Q: A business wants to model its profit $$P(x) as a $$function of the number of units sold $$x. If$$ $$P(x) = 200x - 0.5x^2$, what is the marginal profit when 100 units are sold?

$$P'(100) = 200 - 1 \times 100 = 100$$

Q: If a population of bacteria grows according to $$P(t) = 100e$$^{0.05t}$$, where t$ is in hours, what is the rate of growth at t = 10$ hours?

$$P'(10) = 100 \times 0.05 \times e$$^{0.05 \times 10}$$ = 5e$$^{0.5}$$

Q: A manufacturer’s cost function is $$C(x) = 1000 + 20x + 0.1x^2$. What is the marginal cost when producing 50 units?

$$C'(50) = 20 + 0.2 \times 50 = 30$$

Q: If $$f(x) = x^3$ - 6x^2$ + 9x$$, what is the value of $$f''(2)$$?

$$f''(2) = 6 \times 2 - 12 = 0$$

Question 1

Which of the following best describes the primary focus of Elementary Applied Calculus (MATH 150) as outlined in the course syllabus?

Accepted Answer

The application of differential and integral calculus to problems in business, economics, life sciences, and social sciences.

Question 2

According to the syllabus, which calculator is recommended for use in MATH 150?

Accepted Answer

A graphing calculator such as TI-83 or TI-84.

Question 3

A business wants to model its profit $$P(x) as a $$function of the number of units sold $$x. If$$ $$P(x) = 200x - 0.5x^2$, what is the marginal profit when 100 units are sold?

Accepted Answer

$$P'(100) = 200 - 1 	imes 100 = 100$$

Question 4

Which of the following is NOT permitted during quizzes and exams in MATH 150, according to the syllabus?

Accepted Answer

Using a calculator app on a smartphone.

Question 5

Suppose a company’s revenue $$R(x) is $$given by $$R(x) = 500 \ln(x+1)$$, where $$x is $$the number of items sold. What is the instantaneous rate of change of revenue when $$x = 99$$?

Accepted Answer

$$R'(99) = 500 \div 100 = 5$$

Question 6

If a population of bacteria grows according to $$P(t) = 100e$$^{0.05t}$$, where t$ is in hours, what is the rate of growth at t = 10$ hours?

Accepted Answer

$$P'(10) = 100 	imes 0.05 	imes e$$^{0.05 	imes 10}$$ = 5e$$^{0.5}$$

Question 7

A manufacturer’s cost function is $$C(x) = 1000 + 20x + 0.1x^2$. What is the marginal cost when producing 50 units?

Accepted Answer

$$C'(50) = 20 + 0.2 	imes 50 = 30$$

Question 8

Which of the following is a real-world application of the derivative in business, as discussed in the course?

Accepted Answer

Finding the marginal cost to determine the cost of producing one more unit.

Question 9

If $$f(x) = x^3$ - 6x^2$ + 9x$$, what is the value of $$f''(2)$$?

Accepted Answer

$$f''(2) = 6 	imes 2 - 12 = 0$$

Question 10

A company’s demand function is $$p(x) = 100 - 2x$$, where $$p is $$price and $$x is $$quantity. What is the elasticity of demand at $$x = 20$$?

Accepted Answer

$$E = \frac{x}{p} \cdot \frac{dp}{dx} = \frac{20}{60} \cdot (-2) = -0.666$$

Question 11

Which of the following is a requirement for withdrawal with a grade of 'W' in MATH 150?

Accepted Answer

Withdraw before November 19.

Question 12

A company’s cost and revenue functions are $$C(x) = 500 + 10x$$ and $$R(x) = 20x. $$What is the break-even quantity?

Accepted Answer

$$x = \frac{500}{20 - 10} = 50$$

Question 13

Which of the following is emphasized as a key expectation for participation in MATH 150?

Accepted Answer

Respecting others’ ideas and actively engaging with course materials.

Question 14

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

Accepted Answer

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

Question 15

A company’s profit function is $$P(x) = -x^2$ + 40x - 300. $$What is the maximum profit, and at what quantity $$x$$ does it occur?

Accepted Answer

Maximum at $$x = 20$$, $$P(20) = 100$$

Question 16

Which of the following is NOT a recommended resource for extra help in MATH 150?

Accepted Answer

Calculator apps on smartphones.

Question 17

If $$f(x) = \ln(x^2$ + 1)$$, what is $$f'(x)$$?

Accepted Answer

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

Question 18

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 19

Which of the following is a violation of academic integrity as described in the syllabus?

Accepted Answer

Using unauthorized calculators or sharing answers during quizzes.

Elementary Applied Calculus (Business Calculus) Syllabus and Study Guide

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