Question 1

Evaluate the definite integral $$\int_{-2}^{3} (4x^3$ - 6x + 1) \, dx.$$

Accepted Answer

$$\ 212$$

Question 2

Which of the following is the correct antiderivative of $$4x^3 - 6x + 1$?

Accepted Answer

$$\ x^4$ - 3x^2$ + x + C$$

Question 3

Evaluate the definite integral $$\int_{1}^{4} \$$left(-3x^{-\frac{1}$${2}} + x$$^{\frac{1}$${2}}\right) dx.$$

Accepted Answer

$$\ -3$$

Question 4

A company’s marginal cost function is $$C'(x) = 4x^3$ - 6x + 1. $$What does $$\int_{0}^{10} C'(x) dx$$ represent in a business context?

Accepted Answer

The change in total cost when production increases from 0 to 10 units.

Question 5

Evaluate the definite integral $$\int_{1}^{6} (2x$$^{-1}$$ - e^x) dx.$$

Accepted Answer

$$\ 2\ln(6) - (e^6$ - e^1$)$$

Question 6

Which of the following is the correct antiderivative of $$2x^{-1} - e^x$?

Accepted Answer

$$\ 2\ln|x| - e^x + C$$

Question 7

Suppose the rate of change of a population is given by $$P'(t) = 3(2t+1)e$$^{t^2 + t}$$. What is the general solution for the population function P(t$$)$$?

Accepted Answer

$$\ P(t) = 3e$$^{t^2 + t}$$ + C$$

Question 8

Evaluate $$\int_{-1}^{2} 3(2t+1)e$$^{t^2 + t}$$ dt$$ using substitution.

Accepted Answer

Question 9

Which substitution is most appropriate for evaluating $$\int 3(2t+1)e$$^{t^2 + t}$$ dt$$?

Accepted Answer

$$\ u = t^2$ + t$$

Question 10

A business’s revenue function is $$R(x) = \int_{0}^{x} (4t^3$ - 6t + 1) dt. $$What is $$R'(x)$$?

Accepted Answer

$$\ 4x^3$ - 6x + 1$$

Question 11

If $$\int_{a}^{b} f(x) dx = 0$$, which of the following must be true?

Accepted Answer

The area under $$f(x)$$ from $$a to$$ $$b is $$zero.

Question 12

Which of the following best describes the Fundamental Theorem of Calculus?

Accepted Answer

It relates differentiation and integration, stating that if $$F(x) is an $$antiderivative of $$f(x)$$, then $$\int_{a}^{b} f(x) dx = F(b) - F(a).$$

Question 13

A company’s profit function is $$P(x) = \int_{0}^{x} (2t$$^{-1}$$ - e^t) dt. $$What is the marginal profit at $$x = 5$$?

Accepted Answer

$$\ 2(5$$)^{-1}$$ - e$$^{5}$$

Question 14

Evaluate $$\int_{1}^{4} x$$^{1/2}$$ dx.$$

Accepted Answer

$$\ \frac{2}{3}(4$$^{3/2}$$ - 1)$$

Question 15

Which of the following is the correct antiderivative of $$x^{1/2}$$?

Accepted Answer

$$\ \frac{2}{3}x$$^{3/2}$$ + C$$

Definite Integrals Practice – Business Calculus Guidance

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