
Definition of a Derivative
I copy the definitions of three different types of derivatives from [1]: $ \lim_{\Delta x \rightarrow 0} \frac{f(a+\Delta x) – f(a)}{\Delta x} \equiv \frac{d f(x)}{dx}\big_{a}$ $ \lim_{\Delta x \rightarrow 0^+} \frac{f(a+\Delta x) – f(a)}{\Delta x} \equiv \frac{d f(x)}{dx}\big_{a^+}$ $ \lim_{\Delta x \rightarrow 0^} \frac{f(a+\Delta x) – f(a)}{\Delta x} \equiv \frac{d f(x)}{dx}\big_{a^}$ These definitions are best…

Definition of a Limit
First, here is a video I made about a limit: Understanding what a Limit is. Next, I copy a definition of a limit from [1]. A function $ f(x)$ approaches a limit $ A$ as $ x$ approaches $ a$ if, and only if, for each positive number $ \epsilon$ there is another, $ \delta$, such…

The Tangent Line & the Tangent Function
The following equalities provide context to the definition of the tangent function. Now, what if $y$ is a function symbolized by $y(x)$ such that $y(x)=x$? In this case, $x$ is the independent variable and $y$ is the dependent variable. Then $ \tan\phi = \frac{x}{x}=1$. Interesting… 1 is also the slope of the linear function $…

Spherical Coordinates
The following drawing shows how to convert from Cartesian coordinates to spherical coordinates. This is a short post, but these three equations are pretty useful. I am going to use the end of this post to define the cosine and sine functions: $ \sin\phi \equiv \frac{opp}{hyp}$ $ \cos\phi \equiv \frac{adj}{hyp}$. Here, $ hyp$ is the…