Q: Which of the following statements best describes the instantaneous rate of change of a function $f(x)$ at $x=a$?

It is the limit $\lim_{h \to 0} \frac{f(a+h) - f(a)}{h}$, provided the limit exists.

Q: If the profit function is $P(x) = 40x - 5x^2 - 200$, what is the marginal profit when $x=8$?

$P'(8) = 40 - 10 \times 8 = -40$

Question 1

Which of the following statements best describes the instantaneous rate of change of a function $f(x)$ at $x=a$?

Accepted Answer

It is the limit $\lim_{h 	o 0} \frac{f(a+h) - f(a)}{h}$, provided the limit exists.

Question 2

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

Accepted Answer

The function has a sharp corner or cusp at $x=a$.

Question 3

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

Accepted Answer

$y = 12x - 16$

Question 4

Suppose the cost function for producing $x$ units of a product is $C(x) = 200 + 20x$. What is the marginal cost when $x=10$?

Accepted Answer

$20$

Question 5

If the revenue function is $R(x) = 60x - 5x^2$, what is the marginal revenue when $x=6$?

Accepted Answer

$-0$

Question 6

A company finds that increasing production from 150 to 151 units increases revenue by approximately $30. What concept does this illustrate?

Accepted Answer

Marginal revenue

Question 7

Which of the following is true about the break-even point?

Accepted Answer

It occurs when $C(x) = R(x)$.

Question 8

Given the profit function $P(x) = R(x) - C(x)$, if $R(x) = 60x - 5x^2$ and $C(x) = 200 + 20x$, what is $P(x)$?

Accepted Answer

$P(x) = 40x - 5x^2 - 200$

Question 9

If $f(x)$ is continuous on $[a, b]$ and differentiable on $(a, b)$, which theorem guarantees that there exists $c$ in $(a, b)$ such that $f'(c) = \frac{f(b) - f(a)}{b - a}$?

Accepted Answer

Mean Value Theorem

Question 10

If the cost of producing $x$ units is $C(x) = 10x^2 + 5x + 100$, what is the exact cost of producing the 16th unit?

Accepted Answer

$C(16) - C(15)$

Question 11

Which of the following is NOT a reason for a function to be non-differentiable at a point?

Accepted Answer

The function is linear at the point.

Question 12

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

Accepted Answer

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

Question 13

Suppose the price-demand function is $p = 60 - 5x$. What is the revenue function $R(x)$?

Accepted Answer

$R(x) = 60x - 5x^2$

Question 14

If the cost function is $C(x) = 200 + 20x$, what is the average cost per unit when $x=10$?

Accepted Answer

$22$

Question 15

A factory produces chocolate bars with cost function $C(x) = 10x^2 + 5x + 100$. What is the marginal cost function?

Accepted Answer

$C'(x) = 20x + 5$

Question 16

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

Accepted Answer

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

Question 17

If the profit function is $P(x) = 40x - 5x^2 - 200$, what is the marginal profit when $x=8$?

Accepted Answer

$P'(8) = 40 - 10 	imes 8 = -40$

Question 18

Which of the following best describes the average cost function $AC(x)$?

Accepted Answer

$AC(x) = \frac{C(x)}{x}$

Question 19

If $f(x)$ is continuous on $[1,3]$ but not continuous at $x=2$, which of the following intervals is $f(x)$ continuous on?

Accepted Answer

$[1,2)$ and $(2,3]$

Continuity and Differentiability in Business Calculus

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