I've taught a variety of courses at both UVA and Wellesley. At UVA, I had the opportunity to be a part of the Transforming Calculus program, which began the transition of lower-level courses to a flipped-classroom format, including the pilot program for Calculus II.
This course builds on the material of a typical Calculus I course. We begin by developing the idea of integration, focusing on the idea that integrals allow us to combine an infinite amount infinitessimal information to arrive at a meaningful answer. We see this in action by using integrals to compute some geometric quantities, such as volumes of solids and lengths of curves. We then transition to the discussion of series, which are (discrete) infinite sums that play a very similar role to that of integrals. The capstone result is Taylor series - these are a vast generalization of the idea of tangent lines allow us to write complicated functions as infinite sums of simpler functions. Additional topics include the calculus of parametric functions and polar coordinates.
You can find the course syllabus here.
The main goal of this course is to understand the basics ideas of calculus, differentiability and integration, in the context of complex-valued functions. After developing some intuition for complex functions, we introduce the idea of analytic functions, which play the role of differentiable functions from calculus. From there, we study integration of complex functions and the behavior of contour integrals. We will learn some classical results on the contour integrals of analytic functions. We then introduce some powerful tools for studying complex functions: power series and Laurent series. We will learn how to write reasonably well-behaved functions as series and use this to study complex integrals over regions in which given functions may not be analytic. The capstone result is the so-called Residue Theorem. Time permitting, we will discuss some topics from analytic number theory, such as the prime number theorem and the Riemann Zeta function.
You can find the course syllabus here.
This course introduces students to the central concepts of calculus. We begin by exploring the idea of limits, which allow us to study the behavior of a function at a particular value using the data from "nearby" values. Limits lead us into studying the derivative of a function. Derivatives arise from using limits to compute the slope of a function's graph's tangent line at a particular value and measure the instantaneous rate of change of a function. This tool allows us to study the global behavior of a function, such as its direction, concavity, and extrema. From there, we move on to integration, which describes the accumulation of a function between two input values. Limits allow us to compute these directly. We close with the Fundamental Theorem of Calculus, which link integrals with the study of derivatives.
You can find the course syllabus here.
This serves as a first course in number theory. We begin by discussing basic properties of the integers, such as divisibility and greatest common divisors. From there, we move on to modular arithmetic, where we study congruence classes of integers and the way these classes inform our understanding of the integers at large. Of particular interest will be congruence classes of integers modulo primes \(p\). Topics here include Fermat's Little Theorem, Euler's Theorem, and the Chinese Remainder Theorem. We then study quadratic residues, the analogue of square roots in modular arithmetic. This closely relates to the study of quadratic polynomials in the setting of modular arithmetic. This culminates in the result of quadratic reciprocity, which describes exactly when a quadratic polynomial as solutions \(\mod p\). Time permitting, we will discuss writing integers as sums of squares, Fermat's Last Theorem, Gaussian integers, and RSA encryption.
You can find the course syllabus here.
This course builds on the material of a typical Calculus I course. We begin by developing the idea of integration, focusing on the idea that integrals allow us to combine an infinite amount infinitessimal information to arrive at a meaningful answer. We see this in action by using integrals to compute some geometric quantities, such as volumes of solids and lengths of curves. We then transition to the discussion of series, which are (discrete) infinite sums that play a very similar role to that of integrals. The capstone result is Taylor series - these are a vast generalization of the idea of tangent lines allow us to write complicated functions as infinite sums of simpler functions. Additional topics include the calculus of parametric functions and polar coordinates.
You can find the course syllabus here.
This course introduces students to the central concepts of calculus. We begin by exploring the idea of limits, which allow us to study the behavior of a function at a particular value using the data from "nearby" values. Limits lead us into studying the derivative of a function. Derivatives arise from using limits to compute the slope of a function's graph's tangent line at a particular value and measure the instantaneous rate of change of a function. This tool allows us to study the global behavior of a function, such as its direction, concavity, and extrema. From there, we move on to integration, which describes the accumulation of a function between two input values. Limits allow us to compute these directly. We close with the Fundamental Theorem of Calculus, which link integrals with the study of derivatives.
You can find the course syllabus here.
This course provides an introduction to statistics. We begin with some basic descriptive statistics, such as mean, median, and standard deviation. We then learn about various statistical distributions, with particular attention paid to normal distributions. From here, we transition to learning how to interpret and infer from data. After reviewing some probability, we discuss confidence intervals and hypothesis testing, which allow us to use data to make statements about the results of experiments. The central goal of the course is to equip students with the basic tools to interact with and assess data in a practical way.
You can find the course syllabus here.
This course introduces students to the central concepts of calculus. We begin by exploring the idea of limits, which allow us to study the behavior of a function at a particular value using the data from "nearby" values. Limits lead us into studying the derivative of a function. Derivatives arise from using limits to compute the slope of a function's graph's tangent line at a particular value and measure the instantaneous rate of change of a function. This tool allows us to study the global behavior of a function, such as its direction, concavity, and extrema. From there, we move on to integration, which describes the accumulation of a function between two input values. Limits allow us to compute these directly. We close with the Fundamental Theorem of Calculus, which link integrals with the study of derivatives.
You can find the course syllabus here.
This course builds on the material of a typical Calculus I course. We begin by developing the idea of integration, focusing on the idea that integrals allow us to combine an infinite amount infinitessimal information to arrive at a meaningful answer. We see this in action by using integrals to compute some geometric quantities, such as volumes of solids and lengths of curves. We then transition to the discussion of series, which are (discrete) infinite sums that play a very similar role to that of integrals. The capstone result is Taylor series - these are a vast generalization of the idea of tangent lines allow us to write complicated functions as infinite sums of simpler functions. Additional topics include the calculus of parametric functions and polar coordinates.
You can find the course syllabus here.
This course introduces students to the central concepts of calculus. We begin by exploring the idea of limits, which allow us to study the behavior of a function at a particular value using the data from "nearby" values. Limits lead us into studying the derivative of a function. Derivatives arise from using limits to compute the slope of a function's graph's tangent line at a particular value and measure the instantaneous rate of change of a function. This tool allows us to study the global behavior of a function, such as its direction, concavity, and extrema. From there, we move on to integration, which describes the accumulation of a function between two input values. Limits allow us to compute these directly. We close with the Fundamental Theorem of Calculus, which link integrals with the study of derivatives.
You can find the course syllabus here.
This course builds on the material of a typical Calculus I course. We begin by developing the idea of integration, focusing on the idea that integrals allow us to combine an infinite amount infinitessimal information to arrive at a meaningful answer. We see this in action by using integrals to compute some geometric quantities, such as volumes of solids and lengths of curves. We then transition to the discussion of series, which are (discrete) infinite sums that play a very similar role to that of integrals. The capstone result is Taylor series - these are a vast generalization of the idea of tangent lines allow us to write complicated functions as infinite sums of simpler functions. Additional topics include the calculus of parametric functions and polar coordinates.
You can find the course syllabus here.
This is an introductory course in calculus primarily for students studying business, biology, or social sciences. Early in the course, the key idea is that of a limit, which allows us to study the behavior of a function at a particular value using the data from "nearby" values. Limits lead us into studying the derivative of a function. Derivatives arise from using limits to compute the slope of a function's graph's tangent line at a particular value and measure the instantaneous rate of change of a function. This tool allows us to study the global behavior of a function, such as its direction, concavity, and extrema. We first study these in the context of algebraic functions, such as polynomials and rational functions, and gradually introduce transcendental functions, such as exponential and logarithmic functions, as the course progresses. From there, we move on to integration, which describes the accumulation of a function between two input values. Limits allow us to compute these directly. We close with the Fundamental Theorem of Calculus, which link integrals with the study of derivatives. This course was previously titled Applied Calculus I.
You can find the course syllabus here.
This is an introductory course in calculus primarily for students studying business, biology, or social sciences. Early in the course, the key idea is that of a limit, which allows us to study the behavior of a function at a particular value using the data from "nearby" values. Limits lead us into studying the derivative of a function. Derivatives arise from using limits to compute the slope of a function's graph's tangent line at a particular value and measure the instantaneous rate of change of a function. This tool allows us to study the global behavior of a function, such as its direction, concavity, and extrema. We first study these in the context of algebraic functions, such as polynomials and rational functions, and gradually introduce transcendental functions, such as exponential and logarithmic functions, as the course progresses. From there, we move on to integration, which describes the accumulation of a function between two input values. Limits allow us to compute these directly. We close with the Fundamental Theorem of Calculus, which link integrals with the study of derivatives.
You can find the course syllabus here.
This course builds on the material of a typical Calculus I course. We begin by developing the idea of integration, focusing on the idea that integrals allow us to combine an infinite amount infinitessimal information to arrive at a meaningful answer. We see this in action by using integrals to compute some geometric quantities, such as volumes of solids and lengths of curves. We then transition to the discussion of series, which are (discrete) infinite sums that play a very similar role to that of integrals. The capstone result is Taylor series - these are a vast generalization of the idea of tangent lines allow us to write complicated functions as infinite sums of simpler functions. Additional topics include the calculus of parametric functions and polar coordinates.
You can find the course syllabus here.
This is an introductory course in calculus primarily for students studying business, biology, or social sciences. Early in the course, the key idea is that of a limit, which allows us to study the behavior of a function at a particular value using the data from "nearby" values. Limits lead us into studying the derivative of a function. Derivatives arise from using limits to compute the slope of a function's graph's tangent line at a particular value and measure the instantaneous rate of change of a function. This tool allows us to study the global behavior of a function, such as its direction, concavity, and extrema. We first study these in the context of algebraic functions, such as polynomials and rational functions, and gradually introduce transcendental functions, such as exponential and logarithmic functions, as the course progresses. From there, we move on to integration, which describes the accumulation of a function between two input values. Limits allow us to compute these directly. We close with the Fundamental Theorem of Calculus, which link integrals with the study of derivatives.
You can find the course syllabus here.
This course begins where Applied Calculus I ends. After reviewing some basic integration, we move on to some applications, such as computing volumes of solids and studying probability distributions. We then transition to the discussion of series, which are (discrete) infinite sums that play a very similar role to that of integrals. The capstone result is Taylor series - these are a vast generalization of the idea of tangent lines allow us to write complicated functions as infinite sums of simpler functions. As an additional topic, we study the basics of differential equations.
You can find the course syllabus here.
This is an introductory course in calculus primarily for students studying business, biology, or social sciences. Early in the course, the key idea is that of a limit, which allows us to study the behavior of a function at a particular value using the data from "nearby" values. Limits lead us into studying the derivative of a function. Derivatives arise from using limits to compute the slope of a function's graph's tangent line at a particular value and measure the instantaneous rate of change of a function. This tool allows us to study the global behavior of a function, such as its direction, concavity, and extrema. We first study these in the context of algebraic functions, such as polynomials and rational functions, and gradually introduce transcendental functions, such as exponential and logarithmic functions, as the course progresses. From there, we move on to integration, which describes the accumulation of a function between two input values. Limits allow us to compute these directly. We close with the Fundamental Theorem of Calculus, which link integrals with the study of derivatives.
You can find the course syllabus here.