Integration by Substitution

Example 1. Suppose we want to evaluate the following integral     Of course, one way to do this is to expand and integrate each term. […]

Definite Integrals

In this section, we go over a few elementary examples of evaluating a definite integral. Example 1.     Example 2.     Example 3.   […]

Antiderivatives

Observation 1. Suppose we are asked to find a function such that if we differentiate it, we get . Anyone who has studied differentiation will suggest […]

Exponential & Logarithmic Differentiation

Example 1. Suppose we are given and want to differentiate . We write     Example 2. Suppose we are given and want to differentiate . […]

Related Rates

Example 1. Suppose we are given a cone with height equal times radius at any given height. Assume that we know the rate of change of […]

Implicit Differentiation

Example 1. We are given and that is a function of We would like to find implicitly. Start by differentiating each side with respect to , […]

Extreme Value Theorem

Example 1. Suppose we are given for all in . Since the function is continuous and the given interval is closed, the Extreme Value Theorem guarantees […]

Optimization Examples

Example 1. We would like to find to positive numbers and such that their sum is equal to and their product is maximum. Note that   […]

Second Derivative Test

Example 1. Suppose we are given and we would like to locate any relative maximum or minimum for this function. We follow the following steps: Step […]