# MAT-01366 Mathematics 3, 5 cr

#### Vastuuhenkilö

Sampsa Pursiainen

#### Osaamistavoitteet

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 "shrewd quessing". 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 how to justify their claims using mathematical methods and to present their solutions orally as well as in written form.

#### Sisältö

 Sisältö Ydinsisältö Täydentävä tietämys Erityistietämys 1. Antiderivative and basic integration techniques. Proper and improper integrals. Applications of integration in, e.g., determining areas and volumes of geometrical shapes, and computing the length of a curve. Numerical integration, trapezoid rule and Simpson's formula. Computing the Riemann sums. 2. Ordinary linear differential equations of first and second order. Separable first order differential equations. Higher order differential equations. Modeling specific real world problems, such as growth of populations, with differential equations. Existence and uniqueness results, matrix notation for linear systems. 3. Limit of a sequence, increasing and decreasing sequences. 4. Series (geometric, with positive terms, alternating, Taylor series) and their convergence. Approximating a function with a polynomial. Testing convergence. Computing limits and integrals using series. Estimating the error in polynomial approximations of functions. 5. Using Matlab as a tool in solving the exercise problems.

#### Ohjeita opiskelijalle osaamisen tasojen saavuttamiseksi

For grade 5, the student masters the core course content as well as completementary knowledge. For 3-4, the student is understands the core content and has a working knowledge of the complementary knowledge issues. For 1-2, the student has a working knowledge of the core content.

#### Arvosteluasteikko:

Numerical evaluation scale (0-5)

#### Oppimateriaali

 Tyyppi Nimi Tekijä ISBN URL Lisätiedot Tenttimateriaali Book Calculus 6e, Early Transcendentials, Matrix Version Edwards & Penney Chapters 5, 7, 8 and 10. No

#### Esitietovaatimukset

 Opintojakso P/S Selite MAT-01266 Mathematics 2 Mandatory

#### Vastaavuudet

Opintojakso ei vastaan mitään toista opintojaksoa

 Päivittäjä: Kauhanen Janne, 12.06.2019