Calculus IV

Course Description

A study of functions of several variables: extrema and Lagrange Multipliers, with application to today’s optimization-problems in engineering, business, and ecology. Vector algebra and space geometry; Kepler’s Laws with application to satellite orbital velocity problems and the rendezvous phenomenon, iterated integrals and applications, the Jacobian transformation will be studied. A graphing calculator is required.

Learning Outcomes

• Understanding that a sequence is a function on the integers into R1. The notion of a limit of a sequence. Pattern recognition for sequences, monotonic sequences, and convergence of monotone and bounded sequences. The Fibonacci sequence.
• Partial sums of a series as a sequence. Convergent and divergent series, geometric, harmonic; and the Nth-Term Test as a necessary condition for convergence of a series. The integral and p-series as a
• The integral and p-series as a sequence. Deriving the p-series test, the ration and root tests, direct comparison tests. The alternating series test will be derived by the student.
• Taylor polynomials and approximations to sufficiently smooth functions. Be able to derive the Taylor polynomials of sufficient accuracy as to approximate within a proscribed degree of accuracy.
• Power series definitions, development and radius of convergence. Differentiation and integration of power series (converging uniformly on a contact set). Understand Ramanujan’s contribution to series approximation of p.
• Classification of ordinary and partial differential equations. Recognize a solution (general and singular, as well as a particular solution). Be able to find particular solutions matching pre-stipulated initial value conditions. Recognize ode applicable to the separation of variable methodology. Be able to perform that separability technique, and demonstrate sufficient skills in integration techniques so as to develop confidence and self-sufficiency in its approach.
• Recognize and select an appropriate methodology to solve homogeneous ode of the first order. Exponential growth and decay applications. Exactness of first-order equations; test and solution techniques.
• Generating integrating factors for the 2nd-order equation, if they exist. Recognizing and solving the Bernoulli nonlinear differential equation. Be able to solve generalized, and idealized, differential equations emerging from mechanical and electrical engineering applications.
• Recognize and solve 2nd-order homogeneous linear equations. Know the concept of linear independence of functions over a subset of R1. Know the definition and conceptual considerations regarding basis of a solution space; know that all elements of that solution space can be written as linear combinations of that basis.
• Recognize and solve 2nd-order homogeneous, linear, ordinary differential equations with forcing function. Know the methodology of undetermined coefficients. Be able to form a Wronskian from the basis functions for the corresponding homogenous equations and be able to apply it to the variation of parameters technique.
• Demonstrate proficiency in graphing-calculator usage by investigation of Euler’s technique for numerical approximation of simple initial-value problems. Recognize and perform the enhanced Euler’s technique, and be able to compare rates of convergence. Be familiar with Milne’s method, Runge-Kutta. Understand predictor-corrector techniques of numerical solutions of non-linear initial-value problems of ordinary differential equations.

Prerequisites

• MTH222