Question 1

Which of the following best describes the process shown in the first image for solving integrals using substitution?

Accepted Answer

Start with a problem in $$x$$, make a substitution $$u = g(x)$$, solve the problem in $$u$$, then substitute back to $$x.$$

Question 2

Given the property $$\int_a^b k f(x) dx = k \int_a^b f(x) dx$$, where $$k is a $$constant, what is $$\$$int_1$$^3 7x^2$ dx$$?

Accepted Answer

$$7 \$$int_1$$^3 x^2$ dx$$

Question 3

Evaluate $$\$$int_2$$^2 (3x^2$ - 4x + 5) dx.$$

Accepted Answer

$$0$$

Question 4

Which property of definite integrals is illustrated by $$\int_a^b f(x) dx = -\int_b^a f(x) dx$$?

Accepted Answer

Reversing the limits of integration changes the sign of the integral.

Question 5

Suppose $$\$$int_1$$^4 f(x) dx = 7$$ and $$\$$int_1$$^4 g(x) dx = -2. $$What is $$\$$int_1$$^4 [f(x) + g(x)] dx$$?

Accepted Answer

$$5$$

Question 6

If $$\$$int_1$$^3 f(x) dx = 8$$ and $$\$$int_3$$^5 f(x) dx = 6$$, what is $$\$$int_1$$^5 f(x) dx$$?

Accepted Answer

$$14$$

Question 7

A company’s marginal cost function is $$C'(x) = -0.06x + 23. If $$the cost to produce 175 items is $$8,046.25, what is the cost function C(x$$)$?

Accepted Answer

$$C(x) = -0.03x^2$ + 23x + C$$

Question 8

Evaluate $$\int 4e$$^{4x+8}$$ dx$$ using substitution.

Accepted Answer

$$e^{4x+8} + C$

Question 9

A medical researcher models antibody production as $$N'(t) = \frac{3,000}{10t+20} (in $$thousands per day), with $$N(0) = 0. $$What is the function $$N(t)$$?

Accepted Answer

$$N(t) = 300 \ln(10t+20)$$

Question 10

Find the area between $$f(x) = 2x + 7$$ and $$g(x) = x^2$ - 4x + 12$$ from $$x = 1 to$$ $$x = 3.$$

Accepted Answer

$$\$$int_1$$^3 [(2x+7) - (x^2$-4x+12)] dx$$

Question 11

If $$f(x) is $$continuous and nonnegative on $$[a, b]$$, what does $$\int_a^b f(x) dx$$ represent?

Accepted Answer

The area under $$f(x)$$ from $$x = a to$$ $$x = b$$

Question 12

Which substitution would be most appropriate for evaluating $$\int \frac{x}{\sqrt{x+4}} dx$$?

Accepted Answer

Let $$u = x + 4$$

Question 13

A company’s revenue and cost functions over 15 years are $$R(x) = 90e$$^{0.05x}$$ and C(x$$) = $$0.4x^2 + 80$. What integral gives the total profit over 15 years?

Accepted Answer

$$\$$int_0$$^{15} [90e$$^{0.05x}$$ - (0.4x^2$ + 80)] dx$$

Question 14

Given the demand function $$D(x) = 1,500 - 3x^2$ and equilibrium price p_e = 300$, what is the consumer’s surplus?

Accepted Answer

$$\$$int_0$$^{x_e} [1,500 - 3x^2$ - 300] dx$$

Question 15

If $$\$$int_0$$^2 f(x) dx = 5$$ and $$\$$int_2$$^5 f(x) dx = -3$$, what is $$\$$int_5$$^0 f(x) dx$$?

Accepted Answer

$$-2$$

Question 16

Which of the following is a correct application of the Fundamental Theorem of Calculus?

Accepted Answer

$$\int_a^b f(x) dx = F(b) - F(a)$$, where $$F(x) is an $$antiderivative of $$f(x)$$

Business Calculus Study Guide: Integration and Its Applications

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