Question 1

Which of the following best describes the meaning of the shaded area under the velocity-time graph shown in the first image (constant velocity of 10 m/s from t = 0 to t = 10 seconds)?

Accepted Answer

It represents the total distance traveled by the object in 10 seconds.

Question 2

Given $$f(x) = 3x^2$ - 2$$, which of the following is an antiderivative of $$f(x)$$?

Accepted Answer

$$F(x) = x^3$ - 2x$$

Question 3

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

Accepted Answer

Reversal of limits property

Question 4

Suppose $$\$$int_0$$^2 x dx = 2$$ and $$\$$int_0$$^2 x^2$ dx = \frac{8}{3}. $$What is $$\$$int_0$$^2 (2x - 6x^2$) dx$$?

Accepted Answer

$$4 - 16 = -12$$

Question 5

Which figure (2, 3, or 4) best represents a right Riemann sum approximation for $$\$$int_1$$^5 (0.25x^2$ + 1) dx$$?

Accepted Answer

Figure 3

Question 6

Evaluate $$\$$int_0$$^1 e$$^{2x}$$ dx.$$

Accepted Answer

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

Question 7

A company’s marginal cost function is $$C'(x) = 400 - \frac{1}{3}x$$ for $$0 \leq x \leq 900. $$What is the increase in cost when production increases from 450 to 900 units?

Accepted Answer

$$\int_{450}^{900} (400 - \frac{1}{3}x) dx$$

Question 8

If $$u = 3x^2$ + 4$$, what is $$du in $$terms of $$dx$$?

Accepted Answer

$$du = 6x dx$$

Question 9

Which of the following integrals can be evaluated using the substitution $$u = x^3$ + 5$$?

Accepted Answer

$$\int 4x^2$ \sqrt{x^3$ + 5} dx$$

Question 10

The area between the curves $$y = f(x)$$ and $$y = g(x)$$ from $$x = a to$$ $$x = b (as $$shown in the fifth image) is given by:

Accepted Answer

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

Question 11

Given the Lorenz curve $$y = f(x)$$ for income distribution, what does the point (0.40, 0.11) on the curve represent (see the seventh image)?

Accepted Answer

The bottom 40% of families receive 11% of the total income.

Question 12

If $$\$$int_0$$^1 f(x) dx = 1$$ and $$\$$int_1$$^2 f(x) dx = 3$$, what is $$\$$int_0$$^2 f(x) dx$$?

Accepted Answer

$$4$$

Question 13

Suppose $$f(x) is a $$probability density function on $$[0, 2]. $$Which of the following must be true?

Accepted Answer

$$\$$int_0$$^2 f(x) dx = 1$$

Question 14

The Gini Index for a Lorenz curve $$y = f(x) is $$given by:

Accepted Answer

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

Question 15

If $$\$$int_2$$^5 (11t - 5) dt$$ represents the displacement of a particle, what is the correct interpretation of this definite integral?

Accepted Answer

The total change in position of the particle from $$t = 2 to$$ $$t = 5$$ seconds.

Question 16

A continuous income stream flows at a rate $$f(t) = 5000e$$^{0.04t}$$ dollars per year. What is the total income produced in the first 5 years?

Accepted Answer

$$\$$int_0$$^5 5000e$$^{0.04t}$$ dt$$

Question 17

Which of the following is the correct formula for the average value of a continuous function $$f$$ over $$[a, b]$$?

Accepted Answer

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

Question 18

If $$p = D(x) = 20 - 0.05x$$ and $$p = S(x) = 2 + 0.0002x^2$, what is the equilibrium quantity $$\bar{x}$$?

Accepted Answer

The value of $$x$$ where $$D(x) = S(x)$$

Integration and Its Applications in Business Calculus

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