Question

Explain why we call the collection of antiderivatives of a function f a family. How are the antiderivatives of a function related?

Short answer

Antiderivatives of a function are related to each other so that the given function can be equal to the derivatives of infinitely many functions, where they differ by a constant only so the collection of antiderivatives of a function f is called a family.

Step-by-step solution

Step 1. Given information. 

The given statement states that the collection of antiderivatives of a function f is called a family.

Step 2. Explanation. 

antiderivatives of a function are related to each other so that the given function can be equal to the derivatives of infinitely many functions, where they differ by a constant only so the collection of antiderivatives of a function f is called a family.

the antiderivatives of a function are differed by only a constant.

Consider two functions $f\left(x\right)\&g\left(x\right).$

where $$\begin{gathered} g\left(x\right)=f\left(x\right)+c \\ \mathrm{So}g\left(x\right)-f\left(x\right)=c \end{gathered}$$

So here c is a constant which makes them different.