Question 1

Given the graph in Image 1, what is $$\lim_{x 	o 4^-} f(x)$$?

Accepted Answer

$$4$$

Question 2

Given the graph in Image 1, what is $$\lim_{x 	o 4^+} f(x)$$?

Accepted Answer

$$6$$

Question 3

Given the graph in Image 1, what is $$\lim_{x 	o 4} f(x)$$?

Accepted Answer

The limit does not exist

Question 4

Given the graph in Image 1, what is $$\lim_{x 	o 9} f(x)$$?

Accepted Answer

$$-\infty$$

Question 5

Evaluate $$\lim_{x 	o 0} \frac{\sin 3x}{4x}.$$

Accepted Answer

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

Question 6

Which of the following best describes the discontinuity at $$x = 4 in $$the function from Image 1?

Accepted Answer

Jump discontinuity

Question 7

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

Accepted Answer

$$0$$

Question 8

Given $$f(x) = \sqrt{x+1} - \sqrt{7}$$, evaluate $$\lim_{x 	o 6} \frac{\sqrt{x+1} - \sqrt{7}}{x-6}.$$

Accepted Answer

$$\frac{1}{2\sqrt{7}}$$

Question 9

Which of the following statements about $$\lim_{x 	o \infty} |\sin x| is $$correct?

Accepted Answer

The limit does not exist because $$|\sin x|$$ oscillates between $$0$$ and $$1 as$$ $$x 	o \infty.$$

Question 10

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

Accepted Answer

$$y = -5x + 13$$

Question 11

Use the limit definition of derivative to find $$f'(x)$$ for $$f(x) = -2x^2$ + 9x - 7.$$

Accepted Answer

$$-4x + 9$$

Question 12

Given $$f(x) = \frac{1}{x-3}$$, use the limit definition of derivative to find $$f'(x).$$

Accepted Answer

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

Question 13

Given $$f(x) = x^3$ - 5x^2$, use the limit definition of derivative to find f$$'(x)$$.

Accepted Answer

$$3x^2 - 10x$

Question 14

Given $$f(x) = \sqrt{x-1}$$, use the limit definition of derivative to find $$f'(x).$$

Accepted Answer

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

Question 15

Given $$f(x) = \frac{1}{x}$$, what is the second derivative $$f''(x)$$ using the limit definition?

Accepted Answer

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

Question 16

Given the graph in Image 2, what is the value of the derivative at $$x = 0$$?

Accepted Answer

The derivative does not exist at $$x = 0$$ due to a cusp.

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

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

Accepted Answer

The average rate of change measures the change over an interval, while the instantaneous rate of change measures the change at a single point.

Calculus Exam I Review: Limits and Derivatives Study Guide

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