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.
After completing this course, you should be able to
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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.
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