21-122 Integration and Approximation

Description

This course is a continuation of the ideas in 21-120 Differential and Integral Calculus. This course introduces the ideas of differentiation and integration, respectively.

The course begins with a strengthening of our integration skills. We introduce three new techniques for use in different situations which, when combined with the Method of Substitution and Integration by Parts, allow us to integrate a wide variety of functions. We also extend the range of integration problems we are willing to consider, allowing discontinuities in the integrand, and integration over an interval of infinite extent.

Our second main theme is the study of differential equations, i.e. equations that involve the derivative of a function. To solve such an equation, i.e. to determine the unknown function, usually requires the computation of an integral. Differential equations are ubiquitous in the natural sciences and social sciences, because they are useful in modeling the behavior of systems over time. We will discuss how to write a mathematical model for a physical system, and also how to compute solutions for two fundamental types of equations.

Our third theme is that of Approximation. The idea of approximation shows up in numerical integration, and in Newton's method for finding an approximate root of a function. We shall also devote a substantial portion of the course to finding polynomial approximations to functions. To do so, we will develop the notions of an infinite sequence, and infinite series (a summation with infinitely many terms). We will discuss convergence of these series in terms of limits and derive tests for convergence. We will also see how many functions may be described in terms of a power series.

Learning Objectives

After completing this course, you should be able to

1. Determine which method of integration is appropriate in a given situation and applly the various methods of integration when appropriate.
2. Determine whether a definite integral is proper or improper. Determine the convergence or divergence of improper integrals. Determine the value of a coonvergent improper integral.
3. Use numerical integration techniques to approximate the value of definite integrals.
4. Use the various error estimates to determine the accuracy of a numerical approximation to an integral.
5. Use integration techniques to solve a variety of theoretical and applied problems.
6. Identify and solve separable and linear differential equations (subject to the computation of integrals).
7. Use qualitative techniques to describe the long term behavior of solutions to differential equations.
8. Apply differential equations to model physical processes and population dynamics.
9. Determine the limit of a sequence using the ε-N definition.
10. Discuss convergence of a series in terms of the limit of its sequence of partial sums.
11. Apply the various test for convergence to determine whether a series converges or diverges.
12. Determine the value of certain convergent series.
13. Determine the radius of convergence and interval of convergence for power series.
14. Use Taylor's remainder theorem to find the error in polynomial approximations of a function.
15. Represent the solutions to certain differential equations as a power series.

How to use this site...

Home

The description gives a brief overview of the topics we will discuss this semester. The learning objectives give an itemized list of the skills you should be developing. The list of learning objectives may give you some useful direction in terms of studying for exams.

Course Information

Times and rooms for lecture and recitation sections.

Schedule

List of topics to be covered each week, with links to reading assignments and homework.

Policies

Information about the calculation of grades, dates for exams, policies for late assignments and other matters. Familiarize yourself with these policies early in the semester.