Q: If $$f(x) is $$continuous at $$x = a$$, which of the following must be true?

$$\lim_{x \to a} f(x)$$ exists and equals $$f(a)$$

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

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

Question 1

Find the slope of the secant line through the points $$(-2, f(-2))$$ and $$(1, f(1))$$ for the function $$f(x) = x^2$ + 2x + 3. $$What is the value of the slope?

Accepted Answer

$$m = \frac{f(1) - f(-2)}{1 - (-2)} = \frac{6 - 3}{3} = 1$$

Question 2

For the function $$f(x) = x^2$ + 2x + 3$$, find the value of the difference quotient $$\frac{f(x+h) - f(x)}{h}$$ for $$x = 2$$ and $$h = 0.01. $$Round your answer to two decimal places.

Accepted Answer

$$6.01$$

Question 3

The following chart shows 'living wage' jobs in Rochester per 1000 working age adults over a 5-year period:

Year: 1997, 1998, 1999, 2000, 2001
Jobs: 6.53, 6.95, 7.50, 7.74, 7.25

What is the average rate of change in the number of living wage jobs from 1998 to 2001?

Accepted Answer

$$\frac{7.25 - 6.95}{3} = 0.10$$ jobs per year

Question 4

Given the graph of $$f(x)$$, which of the following is true about $$\lim_{x 	o 2^-} f(x)$$ and $$\lim_{x 	o 2^+} f(x)$$?

Accepted Answer

Both limits exist but are not equal

Question 5

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

Accepted Answer

$$1$$

Question 6

Evaluate the limit: $$\lim_{n 	o \infty} \frac{4n^2$ + 2n}{n^2$}$$

Accepted Answer

$$4$$

Question 7

Evaluate the limit: $$\lim_{x 	o 3} \frac{1 - x}{x^2$ - 3x + 2}$$

Accepted Answer

$$-1$$

Question 8

Given the graph of $$f(x)$$, select all statements below that you agree with:
(a) $$f(2) is $$defined
(b) $$\lim_{x 	o 2} f(x)$$ exists
(c) $$\lim_{x 	o 2} f(x) = f(2)
(d) $$The function is not continuous at $$x = 2$$

Accepted Answer

a, b, and d only

Question 9

Let $$f(x) = \begin{cases} mx - 10 & 	ext{if } x \u003c -2 \ \frac{1}{x^2$ + 3x - 2} & 	ext{if } x \geq -2 \end{cases}. $$For what value of $$m is$$ $$f(x)$$ continuous at $$x = -2$$?

Accepted Answer

$$m = -1$$

Question 10

Suppose $$f(x) = 0$$ for some $$x in $$the interval $$[-4, 0]. $$Which of the following is true?

Accepted Answer

Sometimes true and sometimes false

Question 11

When demonstrating that $$\lim_{x 	o 1} 3x + 2 = 5$$ with $$\epsilon = 0.3$$, which of the following $$\delta-$$values suffices?

Accepted Answer

$$\delta = 0.2$$

Question 12

In formally proving that $$\lim_{x 	o 3} (x + 8) = 11$$, let $$\epsilon \u003e 0 be $$arbitrary. Determine $$\delta as a $$function of $$\epsilon.$$

Accepted Answer

$$\delta = \epsilon$$

Question 13

A chemical reaction's temperature $$T (in$$ $$^\circC) is $$modeled by $$T(t) = 5t^2$ + 3t + 2$$ for time $$t in $$minutes. What is the average rate of change of temperature from $$t = 1 to$$ $$t = 4$$?

Accepted Answer

$$\frac{T(4) - T(1)}{4 - 1} = \frac{90 - 10}{3} = 26.67$$ $$^\circC/$$min

Question 14

A patient’s blood concentration of a drug is given by $$C(t) = 10e$$^{-0.5t}$$, where t$ is in hours. What is the instantaneous rate of change of concentration at t = 2$ hours?

Accepted Answer

$$C'(2) = -5e$$^{-1}$$ \approx -1.84 mg/L/hr$$

Question 15

If $$f(x) is $$continuous at $$x = a$$, which of the following must be true?

Accepted Answer

$$\lim_{x 	o a} f(x)$$ exists and equals $$f(a)$$

Question 16

A car’s position is given by $$s(t) = 4t^3$ - 2t^2$ + t ($$meters). What is the average velocity between $$t = 1$$ and $$t = 3$$ seconds?

Accepted Answer

$$\frac{s(3) - s(1)}{3 - 1} = \frac{97 - 3}{2} = 47 m/s$$

Question 17

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 18

A population of bacteria grows according to $$P(t) = 1000e$$^{0.2t}$$. What is the instantaneous rate of growth at t = 5$ hours?

Accepted Answer

$$P'(5) = 200e$$^{1}$$ \approx 543.66$$ bacteria/hour

Question 19

Which of the following statements about limits is correct?

Accepted Answer

If $$\lim_{x 	o a^-} f(x) 
eq \lim_{x 	o a^+} f(x)$$, then $$\lim_{x 	o a} f(x)$$ does not exist.

Calculus I: Limits, Continuity, and Secant Slopes – Study Guide

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