Differential and Integral Calculus (5cr)

Objectives

On this course the students learn basic techniques such as integration by parts and changing of integration variables in integration of simple functions. The students learn to compute antiderivatives of rational functions and to analyze and compute improper integrals. The students also learn to solve simple separable differential equations, compute general solutions of homogeneous second order differential equations with constant coefficients, and to compute the particular solution of a nonhomogeneous differential equation using the method of undetermined coefficients After the course the students are capable of analyzing the limit of a sequence, computing the sum of a geometric series, and testing the convergence of a series with positive terms. The students also learn how to determine the interval of convergence of a power series, form Taylor polynomials of functions, and simple Taylor series. The students learn present their solutions orally as well as in written form.

Content

Core contentAntiderivative and basic integration techniques. Proper and improper integrals.Ordinary linear differential equations of first and second order. Separable first order differential equations.Limit of a sequence, increasing and decreasing sequences.Series (geometric, with positive terms, alternating, Taylor series) and their convergence.Complementary knowledgeApplications of integration in, e.g., determining areas and volumes of geometrical shapes, and computing the length of a curve.Higher order differential equations. Modeling specific real world problems, such as growth of populations, with differential equations.Approximating a function with a polynomial.Using Matlab as a tool in solving the exercise problems.Specialist knowledgeNumerical integration, trapezoid rule and Simpson's formula. Computing the Riemann sums.Existence and uniqueness results, matrix notation for linear systems.Testing convergence. Computing limits and integrals using series. Estimating the error in polynomial approximations of functions.

Further information

Partial completions of the course must be carried out during the same implementation round.This course belongs to the SEFI 1 level of engineering mathematics.