With the arrival of block four our attention turns to analysis during the afternoons.
At our most introductory level students will be continuing their study of Algebra 1, in order to lay the foundation for later courses. The next two levels examine functions, exponentials, logarithms, and trigonometry, or FELT for short. These comprise the classic suite of analytical functions, with which most natural phenomena can be described. Our final two courses add a further layer, by introducing tools that may be applied to these functions—whether the limit of an increasingly fine sum (the integral) or strategies for numerically computing solutions of equations involving these functions.
— Sam Vandervelde
with Sachi Hashimoto
This course provides a continuation of and conclusion to Algebra 1a. Students will cover topics including simplifying algebraic expressions, the Cartesian plane, systems of two equations in two variables, and rates/ratios/percents/proportion. The course will culminate with a first look at quadratic equations. The purpose of the course is to organically motivate the need for and utility of algebra, while providing adequate training in order to use algebra proficiently.
Exponentials, Logs, and Trig
with Susan Durst
This course begins by instilling a solid understanding of the laws of and relationship between exponential functions and logarithmic functions. Along the way students will apply these enormously useful functions to understand a variety of natural phenomena, such as population growth and compound interest. The second part of the course will be devoted to a comprehensive study of trigonometry, including right triangle trigonometry, the Law of Sines and Law of Cosines, the unit circle, graphs of trigonometric functions, and applications to periodic motion.
with Sam Vandervelde
Having already covered the core precalculus topics of functions, exponentials, logarithms, and trigonometry, students are ready to explore a variety of topics that build on and reinforce this core material. The course begins with a review of polar coordinates and an introduction to cardioids, lemniscates, and more exotic curves. From there we will investigate iterated functions and dynamical systems, functional equations, applications of triangle trigonometry, and other topics at the discretion of the instructor. A primary purpose of the course will be to guide students to discover fascinating, yet unfamiliar areas of mathematics that are accessible with only a solid precalculus background.
with Austin Shapiro
This course focuses on the definite integral and its diverse applications. We will begin with the idea of the Riemann sum as a total of many small contributions arising from the product of two variable factors. Students will learn how to set up Riemann sums, then integrals, representing total change in a function with a known rate of change, displacement and distance traveled by an object, area and volume of regions, and much more. In parallel, we will cover a suite of analytical and numerical methods for evaluating definite integrals and their cousins, antiderivatives. We will round out the course by taking a first glimpse at the world of differential equations. The selection of topics is guided by the syllabus for the AP Calculus AB and BC tests.
with John DeIonno
This course is an introduction to numerical methods and analysis of these methods. We will investigate problems such as approximating solutions to algebraic equations, integral approximations, and approximating solutions to differential equations. Through these problems, the students will be exposed to many aspects of the field, including inventing methods, implementing these methods in code, analyzing the accuracy, efficiency, and stability of the methods, learning how to select the best method for a particular problem, and thinking about how to modify methods for tricky problems where a method won’t work out-of-the-box. The emphasis will be on obtaining a flavor for the field rather than being exhaustive with content.