Lesson Series
We present here some basic rules of absolute values needed for many proofs.
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 show how the first derivative can be used to identify when a function is either increasing or decreasing on an interval.
We provide a mathematically rigorous definition of one-sided limits and prove a very important relationship between two-sided and one-sided limits.
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 define the concept of a set and the notation of set definitions.
We explain how the derivative is the rate of change of one variable in relation to another.
We discuss what it means for a function's range to be bounded or unbounded.
We list here important properties of function limits.
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.
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.
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 $f(g(x))$.
We define both an intuition and a rigorous mathematical definition of continuous functions.
We define the definite integral as the limit of Riemann sums.
