This page presents practice questions on the concepts and properties of antiderivatives in calculus. The questions are designed to help you build a solid understanding of how antiderivatives work and how they relate to differentiation.
Before attempting the questions, it is recommended that you review the definitions and basic theorems related to antiderivatives.
True or False. If \( F(x) \) is an antiderivative of \( f(x) \) and \( c \) is any constant, then \( F(x) + c \) is also an antiderivative of \( f(x) \).
Answer: True.
Differentiating \( F(x) + c \) gives
\[
\frac{d}{dx}\bigl(F(x) + c\bigr) = F'(x) = f(x).
\]
True or False. If \( F(x) \) is an antiderivative of \( f(x) \), then \[ \frac{1}{c} F(cx) \] is an antiderivative of \( f(cx) \), where \( c \neq 0 \).
Answer: True.
Let \( u = cx \). Differentiating,
\[
\frac{d}{dx}\left( \frac{1}{c} F(cx) \right)
= \frac{1}{c} \cdot c \cdot F'(u)
= f(cx).
\]
True or False. An antiderivative of \( f \) plus an antiderivative of \( g \) is an antiderivative of \( f + g \).
Answer: True.
If \( F' = f \) and \( G' = g \), then
\[
\frac{d}{dx}(F + G) = F' + G' = f + g.
\]
True or False. An antiderivative of \( f \) divided by an antiderivative of \( g \) is an antiderivative of \( \dfrac{f}{g} \).
Answer: False.
Differentiating \( \dfrac{F}{G} \) gives
\[
\frac{d}{dx}\left(\frac{F}{G}\right)
= \frac{F'G - FG'}{G^2},
\]
which is not equal to \( \dfrac{f}{g} \) in general.