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

$$P'(10) = 0.05 \times 1000e$$^{0.5}$$ \approx 82$$

Q: A medication decays in the bloodstream according to $$C(t) = 50e$$^{-0.2t}$$. What is the rate of change of concentration at t = 5$?

$$C'(5) = -0.2 \times 50e$$^{-1}$$ \approx -3.68$$

Q: A company’s cost function is $$C(x) = 200 + 5x. $$What is the average cost per item when $$x = 20$$?

$$C_{avg} = \frac{200 + 5 \times 20}{20} = 15$$

Q: A savings account earns interest compounded continuously at a rate of $$6\%. If $$the initial deposit is $$\500$, what is the amount after 5$ years?

$$A = 500e$$^{0.06 \times 5}$$ \approx 674$$

Question 1

Which of the following functions is NOT linear?

Accepted Answer

$$f(x) = x^2$ + 3$$

Question 2

A company’s profit $$P (in $$thousands of dollars) is modeled by $$P(x) = 5x^2$ - 20x + 100$$, where $$x is $$the number of units sold (in thousands). What is the minimum profit, and at how many units sold does it occur?

Accepted Answer

Minimum profit is $$80 at$$ $$x = 2$$

Question 3

Which of the following limits does NOT exist?

Accepted Answer

$$\lim_{x 	o 0} \frac{1}{x}$$

Question 4

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

Accepted Answer

$$f'(x) = 12x^3$ - 4x$$

Question 5

A business’s revenue $$R(x) is $$given by $$R(x) = 100x - x^2$, where x$ is the number of items sold. What is the marginal revenue when x = 10$?

Accepted Answer

$$R'(10) = 80$$

Question 6

Which rule should be used to find the derivative of $$f(x) = (2x^3$ + 1)(x^2$ - 5)$$?

Accepted Answer

Product Rule

Question 7

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

Accepted Answer

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

Question 8

The price $$p$$ and quantity $$q of a $$product are related by the demand function $$p = 50 - 2q. $$What is the elasticity of demand when $$q = 10$$?

Accepted Answer

Elasticity $$E = \frac{2q}{50 - 2q}$$, so $$E = \frac{20}{30} = 0.67$$

Question 9

Which of the following is the correct application of the Chain Rule for $$f(x) = \sin(3x^2$)$$?

Accepted Answer

$$f'(x) = \cos(3x^2$) \cdot 6x$$

Question 10

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

Accepted Answer

$$P'(10) = 0.05 	imes 1000e$$^{0.5}$$ \approx 82$$

Question 11

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

Accepted Answer

$$f''(x) = 6x - 8$$

Question 12

Which of the following is the correct derivative of $$f(x) = \ln(x^2$ + 1)$$?

Accepted Answer

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

Question 13

A medication decays in the bloodstream according to $$C(t) = 50e$$^{-0.2t}$$. What is the rate of change of concentration at t = 5$?

Accepted Answer

$$C'(5) = -0.2 	imes 50e$$^{-1}$$ \approx -3.68$$

Question 14

Evaluate the definite integral $$\$$int_0$$^2 (3x^2$ - 2x + 1) dx.$$

Accepted Answer

$$8$$

Question 15

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

Accepted Answer

$$F(x) = x^4$ + C$$

Question 16

A company’s cost function is $$C(x) = 200 + 5x. $$What is the average cost per item when $$x = 20$$?

Accepted Answer

$$C_{avg} = \frac{200 + 5 	imes 20}{20} = 15$$

Question 17

Which of the following is the correct application of the Quotient Rule for $$f(x) = \frac{x^2$}{x + 1}$$?

Accepted Answer

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

Question 18

If $$f(x) = \sqrt{x}$$, what is $$f'(x)$$?

Accepted Answer

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

Question 19

A savings account earns interest compounded continuously at a rate of $$6\%. If $$the initial deposit is $$\500$, what is the amount after 5$ years?

Accepted Answer

$$A = 500e$$^{0.06 	imes 5}$$ \approx 674$$

Business Calculus: Course Syllabus and Topic Overview

Guida di studio - Domande di pratica

Study guide - Flashcard