Tutor profile: Hannah B.

How do I include graphics in my LaTeX PDF document?

1) Make sure you have \usepackage{graphicx} in the document preamble. 2) Make sure whatever TeX generator you're using (such as Overleaf, which can be nice because of the online storage) has access to the graphic you're trying to load -- in Overleaf, you'll need to upload a .png, .jpeg, .pdf, or .jpg file. 3) When you want to put the graphic into your TeX file, use the following code: \begin{figure} \centering \includegraphics[width=\textwidth]{picture.png} \caption{This is my caption.} \label{fig:thisismylabel} \end{figure} If you don't want to center the graphic, leave out the "\centering" command. The [width=\textwidth] text controls how big the picture is, and can easily be left out too. (Or if you don't want the picture to be textwidth size, you can write [width=.5\textwidth] or [width=.7\textwidth], as examples.) I often delete the \label command, also.

What is the derivative of the function $$f(x) = e^{3\pi x}$$?

It can be tempting to try to differentiate $$f(x) = e^{3\pi x}$$ using the Power Rule, since, well, this is a function with a power involved, right? Actually, the Power Rule stipulates that the base is a variable (for example, we might use the Power Rule to differentiate $$g(x) = x^3 + 7$$). But $$e$$ is, in fact, just a constant. Actually, $$h(x) = ke^x$$ is the unique family of functions where $$f'(x) = f(x)$$, where $$k$$ is an arbitrary real number (notice that the zero function does belong to this family -- take $$k = 0$$!). Instead of using the Power Rule, we actually need to use the Chain Rule here. Why? Well, $$e^{3\pi x}$$ is a more general form of an exponential function. $$e^{3\pi x}$$ is a function ($$e^x$$) evaluated at another function $$(3\pi x)$$. Because of this, to differentiate $$f(x) = e^{3\pi x}$$, we need to use the Chain Rule. Thus, $$f'(x) = 3\pi e^{3\pi x}$$, because $$(3\pi x)' = 3\pi$$.

Why is the field of partial differential equations necessary?

Differential equations are useful because they model change over time. However, in ordinary differential equations classes, we commonly deal with only one variable changing with respect to time. In real life, we're probably going to have a lot more variables to deal with -- hence why we need partial differential equations, which allows us to express equations with multiple variables changing over time.