Lesson Series
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 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.
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.
We provide the intuition behind functions and detail their mathematical properties. Examples with graphs are shown.
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.
We review the concepts of Trigonometry and the Pythagorean Theorem. This lesson focuses on being able to determine the length of all three sides and both angles of a right triangle.
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.
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.
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 provide the intuition and the rigorous mathematical definition of increasing and decreasing functions.
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.
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 describe natural numbers, integers, rational numbers and real numbers.
We define the $\max$ and $\min$ functions here. The $\max$ function returns the largest number and the $\min$ function returns the smallest number.
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.
