Q: Which of the following statements best explains why $$\displaystyle \lim_{x \to 0} \frac{1}{x}$$ does not exist?

The left and right limits as $$x \to 0$$ are not equal; one approaches $$-\infty$$ and the other $$+\infty.$$

Q: Given the graph of $$f(x)$$ below, which of the following is true about $$\displaystyle \lim_{x \to 2^-} f(x) if $$the graph approaches $$3$$ from the left and $$5$$ from the right as $$x$$ approaches $$2$$?

$$\displaystyle \lim_{x \to 2^-} f(x) = 3$$

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

$$\displaystyle \lim_{x \to a} \frac{f(x) - f(a)}{x - a}$$

Q: Suppose $$f(x)$$ has a vertical asymptote at $$x = 2. $$Which of the following statements is true?

$$\displaystyle \lim_{x \to 2} f(x)$$ does not exist because $$f(x)$$ approaches $$\infty or$$ $$-\infty$$

Question 1

Suppose the concentration of a drug in the bloodstream, in mg/L, is modeled by $$C(t) = \frac{10}{t+1}$$, where $$t is $$the time in hours after injection. What is $$\displaystyle \lim_{t 	o 0^+} C(t)$$?

Accepted Answer

$$10$$

Question 2

Evaluate $$\displaystyle \lim_{x 	o 2} \frac{x^2$ - 4}{x - 2}$$ algebraically.

Accepted Answer

$$4$$

Question 3

A function $$f(x) is $$defined as $$f(x) = \begin{cases} x^2$ + k, & x \u003c 3 \ 2x + 1, & x \geq 3 \end{cases}. $$For what value of $$k is$$ $$f(x)$$ continuous at $$x = 3$$?

Accepted Answer

$$k = 6$$

Question 4

Which of the following statements best explains why $$\displaystyle \lim_{x 	o 0} \frac{1}{x}$$ does not exist?

Accepted Answer

The left and right limits as $$x 	o 0$$ are not equal; one approaches $$-\infty$$ and the other $$+\infty.$$

Question 5

Given the graph of $$f(x)$$ below, which of the following is true about $$\displaystyle \lim_{x 	o 2^-} f(x) if $$the graph approaches $$3$$ from the left and $$5$$ from the right as $$x$$ approaches $$2$$?

Accepted Answer

$$\displaystyle \lim_{x 	o 2^-} f(x) = 3$$

Question 6

Which of the following functions is continuous everywhere on $$\mathbb{R}$$?

Accepted Answer

$$f(x) = x^3$ - 2x + 1$$

Question 7

A population of bacteria grows according to $$P(t) = 100e$$^{0.2t}$$. What is the average rate of change of the population from t=0$ to t=5$ hours?

Accepted Answer

$$\frac{P(5) - P(0)}{5}$$

Question 8

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

Accepted Answer

$$\frac{3}{5}$$

Question 9

Which of the following is a removable discontinuity?

Accepted Answer

A hole in the graph at $$x = 1$$ where $$\displaystyle \lim_{x 	o 1} f(x)$$ exists but $$f(1) is $$undefined

Question 10

Given $$f(x) = |x|$$, which of the following statements is true about the derivative at $$x = 0$$?

Accepted Answer

$$f'(0)$$ does not exist because the left and right derivatives are not equal

Question 11

A car's position in meters is given by $$s(t) = 4t^2$ + 2t. $$What is the instantaneous rate of change of position at $$t = 3$$ seconds?

Accepted Answer

$$26$$

Question 12

Which of the following best describes the behavior of $$f(x) = \frac{1}{x^2$} as$$ $$x 	o 0$$?

Accepted Answer

$$f(x)$$ approaches $$\infty$$ from both sides

Question 13

If $$f(x) is $$continuous on $$[1,5]$$ and $$f(1) = 2$$, $$f(5) = 10$$, which of the following must be true by the Intermediate Value Theorem?

Accepted Answer

For every $$k$$ between $$2$$ and $$10$$, there exists $$c \in [1,5]$$ such that $$f(c) = k$$

Question 14

Given the table of values for $$f(x)$$:

$$x$$ | $$1$$ | $$2$$ | $$3$$
---|----|----|----
$$f(x)$$ | $$2$$ | $$4$$ | $$8$$

Estimate $$f'(2)$$ using the symmetric difference quotient.

Accepted Answer

$$3$$

Question 15

Which of the following limits represents the definition of the derivative of $$f(x) at$$ $$x = a$$?

Accepted Answer

$$\displaystyle \lim_{x 	o a} \frac{f(x) - f(a)}{x - a}$$

Question 16

Suppose $$f(x)$$ has a vertical asymptote at $$x = 2. $$Which of the following statements is true?

Accepted Answer

$$\displaystyle \lim_{x 	o 2} f(x)$$ does not exist because $$f(x)$$ approaches $$\infty or$$ $$-\infty$$

Question 17

If $$f(x) is $$continuous everywhere except at $$x = 1$$, where $$f(1) is $$defined but $$\displaystyle \lim_{x 	o 1} f(x)$$ does not exist, what type of discontinuity is this?

Accepted Answer

Jump

Question 18

A patient’s body temperature $$T (in$$ $$^0C) is $$modeled by $$T(t) = 37 + 0.5 \sin(\pi t/12)$$, where $$t is $$hours after midnight. What is the instantaneous rate of change of temperature at $$t = 6$$?

Accepted Answer

$$\frac{\pi}{24}$$

Question 19

Which of the following best describes the relationship between the graph of $$f(x)$$ and its derivative $$f'(x)$$?

Accepted Answer

Where $$f(x) is $$increasing, $$f'(x) is $$positive

Calculus I: Limits, Continuity, and Derivatives – Unit 1 Study Guide

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