Question 1

Which of the following best describes the derivative of a function $$f(x) at a $$point $$x = a$$?

Accepted Answer

The slope of the tangent line to the graph of $$f(x) at$$ $$x = a.$$

Question 2

A company’s cost function is $$C(x) = 500 + 20x. $$What is the marginal cost when producing the 100th item?

Accepted Answer

$$20$$

Question 3

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

Accepted Answer

$$4$$

Question 4

Which of the following functions is continuous for all real numbers?

Accepted Answer

$$f(x) = e^x$$

Question 5

If the revenue function for a product is $$R(x) = 100x - x^2$, what is the maximum revenue, and at what quantity x$ does it occur?

Accepted Answer

Maximum revenue is $$2500 at$$ $$x = 50$$

Question 6

Which of the following is the correct derivative of $$f(x) = x^3$ e^x$$?

Accepted Answer

$$3x^2 e^x$$ + $$x^3 e^x$

Question 7

A business finds that its profit function is $$P(x) = -2x^2$ + 40x - 100. $$What is the production level $$x$$ that maximizes profit?

Accepted Answer

$$x = 10$$

Question 8

Which of the following is the correct indefinite integral of $$\int 5x^4$ dx$$?

Accepted Answer

$$x^5 + C$

Question 9

Suppose the demand function for a product is $$p = 100 - 2x. $$What is the price elasticity of demand when $$x = 20$$?

Accepted Answer

$$-1$$

Question 10

A company’s marginal cost function is $$C'(x) = 6x. If $$the fixed cost is $$100$$, what is the total cost function $$C(x)$$?

Accepted Answer

$$C(x) = 3x^2$ + 100$$

Question 11

Evaluate the definite integral $$\$$int_0$$^2 (4x) dx.$$

Accepted Answer

$$16$$

Question 12

Which of the following statements about the Fundamental Theorem of Calculus is correct?

Accepted Answer

It relates differentiation and integration as inverse processes.

Question 13

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

Accepted Answer

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

Question 14

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

Accepted Answer

$$x = 20$$

Question 15

Which of the following is the correct result for $$\int e$$^{2x}$$ dx$$?

Accepted Answer

$$\frac{1}{2}e$$^{2x}$$ + C$$

Question 16

If $$f(x) = x^2$ - 4x + 3$$, at what value of $$x$$ does $$f(x)$$ attain its minimum?

Accepted Answer

$$x = 2$$

Question 17

Which of the following is the correct application of the chain rule for $$y = \ln(3x^2$ + 1)$$?

Accepted Answer

$$\frac{6x}{3x^2$ + 1}$$

Question 18

A company’s demand function is $$p = 80 - 0.5x. $$What is the equilibrium quantity if the supply function is $$p = 20 + x$$?

Accepted Answer

$$x = 40$$

Question 19

Which of the following best describes the area between two curves $$y = f(x)$$ and $$y = g(x)$$ from $$x = a to$$ $$x = b$$?

Accepted Answer

$$\int_a^b [f(x) - g(x)] dx$$

MAC2233 Calculus for Business, Social and Life Sciences – Syllabus and Study Guide

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