Question 1

Which of the following best describes the meaning of the shaded area under the velocity-time graph 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

Evaluate the indefinite integral $$\int 5t^7$ dt.$$

Accepted Answer

$$\frac{5}{8} t^8$ + C$$

Question 4

Suppose $$f'(x) = x$$^{-3}$$ and f(2) = 9$. Find a formula for f(x$$)$$.

Accepted Answer

$$f(x) = -\frac{1}{2} x$$^{-2}$$ + 9.125$$

Question 5

Which of the following is the correct substitution for $$u in $$the integral $$\int (3x^2$ + 4)^4$ 6x dx$$?

Accepted Answer

$$u = 3x^2$ + 4$$, $$du = 6x dx$$

Question 6

Which figure (2, 3, or 4) best represents a left Riemann sum approximation for the area under $$f(x) = 0.25x^2$ + 1 on$$ $$[1,5]$$?

Accepted Answer

Figure 2

Question 7

Evaluate the definite integral $$\$$int_0$$^2 x dx.$$

Accepted Answer

$$2$$

Question 8

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

Accepted Answer

Reversal of limits property

Question 9

Suppose $$C'(t) = 2$$ and $$R'(t) = 9e$$^{-0.5t}$$ represent the cost and revenue rates for a video game. The useful life of the game is when C$$'(t) = R'(t)$$. What is the value of t$ (rounded to the nearest year)?

Accepted Answer

$$t = 3$$

Question 10

Given the Lorenz curve $$f(x) = x$$^{2.6}$$ for a country, what does the point (0.40$$, 0.11)$$ on the curve represent?

Accepted Answer

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

Question 11

Which formula gives the area between two curves $$y = f(x)$$ and $$y = g(x)$$ from $$x = a to$$ $$x = b$$?

Accepted Answer

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

Question 12

A probability density function $$f(x)$$ must satisfy which of the following conditions?

Accepted Answer

$$f(x) \geq 0$$ for all $$x$$, and $$\int_{-\infty}^{\infty} f(x) dx = 1$$

Question 13

If the marginal cost function for producing bikes is $$C'(x) = 400 - \frac{1}{3}x$$, what is the increase in cost when production increases from 450 to 900 bikes?

Accepted Answer

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

Question 14

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

Accepted Answer

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

Question 15

If $$\$$int_0$$^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

$$2 - 16 = -14$$

Question 16

A vending machine produces income 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

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

Accepted Answer

$$\bar{x} is $$the solution to $$20 - 0.05x = 2 + 0.0002x^2$

Question 18

Which of the following best describes the effect of increasing the number of rectangles in a Riemann sum approximation (as shown in Figure 4)?

Accepted Answer

The approximation becomes more accurate as the number of rectangles increases.

Integration and Its Applications in Business Calculus

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