Free Calculus

Calculus

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These videos cover all topics traditionally covered in a college (or high school AP) level Calculus I and II course. Currently we are still working on filling out the Calculus III videos.

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Calculus Prerequisites

These videos are not all on specific math topics, but most are intended to assist you in different aspects of preparing for your Calculus class such as studying, preparing for tests, etc. There are also a few videos for brushing up on the unit circle.

Limits

Limits form the foundation on which Calculus is built. In mathematics, a limit is the value that a function “approaches” as the input, often a value of x, approaches some value. We discuss the concept of a limit, and different ways to evaluate limits.

Derivatives

Derivatives measure the rate of change in a function over an interval. In algebra class we called this rate of change “slope”. The difference in Calculus is that now we are finding the slope of a curve that changes depending on x, instead of just a straight line. There are more practical uses of derivatives than you can imagine in fields such as physics, engineering, biology, chemistry, and others. This unit is primarily concerned with explaining where derivatives come from and the actual process of how to take a derivative.

Applications of Derivatives

Now that we know the rules for how to take a derivative, we turn our attention to the various uses of derivatives. This unit assumes you are already proficient with the derivative rules, and we primarily focus on bigger picture concepts in regards to derivatives.

Integration

The ultimate goal of integration is to find the area under a curve over a given interval. Interestingly, this is somehow linked to the concept of the derivative from the last unit. We explain these connections as well as how to compute the area under a curve.

Exponential Derivatives and Integrals

We should already be familiar with exponential and logarithmic functions from a previous algebra course. In this unit we begin by reviewing some basic properties of exponentials and logarithms, and finish with how to take derivatives and integrals dealing with exponential and logarithmic functions.

Applications of Integration

In this unit we look at some advanced uses of applications. Here we assume the student is already proficient with the integration rules and we focus primarily on the big picture concepts of different ways integrals can be used.

Inverse Trigonometric and Hyperbolic Functions

Here we examine a few additional advanced families of functions and look at a few derivative and integral rules concerning these functions.

Differential Equations (Condensed Version)

A differential equation in its simplest form is any equation that contains a derivative. Many real life and environmental situations are modeled by a differential equation because they examine how things change over time. We will give an introduction to differential equations, and will look at how to solve some basic differential equations.

Advanced Integration Techniques

Sometimes the basic integration techniques from a few units ago are not sufficient to compute every integral we may encounter. Here are some ‘advanced’ integration techniques to help you integrate expressions that fit certain criteria.

Sequences and Series

Most functions we’ve seen up to this point have been defined on a continuous domain (such as the real numbers). In this unit we look at a mathematical concept called a ‘sequence’, which is basically a function that is defined on a discrete domain. We then discuss the idea of an infinite series, which is what we obtain when we add up all the terms of a sequence. We also examine when infinite series converge and diverge.

Conic Sections

Conic sections are obtained by intersecting a plane, with a double napped cone. In the videos below we will explain the basics of conic sections, as well as specifically discuss the different conic sections.

Parametric Equations

If an equation written in rectangular form represents all points (x,y) an object passes along a path, it does not reveal at what time the object passed those points. Parametric equations introduce a third variable that keeps up with the (x,y) location of an object at a particular time, t.

Polar Coordinates

Instead of plotting points in terms of x and y, we now define points in terms of its angle of elevation from the positive x-axis, and the direct distance to the point.