Question 1

Which of the following best describes the definition of an antiderivative of a function $$f(x)$$?

Accepted Answer

A function $$F(x)$$ such that $$F'(x) = f(x)$$ for all $$x in $$the domain of $$f$$

Question 2

What is the most general antiderivative of $$f(x) = rac{3}{x^2$}$$?

Accepted Answer

$$F(x) = -3x$$^{-1}$$ + C$$

Question 3

If $$F(x) is an $$antiderivative of $$f(x) on an $$interval, what is the most general form of all antiderivatives of $$f(x) on $$that interval?

Accepted Answer

$$F(x) + C$$, where $$C is $$any constant

Question 4

Which of the following is the most general antiderivative of $$f(x) = rac{1}{x}$$?

Accepted Answer

$$F(x) = \ln|x| + C$$

Question 5

Find the most general antiderivative of $$f(x) = x^n$$, where $$n 
eq -1.$$

Accepted Answer

$$F(x) = \frac{x$$^{n+1}$$}{n+1} + C$$

Question 6

Which of the following is the most general antiderivative of $$f(x) = a$$, where $$a is a $$constant?

Accepted Answer

$$F(x) = ax + C$$

Question 7

Which function is an antiderivative of $$f(x) = \cos(x)$$?

Accepted Answer

$$F(x) = \sin(x) + C$$

Question 8

A medication is administered at a constant rate of $$5 mg/$$hour. What is the total amount of medication in the body after $$t$$ hours, assuming no medication is lost?

Accepted Answer

$$5t$$

Question 9

Find all functions $$g(x)$$ such that $$g'(x) = 4\sin(x) + 2x^2$ + x$$^{-2}$$.

Accepted Answer

$$g(x) = -4\cos(x) + \frac{2}{3}x^3$ - x$$^{-1}$$ + C$$

Question 10

Suppose $$f'(x) = e^x$$ and $$f(0) = -4. $$What is $$f(x)$$?

Accepted Answer

$$f(x) = e^x - 4$$

Question 11

Find $$F(x)$$ given that $$F''(x) = 12x^2$ - 6x - 4$$ and $$F(1) = 1$$, $$F'(1) = -4.$$

Accepted Answer

$$F(x) = x^3$ - 3x^2$ - 2x + 4$$

Question 12

Which of the following is NOT a valid antiderivative rule for the function $$f(x) = \sec(x)$$?

Accepted Answer

$$F(x) = 	an(x) + C$$

Question 13

A population of bacteria grows at a rate proportional to its current size, $$\frac{dP}{dt} = kP. $$What is the general solution for $$P(t)$$?

Accepted Answer

$$P(t) = Ce$$^{kt}$$

Question 14

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

Accepted Answer

$$F(x) = e^x + C$$

Question 15

If $$f(x) = x^2$ + 5$$, what is the most general antiderivative of $$f(x)$$?

Accepted Answer

$$F(x) = \frac{x^3$}{3} + 5x + C$$

Question 16

Which of the following statements is true about the constant $$C in $$the general antiderivative $$F(x) + C$$?

Accepted Answer

It represents any real number and accounts for all possible vertical shifts of the antiderivative.

Question 17

Which of the following is the most general antiderivative of $$f(x) = \sin(x)$$?

Accepted Answer

$$F(x) = -\cos(x) + C$$

Question 18

A car accelerates at a constant rate $$a. If $$its initial velocity is $$v_0$$, what is its velocity after $$t$$ seconds?

Accepted Answer

$$v(t) = v_0$ + at$$

Antiderivatives and Indefinite Integration: Study Notes

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