted Answer

$$0$$

Question 6

Given the graph in $$image_2$$, what is the value of the derivative at $$x = 0$$?

Accepted Answer

Undefined

Question 7

Which of the following best describes the difference between an average rate of change and an instantaneous rate of change?

Accepted Answer

Average rate of change is the slope between two points, while instantaneous rate of change is the slope of the tangent line at a point.

Question 8

Evaluate $$\lim_{x 	o 0} \frac{1 - \cos x}{x}.$$

Accepted Answer

$$0$$

Question 9

For the function $$f(x) = x^3$ - 5x^2$, what is f$$'(x)$$ using the limit definition of derivative?

Accepted Answer

$$3x^2 - 10x$

Question 10

What is the value of $$\lim_{x 	o 0} |\csc x|$$?

Accepted Answer

$$\infty$$

Question 11

Given the graph in $$image_1$$, what is $$\lim_{x 	o 4^-} f(x)$$?

Accepted Answer

$$4$$

Question 12

What is the second derivative $$f''(x) of$$ $$f(x) = 1/x$$ using the limit definition?

Accepted Answer

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

Question 13

What is the end behavior (horizontal asymptote) of $$f(x) = \frac{3x^2$}{\sqrt{x^4$ + 7}} as$$ $$x 	o \infty$$?

Accepted Answer

$$3$$

Question 14

Given the graph in $$image_1$$, what is $$\lim_{x 	o 9} f(x)$$?

Accepted Answer

$$-\infty$$

Question 15

For the function $$f(x) = \sqrt{x-1}$$, what is $$f'(x)$$ using the limit definition of derivative?

Accepted Answer

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

Question 16

What is the value of $$\lim_{x 	o -\infty} \frac{5x^3$ + 3x - 4}{4x^3$ - 6x^2$ + x}$$?

Accepted Answer

$$\frac{5}{4}$$

Question 17

Given the graph in $$image_3$$, estimate the slope of the tangent line at $$x = 6.$$

Accepted Answer

$$\frac{1}{4}$$

Question 18

What is the value of $$\lim_{x 	o 0} \ln|x|$$?

Accepted Answer

$$-\infty$$

Question 19

For the function $$f(x) = 3x^2$, what is f$$'(x)$$ using the limit definition of derivative?

Accepted Answer

$$-6x$$

Question 20

Given the graph in $$image_1$$, what is $$\lim_{x 	o \infty} f(x)$$?

Accepted Answer

$$6$$

Calculus Exam Review: Limits and Derivatives Guidance

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