I’m writing this blog post because when I first came across Taylor series I found that a lot of my previous intuitions for mathematics were suddenly inadequate. It took time for me to build intuition not only for how these things work but why they are important. I hope this post will be illuminating for those just beginning to learn about Taylor series and also those who have some experience with them but haven’t quite wrapped their heads around them yet.

Taylor series can often seem a bit mysterious the first time that we learn about them. The formula for the Taylor series of a function f(x) around a point x=a is given by

One topic that seemed a bit mysterious and magic to me when I first learned calculus was implicit differentiation. In this post, we’ll start by reviewing some examples of implicit differentiation and then discuss why implicit differentiation works.

The symbol “dx” comes up everywhere in calculus. For example:

If you type √5 into your calculator, it’ll output something like 2.2360679775. But how did your calculator find that answer? Is there any way you could have found it by hand?

If you want help deciding whether to take Math 1 or Math 2 Subject Tests, there are a dozen websites that will guide you through the decision process. But once you’ve decided on Math 1, how do you know what to study? Chances are you’ve already been studying for the Math Section of the SAT, so you might want to know what topics you have to add. The College Board website lists the topics on the Math 1 Subject Test, but a cursory glance reveals that all of those topics also feature on the regular SAT. Yet, the two tests are not the same, and knowing the differences can help you master your Math 1 Subject Test and round out your college application. Fortunately, as an experienced tutor in both levels of Subject Test and the SAT Math Test, I’m here to help.

Probably the most common challenge that I see my students struggle with is understanding and writing out mathematical proofs. Although most higher level college math and computer science courses rely heavily on proofs, there aren’t many courses that really prepare students before they’re thrown off the deep end. I wanted to discuss some tips and tricks that’s helped my students become more comfortable with proofs, and some steps you can take to prepare yourself if you are planning on taking such a course.

Suppose that we have many towns spread across the country and we are trying to connect them with a network of roads. If we would like to do so by laying as little road as possible, how do we do it? In this blog post, we will use Calculus to tackle a special case of this optimization problem.

Mathematical induction is a common and very powerful proof technique. At its core, it’s an appeal to an intuitive notion that Induction proofs often pop up in computer science to proof that an algorithm works as intended (correctness) and that is runs in a particular amount of time (complexity). In this tutorial we’ll break down a classic induction problem in mathematics, and in the next post we’ll apply the same techniques to a classic computer science problem. As a warmup we can look at a classic example used to teach induction, namely the proof that the sum of integers from 1 to n is equal to

Preparing for both mathematics sections on the SAT can be a bit intimidating. You can’t expect yourself to know every topic that might come up, and the time limit adds to the stress. Much more efficient than trying to learn everything you might come across is to start with what you have already learned in high school and use examples to apply it more widely in the SAT. The most classic sort of example is what teachers have been drilling for ages: plug in numbers. However, mathematics is all about generalizing rules and strategies, so I want to talk about how to expand plugging in numbers to the art of creating your own examples.