Lesson Series
Here we describe how a definite integral and the area it represents can be negative. There are typically two ways this can happen. The first is when the function $f(x)$ is negative. The second is when we integrate in the opposite direction so that
\begin{equation}
\int_a^b\,f(x)dx=-\int_b^a\,f(x)dx.
\end{equation}
This lesson provides a list of continuous functions.
We provide some useful examples and a graph that summarizes what we have learned about function limits.
The First Derivative Test is a method for determining whether there is a maximum or minimum in some interval.
The Fundamental Theorem of Integral Calculus is the main technique we use to determine the value of the definite integral $\int_a^b\,f(x)dx$. The theorem states that
\[
\int_a^b\,f(x)dx=F(b)-F(a)
\]
where $F(x)$ is the antiderivative of $f(x)$.
We present the concept and notation of open and closed intervals on the real number line.
We highlight what we have learned with regard to function limits of quotients of functions whose limits go to zero.
The Second Derivative Test allows us to find out whether a critical point is a maximum or minimum by knowing the sign of the second derivative.
We define what it means for a set of numbers to be bounded.
In this lesson we define the concept of concavity and show in pictures when a function is concave up or concave down.
The length of the intervals $[a,b]$, $[a,b)$, $(a,b]$, and $(a,b)$ are all $b-a$.
Concavity is defined by whether a function's first derivative is increasing or decreasing. We show how to use a function's second derivative to determine if it's first derivative is increasing or decreasing.
An inflection point is where a function's concavity changes. For example, the point where a function goes from being concave up to concave down is an inflection point. In this lesson we define an inflection point, show how they can be identified, and provide examples.
The Factorial Function is denoted by $n!$ and means we multiply the numbers $1$ through $n$ together.
We discuss the precise definition of the function limit
\begin{equation}
\lim_{x\to\infty}\,f(x)=L.
\end{equation}
When this limit exists the number $L$ is called a horizontal asymptote.
