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

$\lim\limits_{h \to 0} \frac{f(a+h) - f(a)}{h}$

Q: Suppose $f(x)$ is continuous at $x = 2$. Which of the following must be true?

$\lim\limits_{x \to 2} f(x) = f(2)$

Q: If $f(x)$ is not continuous at $x = 2$, which of the following statements is necessarily true?

$\lim\limits_{x \to 2} f(x) \neq f(2)$

Question 1

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

Accepted Answer

$\lim\limits_{h 	o 0} \frac{f(a+h) - f(a)}{h}$

Question 2

Given $f(x) = x^2 - 3x + 5$, use the limit definition of the derivative to find $f'(x)$. Which of the following is correct?

Accepted Answer

$2x - 3$

Question 3

A company’s cost to produce $x$ thousand calculators is $C(x) = x^3 - 3x^2 + 28$ (in thousands of dollars). What is the marginal cost when $x = 4$?

Accepted Answer

$24$ thousand dollars per thousand calculators

Question 4

Find the average rate of change of $f(x) = x^2 - 3x + 5$ from $x = 1$ to $x = 2$.

Accepted Answer

$-1$

Question 5

Evaluate the limit: $\lim\limits_{x 	o 2} \frac{x^2 - x - 2}{x-2}$

Accepted Answer

$3$

Question 6

Given the function $f(x) = \frac{2x+3}{x^2}$, for which value of $x$ is $f(x)$ not defined?

Accepted Answer

$x = 0$

Question 7

Suppose $f(x)$ is continuous at $x = 2$. Which of the following must be true?

Accepted Answer

$\lim\limits_{x 	o 2} f(x) = f(2)$

Question 8

If $f(x) = 3x^2 - 12x + 8$, at which $x$-value(s) is the tangent line to the curve horizontal?

Accepted Answer

$x = 2$

Question 9

Given $f(x) = x^2 - 3x + 5$, what is the equation of the tangent line at $x = 1$?

Accepted Answer

$y = -1x + 3$

Question 10

A company’s total cost to produce $x$ thousand calculators is $C(x) = x^3 - 3x^2 + 28$. What is the average cost per thousand calculators when $x = 4$?

Accepted Answer

$3.25$ thousand dollars

Question 11

Given $f(x) = x^2 - 3x + 5$, what is $f''(x)$, the second derivative?

Accepted Answer

$2$

Question 12

If $f(x)$ is not continuous at $x = 2$, which of the following statements is necessarily true?

Accepted Answer

$\lim\limits_{x 	o 2} f(x) 
eq f(2)$

Question 13

Given $f(x) = x^2 - 3x + 5$, what is the instantaneous rate of change at $x = 3$?

Accepted Answer

$3$

Question 14

For the function $f(x) = \sqrt{x-4}$, what is the domain?

Accepted Answer

$x \geq 4$

Question 15

If $f(x) = x^2 - 3x + 5$, what is $f(0)$?

Accepted Answer

$5$

Question 16

Given $f(x) = x^2 - 3x + 5$, what is the $x$-intercept of the function?

Accepted Answer

$\frac{3 \pm \sqrt{9 - 20}}{2}$

Question 17

If $f(x) = x^2 - 3x + 5$, what is the minimum value of $f(x)$?

Accepted Answer

$\frac{7}{4}$

Question 18

Given $f(x) = x^2 - 3x + 5$, what is the vertex of the parabola?

Accepted Answer

$(\frac{3}{2}, \frac{7}{4})$

Question 19

If $f(x) = x^2 - 3x + 5$, what is the slope of the tangent line at $x = 0$?

Accepted Answer

$-3$

Question 20

A company’s cost to produce $x$ thousand calculators is $C(x) = x^3 - 3x^2 + 28$. At what production level is the total cost increasing at a rate of $9$ thousand dollars per one thousand calculators?

Accepted Answer

$x = 3$

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