## Section1.1Antiderivatives

You have probably taken a course in Differential Calculus, where you have solved problems of the following nature:

• Given $f(x)=\sin(x)+3x^5-12\text{,}$ find $f'(x)\text{.}$

• Given the position function of an object, determine its velocity function.

• Given the demand function, calculate the elasticity of demand.

To solve any of these problems we need the concept of the derivative, which provides us with information about the rate of change of the quantity involved that leads to the solution. In other words, Differential Calculus allows us to solve problems that are concerned with finding the rate of change of one quantity with respect to another quantity.

The next three chapters are based on the idea of the antiderivative, which basically helps us solve problems that are the reverse of the above problems such as

• Given $f'(x)=\sin(x)+3x^5-12\text{,}$ find $f(x)\text{.}$

• Given the velocity function of an object, determine its position function.

• How much do consumers benefit by purchasing some manufactured goods at the price determined by supply and demand?

We will develop tools for finding the antiderivative, which is the process of antidifferentiation also known as integration. Hence, these kinds of problems fall under the topic of Integral Calculus. In summary, Integral Calculus allows us to solve problems, where the rate of change of one quantity is given with respect to another quantity and we are concerned with finding the relationship between these two quantities.

###### Definition1.1. Antiderivative.

A function $F$ is an antiderivative of $f$ on an interval $I$ if

\begin{equation*} F'(x)=f(x) \end{equation*}

for all $x$ in $I\text{.}$