Question 1

Which of the following best describes an antiderivative of a function $$f on an $$interval $$I$$?

Accepted Answer

A function $$F$$ such that $$F'(x) = f(x)$$ for all $$x in$$ $$I.$$

Question 2

According to Theorem 8, what is the most general antiderivative of $$f(x) on an $$interval $$I$$?

Accepted Answer

$$F(x) + C$$, where $$C is an $$arbitrary constant.

Question 3

Which of the following is the general antiderivative of $$f(x) = x^2$?

Accepted Answer

$$\frac{1}{3}x^3$ + C$$

Question 4

What is the general antiderivative of $$f(x) = \sin x$$?

Accepted Answer

$$-\cos x + C$$

Question 5

Which formula from Table 4.2 gives the general antiderivative for $$f(x) = e$$^{kx}$$, where k$$ 
eq 0$$?

Accepted Answer

$$\frac{1}{k}e$$^{kx}$$ + C$$

Question 6

What is the general antiderivative of $$f(x) = \frac{1}{x}$$ for $$x 
eq 0$$?

Accepted Answer

$$\ln|x| + C$$

Question 7

Which rule allows you to find the antiderivative of $$kf(x)$$, where $$k is a $$constant?

Accepted Answer

Constant Multiple Rule: $$\int kf(x) dx = kF(x) + C$$

Question 8

Which of the following is the indefinite integral of $$f(x)$$ with respect to $$x$$?

Accepted Answer

$$\int f(x) dx$$

Question 9

According to the Fundamental Theorem of Calculus, Part 1, if $$f is $$continuous on $$[a, b]$$, what is the derivative of $$F(x) = \int_a^x f(t) dt$$?

Accepted Answer

$$F'(x) = f(x)$$

Question 10

If $$F(x) is an $$antiderivative of $$f(x)$$, what does the Fundamental Theorem of Calculus, Part 2, state about $$\int_a^b f(x) dx$$?

Accepted Answer

$$\int_a^b f(x) dx = F(b) - F(a)$$

Question 11

Find the area under the curve $$y = x^2$ from x = 0$ to x = 1$.

Accepted Answer

$$\frac{1}{3}$$

Question 12

Evaluate $$\$$int_3$$^6 \frac{1}{x} dx.$$

Accepted Answer

$$\ln 2$$

Question 13

What is the area under the curve $$y = \cos x$$ from $$x = 0 to$$ $$x = \frac{\pi}{2}$$?

Accepted Answer

$$1$$

Question 14

Which of the following statements about the Fundamental Theorem of Calculus is correct?

Accepted Answer

It connects differentiation and integration, showing that integration and differentiation are inverse processes.

Question 15

If $$f(x) = 3x^2$ + 5e$$^{5x}$$ and f(0) = -1$, what is the value of the constant C$ in the general antiderivative f(x$$) = $$x^3$$ + $$e^{5x} + C$?

Accepted Answer

$$C = -2$$

Question 16

Which of the following is NOT a valid application of the Fundamental Theorem of Calculus?

Accepted Answer

Applying it to a function with an infinite discontinuity inside the interval of integration.

Question 17

If $$F(x) is a $$differentiable function, what does the Net Change Theorem state?

Accepted Answer

$$F(b) - F(a) = \int_a^b F'(x) dx$$

Question 18

Which of the following is the general antiderivative of $$f(x) = \sec^2$ x$$?

Accepted Answer

$$	an x + C$$

Question 19

If $$f(x) = x^n$$ with $$n 
eq -1$$, what is the general antiderivative?

Accepted Answer

$$\frac{1}{n+1}x$$^{n+1}$$ + C$$

Antiderivatives and the Fundamental Theorem of Calculus

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