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 the function $$f(x) = 0.25x^2$ + 1 as $$shown in Figures 2, 3, and 4, which figure illustrates the use of left rectangles to approximate the area under the curve from $$x = 1 to$$ $$x = 5$$?

Accepted Answer

Figure 2

Question 3

In the context of Riemann sums, what is the main difference between the approximations shown in Figure 2 and Figure 3 for the function $$f(x) = 0.25x^2$ + 1$$?

Accepted Answer

Figure 2 uses left endpoints for rectangle heights, while Figure 3 uses right endpoints.

Question 4

As the number of rectangles increases in a Riemann sum approximation (as shown in Figure 4), what happens to the accuracy of the area estimation under the curve $$f(x) = 0.25x^2$ + 1$$?

Accepted Answer

The approximation becomes more accurate and approaches the exact area.

Question 5

Suppose you want to find the area between two curves $$y = f(x)$$ and $$y = g(x) on $$the interval $$[a, b] as $$shown in the fifth image. Which integral correctly represents this area?

Accepted Answer

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

Question 6

In the sixth image, what does the shaded region labeled 'A' represent?

Accepted Answer

The area between $$f(x) = \frac{1}{2}x + 3$$ and $$g(x) = -x^2$ + 1$$ from $$x = -2 to$$ $$x = 1$$

Question 7

Refer to the Lorenz curve and the line $$y = x in $$the seventh image. What does the area between the Lorenz curve and the line $$y = x$$ represent?

Accepted Answer

Income inequality in the population

Question 8

Given the Lorenz curve $$f(x)$$ for a country, the Gini Index is defined as $$G = 2 \$$int_0$$^1 [x - f(x)] dx. $$What does a Gini Index of 0 indicate?

Accepted Answer

Absolute equality in income distribution

Question 9

In the eighth image, which integral would give the total area between $$y = f(x)$$ and $$y = g(x)$$ from $$x = -2 to$$ $$x = 3$$?

Accepted Answer

$$\int_{-2}^3 |f(x) - g(x)| dx$$

Question 10

Suppose the velocity of a car is given by $$v(t) = 10 m/s $$for $$0 \leq t \leq 10$$ seconds. What is the total distance traveled by the car during this time interval?

Accepted Answer

$$100$$ meters

Question 11

If $$f(x) = 3x^2$ - 2$$, what is an antiderivative $$F(x) of$$ $$f(x)$$?

Accepted Answer

$$x^3 - 2x$

Question 12

Which of the following is the correct formula for the definite integral of a continuous function $$f(x)$$ from $$a to$$ $$b$$?

Accepted Answer

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

Question 13

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

$$4 - 16 = -12$$

Question 14

Suppose $$p = D(x) = 20 - 0.05x$$ and $$p = S(x) = 2 + 0.0002x^2$ are the demand and supply equations for a product. How do you find the equilibrium quantity $$\bar{x}$$?

Accepted Answer

Set $$D(x) = S(x)$$ and solve for $$x$$

Question 15

If $$f(x) is a $$probability density function on $$[a, b]$$, what must be true?

Accepted Answer

$$\int_{a}^{b} f(x) dx = 1$$

Integration and Its Applications in Business Calculus

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