Antiderivatives in Calculus

Questions on the concepts and properties of antiderivatives in calculus are presented. These questions have been designed to help you better understand the concept and properties of antiderivatives. In order to answer the questions below, you first need to review the definitions and theorems related to antiderivatives.

Question 1:
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.
Differentiate F(x) + c.

Question 2:
True or False. If F(x) is an antiderivative of f(x), then
(1/c) F(c x) is an antiderivative of f(c x), where c is any non zero constant.
Answer :
True.
Let u = c x and Differentiate (1 / c) F(c x) with respect to x
d/dx( (1 / c) F(c x) )
= (1 / c) d(u)/dx dF/du (chain rule)
= (1/c) c f(u) = f(c x)

Question 3:
True or False. An antiderivative of function f plus an antiderivative of function g is an antiderivative of function f + g.
Answer :
True. Use the rule of differentiation to differentiate F + G, where F is the antiderivative of f and G is the antiderivative of g, and see that you can get f + g.

Question 4:
True or False. An antiderivative of function f divided by an antiderivative of function g is an antiderivative of function f / g.
Answer :
False. Use the rule of differentiation to differentiate F / G, where F is the antiderivative of f and G is the antiderivative of g, and see that you cannot get f / g.

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