Lesson Series
There are two important scenarios for which a function limit does not exist. The first is when the limit produces two different values dependent upon whether we approach from the left or the right. The second is when the function approaches positive or negative infinity.
We show how marginal cost, marginal revenue and marginal profit in business economics are related to derivatives.
We describe how a function continuous on $[a,b]$ will take on both a maximum value and a minimum value on $[a,b]$.
Here we show that a derivative is a function limit. We provide some simple examples of how the function limit definition is used.
A power function is a function of the form $Cx^r$ where $C$ and $r$ are real numbers.
We review inequalities and outline some basic methods for solving them.
In this lesson we derive the formula for the tangent line given that we know the function and its derivative.
We review the Quadratic Equation and find the two values of $x$ for which $ax^2+bx+c=0$ is true.
The Binomial Theorem is a formula for determining the answer to $(x+y)^n$.
We define the upper and lower sums of a bounded function and a partition on it. From the upper and lower sums we define the definite integral and provide examples.
A polynomial is a function like $a_nx^n+a_{n-1}x^{n-1}+\cdots+a_1x+a_0$ where $n$ is a positive integer.
Here we provide the mathematically rigorous definition of a function limit and provide explicit steps on how to prove a function limit exists.
We discuss the Mean-Value Theorem which states that for a continuous function the slope of a secant line is equal to the derivative at some point.
We discuss here how a function is continuous at a point if it is differentiable at that point. However, the converse is not true. If a function is continuous at a point it is not guaranteed that it will be differentiable there.
The Chain Rule shows how to find the derivative of a function composition. The Chain Rule states that \[\frac{d}{dx}\left[f\circ g\right] =\frac{df}{dg}\cdot\frac{dg}{dx}.\]
