Continues from MA 115 with improper integrals, infinite series, Taylor series,
and Taylor polynomials. Vectors operations in 3-space, mathematical
descriptions of lines and planes, and single-variable calculus for parametric
curves. Introduction to calculus for functions of two or more variables
including graphical representations, partial derivatives, the gradient vector,
directional derivatives, applications to optimization, and double integrals in
rectangular and polar coordinates.
Course Web Site: Spring 2010 | Spring 2009
This two-semester introduction to calculus, MA 115 and MA 116, prepares students for sophomore-level topics in mathematical analysis (differential equations and vector calculus), and calculus-based subjects in science and engineering. Upon completion of the course, students will have a working knowledge of the fundamental definitions and theorems of elementary calculus, be able to complete routine derivations associated with calculus, recognize elementary applications of differential and integral calculus, and be literate in the language and notation of calculus.
- Term I (MA115) develops the differential and integral calculus for functions of one variable.
- Term II (MA116) is an introduction to differential and integral calculus for parametric curves, multi-variable functions, and power series representations.
Upon completing this course, it is expected that a student will be able to do the following:
1. Mathematical Foundations
- Improper Integrals: Recognize and evaluate improper integrals of Type I and Type II.
- Infinite Series: Explain clearly the definition of an infinite series as the limit of a sequence of partial sums.
- Geometric Series: Recognize a geometric series and correctly apply the convergence theorem.
- Power Series: Apply the ratio test to determine the radius of convergence for a power series.
- Taylor Polynomials: Derive the leading terms in the Taylor Polynomial for a function of one variable.
- Vector Products: Calculate dot products and cross products and interpret them geometrically.
- Lines and Planes: Derive the equations of lines and planes given appropriate information.
- Functions of Two Variables: Determine the maximal domain for functions of two variables, and construct level curves as a tool for visualizing a functionís graph.
- Partial Derivatives: Evaluate partial derivatives, including higher order derivatives and simple cases of the chain rule, and recognize the various notations used for partial derivatives.
- Directional Derivatives: Construct the gradient vector for multivariable functions and determine the derivative in a given direction.
- Tangent Plane: Derive the equation of the tangent plane and use the tangent plane as a local linear approximation to a surface.
- Double Integrals: Formulate and evaluate iterated double integrals using rectangular or polar coordinates.
2. Applications of Mathematics:
- Initial Value Problems: Solve simple initial value problems for trajectories in 3-space where the solution is recovered via direct integration.
- Optimization: Formulate equations for solving elementary constrained optimization in two and three variables, and characterize critical points for functions of two variables.
Stewart, James, Calculus: Concepts and Contexts, 4th Edition, Brooks/Cole Pub., 2008.
Research Associate Professor, Deputy Director
Course Web Sites