# Lessons

This lesson provides a list of continuous functions.

Calculus is the study of two different types of geometries and their applications. The first is differential calculus which is the study of slopes and rates of change. The second is integral calculus which is the study of areas and summations.

An antiderivative of the function $f(x)$ is another function $F(x)$ where $F\,'(x)=f(x)$.

We discuss what it means for a function's range to be bounded or unbounded.

In this lesson we review what a functions is, important terms related to them, and provide you with a tool to graph them.

The quotient rule is a quick way to find the derivative of the division of two functions. The quotient rule states that
\[
\frac{d}{dx}\left(\frac{f}{g}\right)=\frac{1}{g^2}\left(g\frac{df}{dx}-f\frac{dg}{dx}\right).
\]

The Chain Rule shows how to find the derivative of a function composition. The Chain Rule states that \[\frac{d}{dx}\left[f\circ g\right] =\frac{df}{dg}\cdot\frac{dg}{dx}.\]

In this lesson we want to show how we attach a real world meaning to variables and functions.

In this lesson we discuss how a derivative is a rate of change and show how speed is a derivative.

Second derivatives are calculated from first derivatives in the same way that derivatives are calculated from functions. Second derivatives explain how fast something is increasing or decreasing and geometrically describes the concavity of a function.

The Mean-Value Theorem states that given a secant line between two points there is a point where the slope of the function is the same as the slope of the secant line.

As we move from left to right an increasing function will move up and a decreasing function will move down.

Derivatives can be used to find when a function increases and decreases. If between two points $a$ and $b$ $f\,'(x)\gt 0$ then the function is increasing and if $f\,'(x)\lt 0$ then it is decreasing.

A critical point is where $f\,'(x)=0$.

A maximum is the top of a hill and a minimum is the bottom of a valley.

In this lesson we review the concept of a straight line and its two main features - slope and y-intercept. We discuss polynomials and provide interactive tools to graph straight lines and polynomials.

We discuss the relationship between critical points, maxima, and minima. In particular, at every maximum and minimum there is a critical point, however, not every critical point has a maximum or minimum.

At a critical point a maximum is when the slope (first derivative) goes from positive to negative and at a minimum the slope goes from negative to positive.

The second derivative can be used to determine when a function is concave up and concave down. When $f\,''(x)\gt 0$ then the function is concave up and when $f\,''(x)\lt 0$ then the function is concave down.

An inflection point is where a function's concavity changes. We show how second derivatives can be used to find inflection points.

An integral is the area under the function $f(x)$. We can approximate the area as the sum of rectangles. There are two type of integrals: definite and indefinite.

A function's antiderivative is another function, $F(x)$, such that $F\,'(x)=f(x)$.

The Fundamental Theorem of Integral Calculus is the main technique we use to solve integrals. It states that
\[
\int_a^b\,f(x)dx=F(b)-F(a)
\]
where $F(x)$ is an antiderivative of $f(x)$.

A definite integral can be negative beause either the function is negative or because the limits of integration have been switched.

We discuss the different ways that functions can be combined.

The average value of the function between $a$ and $b$ is
\[
\mbox{Average Value of }f(x)=\frac{1}{b-a}\int_a^bf(x)\,dx.
\]

We explain the exponential and natural logarithm functions.

In this lesson we state the derivatives and integrals of the exponential and natural logarithm functions. The most important result is the derivative of the natural logarithm which allows us to integrate $1/x$.

We briefly describe differential calculus and integral calculus. We provide pictures and basic notation of both.

Here we cover the four most important derivatives that you will need.

The Tangent Line is a straight line at a specific point, $x_0$, whose slope is $f\,'(x_0)$ that takes on the value $f(x_0)$ at $x_0$. In this lesson we derive the formula for the tangent line given that we know the function and its derivative.

In this lesson we tell you the derivative of a polynomial and give you a tool for calculating its derivative both symbolically and numerically at any point.

The product rule gives us a quick way of finding the derivative of the multiplication of two functions. The product rule is
\[
\frac{d}{dx}fg=g\frac{df}{dx}+f\frac{dg}{dx}.
\]

We provide a summary of common shapes such as triangles, squares, rectangles, circles, spheres, etc. We provide formulas for their area, volume, and perimeter lengths where appropriate.

Differential calculus is the study of the slopes of functions and their application to real world problems. The slope of a function $f(x)$ at a specific point is called a derivative and is denoted by two common notations: $f'(x)$ and $\frac{df}{dx}$. We provide a listing of real world examples where derivatives are used to describe real world phenomenon.

Here we show that a derivative is a function limit. We provide some simple examples of how the function limit definition is used.

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.

We present here the most important differentiation formulas. We include derivatives of $x^n$, the product rule, the quotient rule, and the chain rule.

We explain how the derivative is the rate of change of one variable in relation to another.

We motivate the fact that we can find as many other derivatives of a function as possible and emphasize the use of the Leibniz notation to describe them. We calculate all of the derivatives of $f(x)=x^2$ and $f(x)=e^x$. We provide links to lessons that detail the meaning of the second derivative.

We define the composition of two functions where the results of one function are applied to another. Function composition is denoted by $(f\circ g)(x)$ and is equal to $f(g(x))$.

Function limits show how close a function can get to a point on the y-axis as it approaches a number on the x-axis. Here we introduce notation and begin developing intuition through pictures.

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.

Here we provide the mathematically rigorous definition of a function limit and provide explicit steps on how to prove a function limit exists.

We provide a mathematically rigorous definition of one-sided limits and prove a very important relationship between two-sided and one-sided limits.

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.

We list here important properties of function limits.

We define both an intuition and a rigorous mathematical definition of continuous functions.

We provide some useful examples and a graph that summarizes what we have learned about function limits.

We highlight what we have learned with regard to function limits of quotients of functions whose limits go to zero.

We provide the intuition behind functions and detail their mathematical properties. Examples with graphs are shown.

We provide the intuition and the rigorous mathematical definition of increasing and decreasing functions.

We describe how a function continuous on $[a,b]$ will take on both a maximum value and a minimum value on $[a,b]$.

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 show how the first derivative can be used to identify when a function is either increasing or decreasing on an interval.

In this lesson we define the concept of concavity and show in pictures when a function is concave up or concave down.

A series of lessons detailing the geometry of functions. We discuss extreme values, mean values, maximums, minimums, critical points, concavity and inflection points.

We provide the definition and graphical intuition behind the concepts of maximum and minimum. A maximum is the top of a hill and a minimum is the bottom of a valley.

We discuss how critical points are defined as those points where $f\,'(x)=0$. We discuss the Critical Point Theorem that discusses the relationship between critical points and maxima and minima of functions.

The First Derivative Test is a method for determining whether there is a maximum or minimum in some interval.

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.

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.

Integral Calculus is the study of areas under curves and their application to real world problems. The area under a curve between points $a$ and $b$ are denoted by \begin{equation}\mbox{Area between }a\mbox{ and }b=\int_a^bf(x)\,dx.\end{equation}
We discuss definite and indefinite integrals and take a quick look at the Fundamental Theorem of Integral Calculus.

We divide the $x$-axis between the limits of integration into an arbitrary set of closed intervals. We provide notation and use pictures to show several arbitrary partitions.

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.

We define the definite integral as the limit of Riemann sums.

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}

The Leibniz notation is where we denote a function's derivative by $\frac{df}{dx}$. We provide an explanation of where the Leibniz notation comes from.

A list of figures for Calculus in 5 Hours

We show how marginal cost, marginal revenue and marginal profit in business economics are related to derivatives.

We present the concept and notation of open and closed intervals on the real number line.

In this lesson we define the term partition limit which we denote by
\[
\lim_{||P\,||\to 0}F(P)=L.
\]

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.

A power function is a function of the form $Cx^r$ where $C$ and $r$ are real numbers.

We summarize the properties of continuous functions.

In this lesson we provide proofs of some of the most common properties of integrals that every student should know.

We present here some basic rules of absolute values needed for many proofs.

We review inequalities and outline some basic methods for solving them.

A sequence is a function whose domain is the natural numbers and whose range is the real numbers. This lesson defines common terms and notation for a sequence.

We define the concept of a set and the notation of set definitions.

We review the economic concepts of supply and demand. We describe both graphically and mathematically how to find the equilibrium price and quantity that businesses and consumers agree upon.

We define the $\max$ and $\min$ functions here. The $\max$ function returns the largest number and the $\min$ function returns the smallest number.

The Binomial Theorem is a formula for determining the answer to $(x+y)^n$.

The Factorial Function is denoted by $n!$ and means we multiply the numbers $1$ through $n$ together.

In this lesson we derive the formula for the tangent line given that we know the function and its derivative.

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 $f(x)$.

The Intermediate-Value Theorem is the theorem that really proves our original intuition that a continuous function is one that can be drawn on a piece of paper without having to lift the pencil from the paper to draw it.

The length of the intervals $[a,b]$, $[a,b)$, $(a,b]$, and $(a,b)$ are all $b-a$.