Q: Suppose the cost function for producing $$x$$ units is $$C(x) = 500 + 20x + 0.5x^2$. What is the marginal cost when x = 100$?

$$C'(100) = 20 + 2 \times 100 \times 0.5 = 120$$

Q: A population grows according to the model $$P(t) = 1000e$$^{0.03t}$$. What is the instantaneous rate of growth at t = 10$?

$$P'(10) = 0.03 \times 1000e$$^{0.03 \times 10}$$

Question 1

Which of the following best describes the main focus of Calculus for Business and Social Sciences (MAC 2233) at FSCJ?

Accepted Answer

The application of calculus concepts to business, economics, and social sciences using algebraic, graphical, and numerical methods.

Question 2

Which of the following is NOT a required material for MAC 2233 at FSCJ?

Accepted Answer

A TI-83/84 graphing calculator.

Question 3

A business wants to maximize its profit, $$P(x)$$, where $$x is $$the number of units sold. Which calculus concept is most directly used to find the value of $$x$$ that maximizes profit?

Accepted Answer

Setting the derivative $$P'(x)$$ equal to zero and analyzing the second derivative.

Question 4

Which rule would you use to differentiate the function $$f(x) = x^3$ e^x$$?

Accepted Answer

The product rule.

Question 5

Suppose the cost function for producing $$x$$ units is $$C(x) = 500 + 20x + 0.5x^2$. What is the marginal cost when x = 100$?

Accepted Answer

$$C'(100) = 20 + 2 	imes 100 	imes 0.5 = 120$$

Question 6

Which of the following statements about definite integrals is TRUE in the context of business applications?

Accepted Answer

A definite integral can represent the area under a revenue curve, which may correspond to total revenue over a period.

Question 7

A company’s revenue function is $$R(x) = 100x - x^2$. What is the maximum revenue, and at what value of x$ does it occur?

Accepted Answer

Maximum at $$x = 50$$, $$R(50) = 2500$$

Question 8

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

Accepted Answer

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

Question 9

Which of the following best describes the chain rule for differentiation?

Accepted Answer

If $$y = f(g(x))$$, then $$y' = f'(g(x)) \cdot g'(x).$$

Question 10

A population grows according to the model $$P(t) = 1000e$$^{0.03t}$$. What is the instantaneous rate of growth at t = 10$?

Accepted Answer

$$P'(10) = 0.03 	imes 1000e$$^{0.03 	imes 10}$$

Question 11

Which of the following is the correct interpretation of the marginal revenue function $$R'(x) in $$economics?

Accepted Answer

It gives the additional revenue from selling one more unit when $$x$$ units have already been sold.

Question 12

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

Accepted Answer

4

Question 13

Which of the following is the correct antiderivative of $$f(x) = 3x^2$?

Accepted Answer

$$F(x) = x^3$ + C$$

Question 14

A company’s cost function is $$C(x) = 1000 + 5x. $$What does the constant 1000 represent in a real-world context?

Accepted Answer

The fixed cost, or the cost incurred even if no units are produced.

Question 15

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

Accepted Answer

$$f'(x) = 2e$$^{2x}$$

Question 16

Which of the following best describes the area under the curve $$y = f(x)$$ from $$x = a to$$ $$x = b$$?

Accepted Answer

It is given by $$\int_a^b f(x) dx$$ and represents the total accumulated value between $$x = a$$ and $$x = b.$$

Question 17

A company’s profit function is $$P(x) = -2x^2$ + 40x - 100. $$What is the maximum profit?

Accepted Answer

$$P(10) = -2(10)^2$ + 40(10) - 100 = 100$$

Question 18

Which of the following is the correct derivative of $$f(x) = \ln(5x)$$?

Accepted Answer

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

Syllabus and Course Guide: Calculus for Business and Social Sciences (MAC 2233)

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