Question 1

Which of the following is the correct Power Rule for antiderivatives, as shown in the first image?

Accepted Answer

$$\int x^n dx = \frac{1}{n+1} x$$^{n+1}$$ + C$$

Question 2

In the context of Riemann sums (see image 2), what happens to the approximation of the area under a curve as the number of subintervals increases?

Accepted Answer

The approximation becomes more accurate.

Question 3

Given the function $$f(x) = x^2$, what is the exact area under the curve from x = 0$ to x = 2$ (see image 4)?

Accepted Answer

$$\$$int_0$$^2 x^2$ dx = \frac{8}{3}$$

Question 4

If $$f(x) = x^2$ - 1 ($$see image 5), what does the definite integral $$\$$int_0$$^2 f(x) dx$$ represent?

Accepted Answer

The area above the x-axis minus the area below the x-axis.

Question 5

Which of the following is the correct formula for consumer surplus at the equilibrium point $$(x_E, p_E)$$, as illustrated in image 7?

Accepted Answer

$$\$$int_0$$^{x_E} D(x) dx - x_E p_E$$

Question 6

For the marginal profit function $$P'(x) = \frac{9000 - 3000x}{(x^2$ - 6x + 10)^2$} ($$see image 6), what does the area under the curve from $$x = 3 to$$ $$x = 7$$ represent?

Accepted Answer

The total profit earned between $$x = 3$$ and $$x = 7$$ units.

Question 7

Which integration technique is most appropriate for evaluating $$\int 3x^2$(x^3$$ + $$1)^{10} dx$?

Accepted Answer

Substitution (u-substitution)

Question 8

If $$P_0 is $$invested at interest rate $$k$$ compounded continuously for $$t$$ years, which formula gives the future value?

Accepted Answer

$$P(t) = P_0$ e$$^{kt}$$

Question 9

Given a continuous income stream $$R(t)$$ invested at rate $$k$$ for $$T$$ years, what is the formula for the accumulated present value if $$R(t) is $$constant?

Accepted Answer

$$B = \frac{R(t)}{k} (1 - e$$^{-kT}$$)$$

Question 10

For the Cobb-Douglas production function $$p(x, y) = Ax^a y$$^{1-a}$$, what does $$\frac{\partial p}{\partial x}$$ represent?

Accepted Answer

Marginal productivity of labor

Question 11

If $$\$$int_1$$^4 x^2$ dx is $$evaluated, what is the result?

Accepted Answer

$$\frac{4^3$}{3} - \frac{1^3$}{3} = \frac{64}{3} - \frac{1}{3} = \frac{63}{3} = 21$$

Question 12

Which of the following best describes the process shown in image 3, where more subintervals are used?

Accepted Answer

The rectangles become narrower and the approximation improves.

Question 13

If $$\int (2x + 3) dx$$ and $$F(1) = -2$$, what is the value of the constant of integration $$C$$?

Accepted Answer

$$C = -4$$

Question 14

Which formula gives the present value of an amount $$P$$ due $$t$$ years later at interest rate $$k$$ compounded continuously?

Accepted Answer

$$P_0 = P$$ $$e^{-kt}$$

Question 15

If $$\int \frac{1}{x} dx is $$evaluated for $$x \u003e 0$$, what is the result?

Accepted Answer

$$\ln x + C$$

Question 16

For the function $$f(x, y) = x^2$ y^3$ + xy + 4y^2$, what is $$\frac{\partial f}{\partial x}$$?

Accepted Answer

$$2xy^3 + y$

Question 17

If $$R(t) = 275,000$$ and $$k = 0.033$$, what is the accumulated future value of the income stream over 8 years?

Accepted Answer

$$A = \frac{R(t)}{k} (e$$^{kT}$$ - 1)$$

Question 18

Which of the following is NOT a property of antiderivatives?

Accepted Answer

The antiderivative of a product is the product of the antiderivatives.

Business Calculus: Integration, Applications, and Multivariable Calculus

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