Q: A function $$f(x)$$ has a jump discontinuity at $$x = 3. $$Which of the following best describes this situation?

$$\displaystyle \lim_{x \to 3^-} f(x) \neq \lim_{x \to 3^+} f(x)$$

Q: Which of the following limits can be evaluated using the Squeeze Theorem?

$$\displaystyle \lim_{x \to 0} x^2$ \sin\left(\frac{1}{x}\right)$$

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

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

Q: Which of the following best describes a removable discontinuity?

A point where $$\displaystyle \lim_{x \to a} f(x)$$ exists, but $$f(a) is $$either undefined or not equal to the limit.

Question 1

Which of the following best describes the average rate of change of a function $$f(x)$$ over the interval $$[a, b]$$?

Accepted Answer

$$\displaystyle \frac{f(b) - f(a)}{b - a}$$

Question 2

Suppose $$f(x)$$ models the temperature (in $$^\circC) of a $$chemical reaction at time $$x (in $$minutes). What does the instantaneous rate of change of $$f at$$ $$x = 10$$ represent?

Accepted Answer

The rate at which the temperature is changing at exactly 10 minutes.

Question 3

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

Accepted Answer

2

Question 4

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

Accepted Answer

$$f'(a)$$ exists

Question 5

A function $$f(x)$$ has a jump discontinuity at $$x = 3. $$Which of the following best describes this situation?

Accepted Answer

$$\displaystyle \lim_{x 	o 3^-} f(x) 
eq \lim_{x 	o 3^+} f(x)$$

Question 6

Which of the following statements about vertical asymptotes is TRUE?

Accepted Answer

A vertical asymptote occurs at $$x = a if$$ $$\displaystyle \lim_{x 	o a} f(x) = \infty or$$ $$-\infty$$

Question 7

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

Accepted Answer

$$y = 2$$

Question 8

Which of the following limits can be evaluated using the Squeeze Theorem?

Accepted Answer

$$\displaystyle \lim_{x 	o 0} x^2$ \sin\left(\frac{1}{x}ight)$$

Question 9

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

Accepted Answer

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

Question 10

A car's position at time $$t is $$given by $$s(t) = 3t^2$ + 2t. $$What is the instantaneous velocity at $$t = 4$$?

Accepted Answer

26

Question 11

Which of the following functions is NOT differentiable at $$x = 0$$?

Accepted Answer

$$f(x) = |x|$$

Question 12

Which of the following is the correct statement of the Intermediate Value Theorem?

Accepted Answer

If $$f is $$continuous on $$[a, b]$$ and $$N is $$between $$f(a)$$ and $$f(b)$$, then there exists $$c in$$ $$(a, b)$$ such that $$f(c) = N.$$

Question 13

Find the equation of the tangent line to $$f(x) = x^2$ - 3x + 4 at$$ $$x = 1.$$

Accepted Answer

$$y = -1x + 2$$

Question 14

Which of the following best describes a removable discontinuity?

Accepted Answer

A point where $$\displaystyle \lim_{x 	o a} f(x)$$ exists, but $$f(a) is $$either undefined or not equal to the limit.

Question 15

Evaluate $$\displaystyle \lim_{x 	o \infty} \frac{5x^3$ - 2x + 1}{2x^3$ + 7}.$$

Accepted Answer

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

Question 16

Suppose $$f(x) is $$continuous on $$[1, 5]$$ and $$f(1) = -2$$, $$f(5) = 6. $$Which of the following must be true by the Intermediate Value Theorem?

Accepted Answer

There exists $$c in$$ $$(1, 5)$$ such that $$f(c) = 0$$

Question 17

If $$f(x) = \sqrt{x}$$, what is the slope of the tangent line at $$x = 4$$?

Accepted Answer

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

Question 18

Which of the following best describes the difference between a vertical and a horizontal asymptote?

Accepted Answer

A vertical asymptote occurs when $$f(x)$$ approaches $$\infty or$$ $$-\infty as$$ $$x$$ approaches a finite value; a horizontal asymptote occurs when $$f(x)$$ approaches a constant as $$x$$ approaches $$\infty or$$ $$-\infty.$$

Question 19

If $$f(x) = \frac{x^2$ - 9}{x - 3}$$, what is the value of $$\displaystyle \lim_{x 	o 3} f(x)$$?

Accepted Answer

6

Calculus I Exam 1 Study Guide: Limits, Continuity, Asymptotes, and Derivatives

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