Q: A business models its revenue with the function $$R(x) = 100x - 2x^2$, where x$ is the number of units sold. What is the marginal revenue when x = 10$?

$$R'(10) = 100 - 4 \times 10 = 60$$

Q: A company’s cost function is $$C(x) = 50 + 5x + 0.1x^2$. What is the marginal cost when x = 20$?

$$C'(20) = 5 + 0.2 \times 20 = 9$$

Q: Which of the following statements about limits is TRUE?

If $$f(x) is $$continuous at $$x = a$$, then $$\lim_{x \to a} f(x) = f(a)$$

Q: A company invests $$\1000$ at a continuous compound interest rate of 5$$\%$ per year. What is the value after 3 years?

$$1000 \cdot e$$^{0.05 \times 3}$$ \approx 1000 \cdot e$$^{0.15}$$ \approx 1161.83$$

Q: A company’s revenue function is $$R(x) = 200x - x^2$. What is the maximum revenue?

$$R(100) = 200 \times 100 - 100^2$ = 10,000$$

Question 1

Which of the following functions is NOT continuous everywhere on its domain?

Accepted Answer

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

Question 2

A business models its revenue with the function $$R(x) = 100x - 2x^2$, where x$ is the number of units sold. What is the marginal revenue when x = 10$?

Accepted Answer

$$R'(10) = 100 - 4 	imes 10 = 60$$

Question 3

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

Accepted Answer

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

Question 4

A company’s cost function is $$C(x) = 50 + 5x + 0.1x^2$. What is the marginal cost when x = 20$?

Accepted Answer

$$C'(20) = 5 + 0.2 	imes 20 = 9$$

Question 5

Which of the following statements about limits is TRUE?

Accepted Answer

If $$f(x) is $$continuous at $$x = a$$, then $$\lim_{x 	o a} f(x) = f(a)$$

Question 6

A manufacturer finds that the demand function for its product is $$p(x) = 100 - 2x$$, where $$p is $$price and $$x is $$quantity. What is the elasticity of demand at $$x = 20$$?

Accepted Answer

$$E = \left| \frac{p}{x} \cdot \frac{dp}{dx} \right| = \left| \frac{60}{20} \cdot (-2) \right| = 6$$

Question 7

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

Accepted Answer

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

Question 8

A business wants to maximize its profit, given by $$P(x) = -2x^2$ + 40x - 100. $$What is the value of $$x$$ that maximizes profit?

Accepted Answer

$$x = 10$$

Question 9

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

Accepted Answer

$$\left[ x^3$ + x^2$ \right]_0^2$ = (8 + 4) - (0 + 0) = 12$$

Question 10

Which of the following is the correct application of the chain rule for $$f(x) = (5x + 2)^4$?

Accepted Answer

$$f'(x) = 4(5x + 2)^3$ \cdot 5$$

Question 11

A company invests $$\1000$ at a continuous compound interest rate of 5$$\%$ per year. What is the value after 3 years?

Accepted Answer

$$1000 \cdot e$$^{0.05 	imes 3}$$ \approx 1000 \cdot e$$^{0.15}$$ \approx 1161.83$$

Question 12

If $$f(x) = x^3$ - 6x^2$ + 9x + 15$$, what are the critical points?

Accepted Answer

$$x = 1$$ and $$x = 3$$

Question 13

Which of the following is the correct indefinite integral of $$f(x) = 4x^3$?

Accepted Answer

$$\int 4x^3$ \, dx = x^4$ + C$$

Question 14

A company’s revenue function is $$R(x) = 200x - x^2$. What is the maximum revenue?

Accepted Answer

$$R(100) = 200 	imes 100 - 100^2$ = 10,000$$

Question 15

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

Accepted Answer

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

Question 16

A business wants to minimize its cost function $$C(x) = x^2$ - 8x + 20. $$What is the minimum cost and at what value of $$x$$ does it occur?

Accepted Answer

Minimum at $$x = 4$$, $$C(4) = 4^2$ - 8 	imes 4 + 20 = 4$$

Question 17

Which of the following is the correct application of L'Hôpital's Rule for $$\lim_{x 	o 0} \frac{\sin x}{x}$$?

Accepted Answer

$$\lim_{x 	o 0} \frac{\sin x}{x} = \lim_{x 	o 0} \frac{\cos x}{1} = 1$$

Question 18

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

Accepted Answer

$$f'(x) = 2x e^x + x^2$ e^x$$

Question 19

A company’s profit function is $$P(x) = 50x - 0.5x^2$. What is the maximum profit?

Accepted Answer

$$P(50) = 50 	imes 50 - 0.5 	imes 50^2$ = 1250$$

Business Calculus: Course Syllabus and Weekly Study Guide

Guida di studio - Domande di pratica

Study guide - Flashcard