Q: Which of the following is the correct definition of the derivative of $$f(x) at$$ $$x = a$$?

$$f'(a) = \lim_{h \to 0} \frac{f(a + h) - f(a)}{h}$$

Q: Which of the following is the correct definition of a vertical asymptote for a function $$f(x)$$?

If $$\lim_{x \to a} f(x) = \infty or$$ $$-\infty$$, then $$x = a is a $$vertical asymptote.

Question 1

Suppose the position of a car is given by $$s(t) = 3t^2$ + 2t (in $$meters) where $$t is in $$seconds. What is the average velocity of the car between $$t = 1$$ and $$t = 4$$ seconds?

Accepted Answer

$$14 m/s$$

Question 2

Which of the following best describes the difference between the average rate of change and the instantaneous rate of change of a function $$f(x)$$?

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 3

Evaluate the limit: $$\lim_{x 	o 2} \frac{x^2$ - 4}{x - 2}$$

Accepted Answer

$$4$$

Question 4

Which of the following is NOT a necessary condition for a function $$f(x) to be $$continuous at $$x = a$$?

Accepted Answer

$$f(x) is $$differentiable at $$x = a$$

Question 5

Which of the following functions has a vertical asymptote at $$x = 3$$?

Accepted Answer

$$f(x) = \frac{1}{x - 3}$$

Question 6

Given $$f(x) = \sqrt{x}$$, what is the instantaneous rate of change of $$f at$$ $$x = 4$$?

Accepted Answer

$$\frac{1}{2}$$

Question 7

Which statement about jump discontinuities is correct?

Accepted Answer

A jump discontinuity occurs when the left and right limits at a point are not equal.

Question 8

The Squeeze Theorem is most useful for evaluating limits of which type of functions?

Accepted Answer

Functions that oscillate or are bounded between two other functions

Question 9

Find the equation of the tangent line to $$f(x) = x^2$ at x = 1$.

Accepted Answer

$$y = 2x - 1$$

Question 10

Which of the following is the correct definition of the derivative of $$f(x) at$$ $$x = a$$?

Accepted Answer

$$f'(a) = \lim_{h 	o 0} \frac{f(a + h) - f(a)}{h}$$

Question 11

If $$f(x) = \frac{2x^2$ + 3x - 5}{x - 1}$$, what is the horizontal asymptote of $$f(x)$$?

Accepted Answer

$$y = 2$$

Question 12

Which of the following is an example of a removable discontinuity?

Accepted Answer

$$f(x) = \frac{x^2$ - 1}{x - 1} at$$ $$x = 1$$

Question 13

The Intermediate Value Theorem guarantees that if $$f(x) is $$continuous on $$[a, b]$$ and $$f(a) \u003c 0 \u003c f(b)$$, then:

Accepted Answer

There exists $$c in$$ $$[a, b]$$ such that $$f(c) = 0$$

Question 14

Which of the following is the correct factorization of $$x^2 - 9$?

Accepted Answer

$$(x + 3)(x - 3)$$

Question 15

A function $$f(x) is $$not differentiable at $$x = a if$$:

Accepted Answer

$$f(x) is $$not continuous at $$x = a$$

Question 16

Evaluate the limit: $$\lim_{x 	o \infty} \frac{5x^2$ + 3x}{2x^2$ - x}$$

Accepted Answer

$$\frac{5}{2}$$

Question 17

Which of the following graphs represents a function with a vertical asymptote at $$x = 0$$ and a horizontal asymptote at $$y = 0$$?

Accepted Answer

$$f(x) = \frac{1}{x}$$

Question 18

If $$f(x) = |x|$$, which statement is true about the differentiability of $$f(x) at$$ $$x = 0$$?

Accepted Answer

$$f(x) is $$not differentiable at $$x = 0$$

Question 19

Which of the following is the correct definition of a vertical asymptote for a function $$f(x)$$?

Accepted Answer

If $$\lim_{x 	o a} f(x) = \infty or$$ $$-\infty$$, then $$x = a is a $$vertical asymptote.

Calculus I Exam 1 Study Guide: Limits, Continuity, and the Derivative

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