Alexander Paulin apaulin@berkeley.edu  Department of Mathematics 796 Evans Hall University of California, Berkeley me  |  research  |  teaching  |  CV

I love all things mathematical and I especially love teaching all things mathematical. If you want to know about my experience and general teaching philosophy, here is my teaching portfolio.

 The central objects of study in calculus are functions. Here we introduce various ways to visualize them. There are far too many functions to study them all. Here we introduce several important classes of function, the ones we will study in most depth. Here we look at how functions can be altered and how it affects their graphs. Exponential functions are a deeply important class of functions, apprearing all throughout nature. Here we introduce them along with the natural base e. Under what circumstances does a function admit an inverse? Here we explore this question as well as asking what graphs of inverse functions look like. The classical definiton of average velocity is the distance traveled divided by time taken to do so. This doesn't make sense if we ask what velocity means at a single moment in time. Here we explore this deep question. To make sense of velocity at a single moment we had to think about how functions behaved as the input approaches (but does not equal) a specific value. Here we formalize this idea with the concept of a limit, the central technical tool in all of calculus. With the concept of a limit definited (at least informally), we develop methods to compute them for a wide class of functions. The informal approach to limits is all well and good, but if we are to study functions in more depth we need a much more precise defintion. Here we introduce the rigorous, so called epsilon-delta defintion of a limit. A function is continuous at a point if the limit there can be calculated in the most straightforward way possible. We carefully introduce this concept and relate it back to our core functions. We introduce a new type of limit, considering how functions behave as the input grows without bound. We consider how this affects the shape of the graph. Using the langage of limits we formally introduce the fundamental concept of the derivative of a function at a point. Informally we can interpret this as a slope of a tangent line. Having calculated the slope of the tangent line at a point, we observe this gives rise to a new function, namely the derivative. This is the main object of study in differential calculus. We compute derivatives of polynomial, power and exponential functions, three of our core classes. If we know the derivatives of two functions, is there a simple way to calculate the derivative of their product or quotient? The answer will turn out to be quite unexpected. We compute derivatives of trigonometric functions, one of our core classes. If we know the derivatives of two functions, is there a simple way to calculate the derivative of their composite? The answer will turn out to be quite elegent. If a function is only known to satisfy some equation (involving its input and output), is there some way to calculate its derivative? Here we calculate derivatives of logarithmic functions using implicit methods. We also introduce the useful technique of logarithmic differentiation. We explore various ways the derivative manifests itself in nature. We study functions which satisfy natural growth/decay, and explore how they occur in nature. We use the derivative to understand extreme values of functions. If I travel 100 miles in one hour, even if my speed is fluctuating, there has to be a moment when I'm travelling at exactly 100mph. This is the mean value theorem. Making this precise will give us deeper understanding of a differentiable functions. We use the derivative to understand when a function is increasing or decreasing, along with its convavity. We apply these ideas to finding extreme values. This gives us a useful tool for computing certain types of limit. (Interesting Fact: This result is actually due to Johann Bernoulli, who was paid by l'Hospital to give him his mathematical discoveries.) We put everything we've learned together to sketch the graphs of functions. We use our techniques for finding extreme values of functions to calculate optimal solutions to various physical problems. We consider the problem for finding functions with specific derivatives. We call these antiderivatives. We consider how to calcuate the area bounded by the graph of a positive function and the x-axis. We introduce the fundamental concept in integral calculus: the definite integeral. We introduce the fundamental theorem, a deep connection between differential and integral calculus. Integration by substitution is a systematic way to reverse the chain rule. It's a fundamental technique used to find antiderivatives. We use the definite integral and the fundamental theorem to calculate areas between curves. We calculate the volumes of solid shapes by integrating cross-sectional area functions.

 When we think about numbers, we think about the number line. We call these the real numbers. Real numbers come with two fundamental operations: addition and multiplication. If we sit the number line in a plane, is there some natural way to extend these operations, preserving all the usual properties of arithmetic?
 A complex number is a point in the Cartesian plane. We justify the term number because we can naturally add and multiply points in the plane in a way that satisfies all the usual properties of arithemtic. Once we transition from the line to the plane, what new properties emerge? Can we make sense of familiar concepts like exponentials?