Elementary Calculus, First Midterm, Copy 2

Elementary Calculus, First Midterm, Copy 2 1. Suppose is a given function and and are real numbers. State what we mean by (1)     Solution: […]

Elementary Calculus, First Midterm, Copy 1

Elementary Calculus, First Midterm, Copy 1 1. Suppose is a given function and is a real number. State what we mean by (1)     Solution: […]

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 […]