Question 1

Which of the following is the correct general formula for the antiderivative of $$f(x) = x^n$$ where $$n 
eq -1$$?

Accepted Answer

$$\int x^n dx = \frac{x$$^{n+1}$$}{n+1} + C$$

Question 2

A company’s revenue function is $$R(x) = 600x - x^2$, where x$ is the number of units sold. What is the total revenue generated as x$ increases from 0 to 600?

Accepted Answer

$$\$$int_0$$^{600} (600x - x^2$) dx$$

Question 3

Which rule of anti-differentiation is most appropriate for evaluating $$\int e$$^{2x}$$ dx$$?

Accepted Answer

Substitution, with $$u = 2x$$

Question 4

The Fundamental Theorem of Calculus states that if $$f(x) is $$continuous on $$[a, b]$$ and $$F(x) is $$any antiderivative of $$f(x)$$, then:

Accepted Answer

$$\int_a^b f(x) dx = F(b) - F(a)$$

Question 5

Given the velocity function $$v(t) = 3t$$, what is the distance traveled by a particle from $$t = 1 to$$ $$t = 4$$ hours?

Accepted Answer

$$\$$int_1$$^4 3t dt = \left. \frac{3}{2} t^2$ \right|_1^4$

Question 6

If $$f(x) = 2x + 1$$ and $$g(x) = x^2$ + 1$$, what is the area of the region bounded by these two curves from $$x = 0 to$$ $$x = 2$$?

Accepted Answer

$$\$$int_0$$^2 [(2x + 1) - (x^2$ + 1)] dx$$

Question 7

Which of the following is the correct formula for integration by parts?

Accepted Answer

$$\int u dv = uv - \int v du$$

Question 8

A population of Phoenix, Arizona is modeled by $$P'(t)$$, the rate of change of population, where $$t is $$years since 1960. If $$P(0) = 790,000$$, which integral gives the population in year $$t$$?

Accepted Answer

$$P(t) = 790,000 + \$$int_0$$^t P'(s) ds$$

Question 9

What is the average value of $$f(x) = x^2$ over the interval $$[0,2]$$?

Accepted Answer

$$\frac{1}{2} \$$int_0$$^2 x^2$ dx$$

Question 10

Which substitution is appropriate for evaluating $$\int x e$$^{x^2}$$ dx$$?

Accepted Answer

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

Question 11

Given the piecewise function $$f(x) = \begin{cases} 9, & x \u003c 3 \ x^2$, & x \geq 3 \end{cases}$$, what is the area under $$f(x)$$ from $$x = -3 to$$ $$x = 6$$?

Accepted Answer

$$\int_{-3}^3 9 dx + \$$int_3$$^6 x^2$ dx$$

Question 12

Which of the following best describes the use of sigma notation in approximating the area under a curve?

Accepted Answer

It represents the sum of the areas of rectangles under the curve over subintervals.

Question 13

For the definite integral $$\int_a^c f(x) dx$$, if $$b is a $$number between $$a$$ and $$c$$, which property of definite integrals is correct?

Accepted Answer

$$\int_a^c f(x) dx = \int_a^b f(x) dx + \int_b^c f(x) dx$$

Question 14

Which of the following is the correct expression for the area between two curves $$f(x)$$ and $$g(x)$$ over $$[a, b]$$?

Accepted Answer

$$\int_a^b [f(x) - g(x)] dx$$

Question 15

Which technique is most appropriate for evaluating $$\int x \ln x dx$$?

Accepted Answer

Integration by parts, with $$u = \ln x$$ and $$dv = x dx$$

Question 16

If $$f(x) = \frac{1}{x}$$, what is the antiderivative of $$f(x)$$?

Accepted Answer

$$\int \frac{1}{x} dx = \ln|x| + C$$

Question 17

A business wants to estimate the area under the curve $$f(x) = 0.1x^3$ - 2.3x^2$ + 12x + 25$$ from $$x = 1 to$$ $$x = 16$$ using 5 subintervals. Which method should they use?

Accepted Answer

Numerical integration using sigma notation and rectangles

Question 18

Which of the following is the correct substitution for evaluating $$\int \frac{1}{x^2$ + 1} dx$$?

Accepted Answer

Let $$x = 	an 	heta$$, so $$dx = \sec^2$ 	heta d	heta$$

Business Calculus: Antiderivatives, Definite Integrals, and Applications

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