Q: If the revenue function is $$R(x) = 17x - 0.01x^2$, what is the marginal revenue when x = 300$?

$$R'(x) = 17 - 0.02x$$,\ R'(300) = 17 - 0.02 \times 300 = 11$

Q: Suppose the cost to produce $$x$$ items is $$C(x) = 5000 + 10x. $$What is the total cost of producing 400 items?

$$C(400) = 5000 + 10 \times 400 = 9000$$

Q: If the profit function is $$P(x) = 7x - 0.01x^2$ - 5000$$, what is the marginal profit when $$x = 300$$?

$$P'(x) = 7 - 0.02x$$,\ P'(300) = 7 - 0.02 \times 300 = 1$

Q: A company finds that the profit from selling the 401st item is approximately equal to the marginal profit at $$x = 400. If$$ $$P(x) = 7x - 0.01x^2$ - 5000$$, what is this value?

$$P'(400) = 7 - 0.02 \times 400 = -1$$

Question 1

Given the piecewise function 
$$
S(t) = \begin{cases} t+1, & t \u003c -1 \ 0, & -1 \leq t \u003c 1 \ t-1, & t \geq 1 \end{cases}
$$
Which of the following statements is TRUE about the continuity of $$S(t)$$?

Accepted Answer

$$S(t) is $$continuous everywhere except at $$t = -1$$ and $$t = 1.$$

Question 2

For the function $$f(x) = \begin{cases} x+1, & x \u003c 0 \ x^2$, & x \geq 0 \end{cases}$$, which of the following is TRUE?

Accepted Answer

$$f(x) is $$continuous at $$x = 0.$$

Question 3

Given the cost function $$C(x) = 5000 + 10x$$ and the revenue function $$R(x) = 17x - 0.01x^2$, what is the profit function P(x$$)$$?

Accepted Answer

$$P(x) = 17x - 0.01x^2$ - (5000 + 10x)$$

Question 4

What is the marginal cost of producing the 401st item if $$C(x) = 5000 + 10x$$?

Accepted Answer

$$10$$

Question 5

If the revenue function is $$R(x) = 17x - 0.01x^2$, what is the marginal revenue when x = 300$?

Accepted Answer

$$R'(x) = 17 - 0.02x$$,\ R'(300) = 17 - 0.02 	imes 300 = 11$

Question 6

Suppose the cost to produce $$x$$ items is $$C(x) = 5000 + 10x. $$What is the total cost of producing 400 items?

Accepted Answer

$$C(400) = 5000 + 10 	imes 400 = 9000$$

Question 7

If the profit function is $$P(x) = 7x - 0.01x^2$ - 5000$$, what is the marginal profit when $$x = 300$$?

Accepted Answer

$$P'(x) = 7 - 0.02x$$,\ P'(300) = 7 - 0.02 	imes 300 = 1$

Question 8

A company finds that the profit from selling the 401st item is approximately equal to the marginal profit at $$x = 400. If$$ $$P(x) = 7x - 0.01x^2$ - 5000$$, what is this value?

Accepted Answer

$$P'(400) = 7 - 0.02 	imes 400 = -1$$

Question 9

Which of the following best describes the meaning of the marginal cost in a business context?

Accepted Answer

The additional cost incurred by producing one more unit of a good.

Question 10

If $$R(x) = 17x - 0.01x^2$ and C(x) = 5000 + 10x$, for what value of x$ does the marginal profit become zero?

Accepted Answer

Set $$P'(x) = R'(x) - C'(x) = 0$$. $$R'(x) = 17 - 0.02x$$, $$C'(x) = 10$$, so $$17 - 0.02x - 10 = 0 \implies x = 350$$

Question 11

Which of the following is the correct derivative for the revenue function $$R(x) = 17x - 0.01x^2$?

Accepted Answer

$$R'(x) = 17 - 0.02x$$

Question 12

A business wants to maximize profit. If the profit function is $$P(x) = 7x - 0.01x^2$ - 5000$$, what value of $$x$$ maximizes profit?

Accepted Answer

Set $$P'(x) = 0 \implies 7 - 0.02x = 0 \implies x = 350$$

Question 13

If $$C(x) = 5000 + 10x$$, what is the average cost per item when 400 items are produced?

Accepted Answer

$$\frac{C(400)}{400} = \frac{9000}{400} = 22.5$$

Business Calculus Study Guidance: Continuity, Cost, Revenue, and Marginal Analysis

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