# MAT-01166 Mathematics 1, 5 cr

Janne Kauhanen

#### Opetus

 Toteutuskerta Periodi Vastuuhenkilö Suoritusvaatimukset MAT-01166 2019-01 1 Janne Kauhanen Final exam, weekly exercises, and peer-reviewed exercises in Moodle.

#### Osaamistavoitteet

On this course the students learn how to compute and use the limit, the derivative, and complex numbers. 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. Sets and set operations. Methods of proof, mathematical induction. Real numbers: algebraic properties, the absolute value, intervals. Quantifiers. The field structure of the real number system. 2. The definition of a function. Real valued functions of a real variable: monotonicity and the inverse function, the composition of functions. The basic properties of elementary functions. Injectivity, surjectivity and bijectivity, the pre-image. The definition of elementary functions. Hyperbolic functions. 3. The limit of a function and the basic properties of the limit. One-sided limits, infinite limits, and limits at infinity. l'Hospital's rule. Continuity. The epsilon-delta definition and proofs. The Squeeze Theorem. Properties of continuous functions: the Intermediate Value Theorem and the Extreme Value Theorem. 4. The derivative: the definition, the linear approximation, algebraic operations, the Chain Rule, the derivative of an inverse function, extrema and critical points, monotonicity. Derivatives of the elementary functions. The Mean Value Theorem, the Taylor polynomial and Taylor's Formula. 5. Complex numbers: definition and algebraic operations, the conjugate, the modulus, the exponential (trigonometric) form, finding the roots. Zeros and factorization of a polynomial with real coefficients. 6. Using Matlab as a tool in solving the exercise problems.

#### Oppimateriaali

 Tyyppi Nimi Tekijä ISBN URL Lisätiedot Tenttimateriaali Book Mathematical Analysis I Claudio Canuto, Anita Tabacco 978-88-470-0875-5 Springer E-kirja/E-book No

#### Vastaavuudet

Opintojakso ei vastaan mitään toista opintojaksoa

 Päivittäjä: Kunnari Jaana, 05.03.2019