Single Variable Calculus And Infinite Series

[Dinah Lianes]

This calculus course covers differentiation and integration of functions of one variable, and concludes with a brief discussion of infinite series. Calculus is fundamental to many scientific disciplines including physics, engineering, and economics.

How did Newton describe the orbits of the planets? To do this, he created calculus. But he used a different coordinate system more appropriate for planetary motion. We will learn to shift our perspective to do calculus with parameterized curves and polar coordinates. And then we will dive deep into exploring the infinite to gain a deeper understanding and powerful descriptions of functions.

How does a computer make accurate computations? Absolute precision does not exist in the real world, and computers cannot handle infinitesimals or infinity. Fortunately, just as we approximate numbers using the decimal system, we can approximate functions using series of much simpler functions. These approximations provide a powerful framework for scientific computing and still give highly accurate results. They allow us to solve all sorts of engineering problems based on models of our world represented in the language of calculus.

Math 18 - Foundations for Calculus (2 units, S/NC, Fall only) covers the mathematical background and fundamental skills necessary for success in calculus and other college-level quantitative work. Topics include ratios, unit conversions, functions and graphs, polynomials and rational functions, exponential and logarithm, trigonometry and the unit circle, and word problems. Class sessions are a mix of lecture and worksheets.

This series covers differential calculus, integral calculus, and power series in one variable. It can be started at any point in the sequence for those with sufficient background. See the detailed list of topics for the Math 20 series.

Covers properties and applications of limits, continuous functions, and derivatives. Calculations involve trigonometric functions, exponentials, and logarithms, and applications include max/min problems and curve-sketching.

Covers properties and applications of integration, including the Fundamental Theorem of Calculus and computations of volumes, areas, and arc length of parametric curves. An introduction to some basic notions related to differential equations (such as exponential growth/decay and separable equations) is also given.

Covers limits at infinity and unbounded functions in the context of integration as well as infinite sums, including convergence/divergence tests and power series. Taylor series and applications are also covered.

The content of Math 21 (improper integrals, infinite series, and power series) is essentially the material of BC-level AP calculus not in the syllabus of AB-level AP calculus nor in IB Higher Level math. The math placement diagnostic results do not waive Math 21 requirements, since the diagnostic has no exam security; its feedback is purely advisory. Knowledge of Math 21 content is fundamental to university-level quantitative work, and is expected by the outside world for anyone earning a degree in a quantitative field here. This is an enforced requirement to enroll in Math 51 or CME 100; for more details, click the button above.

Math 51- Linear Algebra, Multivariable Calculus, and Modern Applications (5 units) covers linear algebra and multivariable differential calculus in a unified manner alongside applications related to many quantitative fields. This material includes the basic geometry and algebra of vectors, matrices, and linear transformations, as well as optimization techniques in any number of variables (involving partial derivatives and Lagrange multipliers).

The unified treatment of both linear algebra (beyond dimension 3 and including eigenvalues) and multivariable optimization is not covered in a single course accessible to non-majors anywhere else. Many students who learn some multivariable calculus before arriving at Stanford find Math 51 to be instructive to take due to its broad scope and synthesis of concepts. If you want transfer credit to substitute for Math 51 then you will likely need two courses (one on multivariable calculus, one on linear algebra).

Math 53- Differential Equations with Linear Algebra, Fourier Methods, and Modern Applications (5 units) develops core concepts, examples, and results for ordinary differential equations, and covers important partial differential equations and Fourier techniques for solving them. This uses both linear algebra and matrix derivative material from Math 51.

This series provides the necessary mathematical background for majors in all disciplines, especially for the Natural Sciences, Mathematics, Mathematical and Computational Science, Economics, and Engineering.

Math 61CM-62CM-63CM- Modern Mathematics: Continuous Methods (5 units each) This proof-oriented three-quarter sequence covers the material of 51, 52, 53, and additional advanced calculus, higher-dimensional geometry, and ordinary and partial differential equations. This provides a unified treatment of multivariable calculus, linear algebra, and differential equations with a different order of topics and emphasis from standard courses. Students should know single-variable calculus very well and have an interest in a theoretical approach to the subject.

This proof-oriented three-quarter sequence covers the same linear algebra and multivariable optimization material as the 60CM-series but draws its motivation from topics in discrete math rather than from the more analytic topics as in the 60CM-series. Its discrete math coverage includes combinatorics, probability, some basic group theory, number theory, and graph theory. Students should have an interest in a theoretical approach to the subject.

Many 100-level mathematics courses assume familiarity with writing proofs, and if you plan to be a Math major then you should learn proof writing as soon as possible. Here is a list of courses to begin learning proof-writing:

Math 104 also provides an introduction to proof-writing, but not at the same level as the above courses (a variety of proofs are covered, but students are not expected to write proofs of their own at the same level as some of those shown in class).

For more information about these courses see ExploreCourses for course descriptions and schedule. If you have further questions about which course to take, contact your academic advisor, or our Director of Undergraduate Studies.

I'm really interested in sequences and series (mainly series). What kind of math branch should I look more into? I understand that sequences and series mostly point toward analysis, but what sub-branch of analysis would I most enjoy?

It's not entirely true that sequences and series are the province of analysis. Generating functions take the form of discrete infinite sums and series and they are important tools in combinatorics and algebra. That being said, the study of sequences and series are mostly in analysis.

You should be warned,though,that as tools in real analysis and calculus, infinite series have largely been supplanted by direct numerical approximation methods.I'm extremely troubled by this drift because I think they are extraordinarily powerful tools in analysis-expressing functions locally as power series in a radius of convergence is very illuminating in "modeling" a function's behavior.But mathematicians-particularly numerical analysts-see infinite series as archaic tools for solving differential equations and look to rely on more rapidly convergent methods of approximation.

In complex analysis, however, where functions have the much stronger conditions of holomorphicity, infinite series and sequences still play an enormously important role to not only quantify analyticity, but to define the properties of contour integrals where singularities exist along paths. The main tool for this is Laurent series. I think you'll find a great deal of fascinating material to mine in complex analysis along these lines, usually presented in graduate complex analysis.

Wilf's book generatingfunctionolgy, available freely online, leads off with a quick list of the ways infinite series are used in combinatorics (where they are called generating functions); he immediately follows this up with some nice examples. The first time one encounters this sort of thing, it seems like magic. Glance through the table of contents and the first chapter to get the flavor.

Many students encounter infinite series for the first time as part of their single-variable calculus coursework. As part of this initial engagement with infinite series convergence, students grapple with infinity in ways that they haven't had to before. For instance, the fact that summing infinitely many terms sometimes yields a finite value, but at other times diverges, poses significant conceptual challenges.

I recently designed and implemented a curriculum for second-semester calculus centered in doing problems to help students develop ideas surrounding infinite series convergence, rather than using direct instruction. The unit design was patterned after a workshop at the Park City Mathematics Institute's Teacher Leadership Program (PCMI TLP).

In this paper, I discuss the design of the three-week curriculum and I discuss what participants in the class learned and how these learnings shape future iterations of the materials.

M408D is the second course in UT's standard first-year calculus sequence. It is directed at students in the natural and social sciences and at engineering students. The emphasis in this course is on problem solving, not on the presentation of theoretical considerations. While the course necessarily includes some discussion of theoretical notions, its primary objective is not the production of theorem-provers. M408D contains a treatment of infinite series, and an introduction to vectors and vector calculus in 2-space and 3-space, including parametric equations, partial derivatives, gradients and multiple integrals.

c80f0f1006