Question 1

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

Accepted Answer

$$\int \ln x\,dx = x\ln x - x + C$$

Question 2

Suppose the marginal profit for producing $$x$$ units is given by $$P'(x) = 6x^2$ - 4x + 10. If $$the profit from producing 0 units is $$P(0) = 500$$, what is the total profit from producing 3 units?

Accepted Answer

$$P(3) = 500 + \$$int_0$$^3 (6x^2$ - 4x + 10)\,dx = 500 + [2x^3$ - 2x^2$ + 10x]_0^3$ = 500 + (54 - 18 + 30) = 500 + 66 = 566$$

Question 3

Given the sales function $$S(t) = 100e$$^{0.05t}$$, where t$ is in weeks, what is the average weekly sales over the first 10 weeks?

Accepted Answer

$$\frac{1}{10} \$$int_0$$^{10} 100e$$^{0.05t}$$\,dt = \frac{1}{10} \left[2000e$$^{0.05t}$$\right]_0$$^{10}$$ = \frac{1}{10}(2000e$$^{0.5}$$ - 2000)$$

Question 4

If the marginal cost function for producing $$x$$ units is $$C'(x) = 5x + 20$$, and the fixed cost is $$C(0) = 1000$$, what is the total cost to produce 10 units?

Accepted Answer

$$C(10) = 1000 + \$$int_0$$^{10} (5x + 20)\,dx = 1000 + (2.5 	imes 100 + 200) = 1000 + 250 + 200 = 1450$$

Question 5

Evaluate $$\int x e$$^{x}$$ dx$$ using integration by parts.

Accepted Answer

$$\int x e$$^{x}$$ dx = x e$$^{x}$$ - \int e$$^{x}$$ dx = x e$$^{x}$$ - e$$^{x}$$ + C$$

Question 6

Find the area between the curves $$y = x^2$ and y = 2x$ from x = 0$ to x = 2$.

Accepted Answer

$$\$$int_0$$^2 (2x - x^2$) dx = [x^2$ - \frac{1}{3}x^3$]_0^2$ = (4 - \frac{8}{3}) - 0 = \frac{4}{3}$$

Question 7

Which substitution would you use to evaluate $$\int 2x \cos(x^2$) dx$$?

Accepted Answer

Let $$u = x^2$, so du = 2x dx$

Question 8

A company’s revenue function is $$R(x) = 50x - 0.5x^2$. What is the accumulated revenue from selling 0 to 20 units?

Accepted Answer

$$\$$int_0$$^{20} (50x - 0.5x^2$) dx = [25x^2$ - \frac{0.5}{3}x^3$]_0$$^{20}$$ = (25 	imes 400 - \frac{0.5}{3} 	imes 8000) = (10000 - 1333.33) = 8666.67$$

Question 9

Evaluate $$\$$int_1$$^e \frac{\ln x}{x} dx.$$

Accepted Answer

$$\frac{1}{2}(\ln x)^2$ \Big|_1^e = \frac{1}{2}(1^2$ - 0^2$) = \frac{1}{2}$$

Question 10

If the average value of a function $$f(x) on$$ $$[a, b] is $$given by $$A = \frac{1}{b-a} \int_a^b f(x) dx$$, what is the average value of $$f(x) = 3x^2$ on $$[0, 2]$$?

Accepted Answer

$$\frac{1}{2} \$$int_0$$^2 3x^2$ dx = \frac{1}{2} [x^3$]_0^2$ = \frac{1}{2}(8 - 0) = 4$$

Question 11

A company’s marginal revenue is $$MR(x) = 20 - 0.2x. If no $$units are sold, revenue is zero. What is the total revenue from selling 50 units?

Accepted Answer

$$\$$int_0$$^{50} (20 - 0.2x) dx = [20x - 0.1x^2$]_0$$^{50}$$ = (1000 - 250) = 750$$

Question 12

Evaluate $$\$$int_0$$^1 (3x^2$ + 2x + 1) dx.$$

Accepted Answer

$$[x^3$ + x^2$ + x]_0^1$ = (1 + 1 + 1) - 0 = 3$$

Question 13

Which of the following is the correct setup to find the area between $$y = x$$ and $$y = x^2$ from x = 0$ to x = 1$?

Accepted Answer

$$\$$int_0$$^1 (x - x^2$) dx$$

Question 14

If $$\int f(x) dx = F(x) + C$$, which of the following statements is true?

Accepted Answer

$$F'(x) = f(x)$$

Question 15

A company’s marginal cost is $$MC(x) = 4x + 100. $$What is the increase in total cost when production increases from 10 to 20 units?

Accepted Answer

$$\int_{10}^{20} (4x + 100) dx = [2x^2$ + 100x]_{10}^{20} = (800 + 2000) - (200 + 1000) = 2800 - 1200 = 1600$$

Question 16

Evaluate $$\int x \ln x dx$$ using integration by parts.

Accepted Answer

$$\int x \ln x dx = \frac{x^2$}{2} \ln x - \frac{x^2$}{4} + C$$

Question 17

Suppose the marginal profit for a product is $$MP(x) = 12 - 0.3x$$, and the profit from selling 0 units is $$P(0) = 100. $$What is the profit from selling 10 units?

Accepted Answer

$$P(10) = 100 + \$$int_0$$^{10} (12 - 0.3x) dx = 100 + [12x - 0.15x^2$]_0$$^{10}$$ = 100 + (120 - 15) = 205$$

Question 18

Which of the following is the correct formula for the average value of $$f(x) on$$ $$[a, b]$$?

Accepted Answer

$$\frac{1}{b-a} \int_a^b f(x) dx$$

Business Calculus Integration and Application Study Guide

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