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Recent Notes

August 12, 2017

More Examples, Linear Systems

Observation 1. Suppose we have a system of linear equations (1)   We are looking for all real numbers such that this system has exactly one […]
August 12, 2017

System of Linear Equations

Observation 1. Consider two lines with equations and in -plane. There are three possible cases: 1. The two lines could intersect in one point. 2. The […]
August 1, 2017

Playing with a Dataset in R

Please upload the data set from this page and let us start editing this data set in R. To begin, we may want to see the first […]
July 31, 2017

Upload a Dataset to R

In this section, we want to start to work with an example in R. We will upload a data set and start to perform basic operations. […]
July 17, 2017

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. […]
July 12, 2017

Definite Integrals

In this section, we go over a few elementary examples of evaluating a definite integral. Example 1.     Example 2.     Example 3.   […]
July 9, 2017

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 […]
July 8, 2017

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 . […]
July 8, 2017

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 […]
July 3, 2017

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 , […]
June 27, 2017

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 […]
June 27, 2017

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   […]