Q: A business finds that its cost function is $$C(x) = 1000 + 5x^2$. What is the average cost per unit when producing x = 10$ units?

$$C_{avg} = \frac{1000 + 5 \times 10^2$}{10} = 60$$

Question 1

A company models its profit $$P as a $$function of price $$x by$$ $$P(x) = 100x - 2x^2$. What price maximizes profit, and what is the maximum profit?

Accepted Answer

Price $$x = 50$$, Maximum profit $$P = 2500$$

Question 2

Which of the following best describes the meaning of the derivative $$f'(x) in a $$business context?

Accepted Answer

The instantaneous rate of change of $$f(x)$$ with respect to $$x$$

Question 3

Evaluate the limit: $$\lim_{x 	o 2} \frac{x^2$ - 4}{x - 2}$$

Accepted Answer

$$4$$

Question 4

A manufacturer finds that the cost $$C to $$produce $$x$$ units is $$C(x) = 500 + 20x. $$What is the marginal cost when $$x = 100$$?

Accepted Answer

$$20$$

Question 5

If $$f(x) = e$$^{2x}$$, what is f$$'(x)$$?

Accepted Answer

$$2e^{2x}$$

Question 6

A population grows according to $$P(t) = 1000e$$^{0.05t}$$. What is the rate of change of the population at t = 10$?

Accepted Answer

$$50e^{0.5}$$

Question 7

Which of the following functions is NOT continuous at $$x = 0$$?

Accepted Answer

$$f(x) = \frac{1}{x}$$

Question 8

A company’s revenue function is $$R(x) = 50x$$, and its cost function is $$C(x) = 30x + 200. $$What is the break-even point?

Accepted Answer

$$x = 20$$

Question 9

Find the derivative of $$f(x) = x^3$ - 4x^2$ + 7x - 2.$$

Accepted Answer

$$f'(x) = 3x^2$ - 8x + 7$$

Question 10

If $$y = x^2$ e^x$$, what is $$\frac{dy}{dx}$$?

Accepted Answer

$$2x e^x + x^2$ e^x$$

Question 11

A business wants to minimize the cost of producing $$x$$ units, where $$C(x) = x^2$ - 10x + 25. $$What is the minimum cost and at what value of $$x$$ does it occur?

Accepted Answer

Minimum cost $$C = 0 at$$ $$x = 5$$

Question 12

Evaluate the definite integral $$\int_{0}^{2} (3x^2$ + 2x) \, dx.$$

Accepted Answer

$$14$$

Question 13

Which of the following is the correct application of the chain rule for $$f(x) = \ln(2x + 1)$$?

Accepted Answer

$$f'(x) = \frac{2}{2x + 1}$$

Question 14

A company’s demand function is $$p = 100 - 2x$$, where $$p is $$price and $$x is $$quantity. What is the revenue function $$R(x)$$?

Accepted Answer

$$R(x) = 100x - 2x^2$

Question 15

If $$f(x) = \ln(x)$$, what is $$f'(x)$$?

Accepted Answer

$$\frac{1}{x}$$

Question 16

A tank is being filled with water at a rate given by $$R(t) = 5t^2$ liters per hour. How much water is added between t = 0$ and t = 3$ hours?

Accepted Answer

$$45$$ liters

Question 17

Which of the following is the correct antiderivative of $$f(x) = 6x$$?

Accepted Answer

$$3x^2 + C$

Question 18

A business finds that its cost function is $$C(x) = 1000 + 5x^2$. What is the average cost per unit when producing x = 10$ units?

Accepted Answer

$$C_{avg} = \frac{1000 + 5 	imes 10^2$}{10} = 60$$

Question 19

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

Accepted Answer

$$2$$

Question 20

A company’s revenue function is $$R(x) = 100x - x^2$. What is the marginal revenue when x = 20$?

Accepted Answer

$$R'(x) = 100 - 2x$$, so $$R'(20) = 100 - 40 = 60$$

Syllabus and Course Structure: Calculus for Business and Social Sciences (MATH 1325)

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