Question 1

Given the graph in Image 1, what type of discontinuity is present at $$x = 6$$?

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

Accepted Answer

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

Question 4

Which of the following best describes the end behavior of the function in Image 1 as $$x 	o \infty$$?

Accepted Answer

The function approaches a horizontal asymptote.

Question 5

Given $$f(x) = -2x^2$ + 9x - 7$$, use the limit definition of the derivative to find $$f'(x). $$Which is correct?

Accepted Answer

$$f'(x) = -4x + 9$$

Question 6

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

Accepted Answer

$$0$$

Question 7

For the piecewise function $$f(x) = \begin{cases} x+2, & x  4 \end{cases}$$, what is $$\lim_{x 	o 0} f(x)$$?

Accepted Answer

$$2$$

Question 8

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

Accepted Answer

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

Question 9

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 10

For $$f(x) = \frac{1}{x-3}$$, what is the derivative $$f'(x)$$ using the limit definition?

Accepted Answer

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

Question 11

Evaluate $$\lim_{x 	o 0} \ln|x|.$$

Accepted Answer

$$-\infty$$

Question 12

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 at a single point.

Question 13

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

Accepted Answer

$$3x^2 - 10x$

Question 14

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

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 3, what is the approximate slope of the tangent line at $$x = 6$$?

Accepted Answer

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

Question 17

Evaluate $$\lim_{x 	o \infty} \frac{x^2$ + 5x + 6}{x^3$ - 2x^2$ - 15x}.$$

Accepted Answer

$$0$$

Question 18

Which of the following is a correct strategy for evaluating limits involving compound fractions?

Accepted Answer

Multiply numerator and denominator by the least common denominator to simplify.

Question 19

Given $$f(x) = 3x^2$, what is f$$'(x)$$ using the limit definition?

Accepted Answer

$$6x$$

Calculus Exam Review: Limits and Derivatives Guidance

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