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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