# Engineering Mathematics 2

Structure Type: | Study unit |
---|---|

Code: | TT00BP65 |

Curriculum: | KT 2024 |

Level: | Bachelor of Engineering |

Year of Study: | 2 (2025-2026) |

Semester: | Autumn |

Credits: | 5 cr |

Responsible Teacher: | Ojanen, Jussi |

Language of Instruction: | Finnish |

Taking the course in advance? See the courses during the academic year 2024-2025.

## Learning Outcomes

The student learns to differentiate an elementary function, and to apply the derivative to, for example, optimization and numerical problems, and hears about the physical applications of the derivative.

The student learns to integrate an elementary function, and to apply the integral to, for example, geometric and numerical problems, and hears about the physical applications of the integral.

The student learns to solve some types of differential equations, both analytically and numerically, and hears about the physical applications of differential equations.

The student learns to expand a function in Taylor's series, gets to know the role of a series in numerical calculation and hears about the importance of series in solving physical problems.

## Student's Workload

135 h, which includes 45 h of contact teaching.

## Prerequisites / Recommended Optional Courses

TT00BP64 Engineering Mathematics 1.

## Contents

Derivative operator, limit and continuity of a function, derivative of a function, tangent line, tangent plane, derivatives of elementary functions (sum, product, quotient, power, root, logarithm, trigonometric x 12), derivative of a combined function, logarithmic derivative, higher derivatives, partial derivative, the differential, extreme values, optimization, Newton's algorithm, implicit function derivative.

Integral operator, indefinite integral, definite integral with geometric and physical applications, integrals of elementary functions, partial integration, variable changing technique (substitution method), rational function integration, numerical integration with polynomial interpolates (rectangle, trapezoid, Simpson), applications, e.g. function value average and quadratic mean value, length of plane and space curves, volume and area of an envelope (at least the body of rotation), center of gravity.

Differential equation (DE), initial condition, directly integrable, separable, linear DE; solution in the time domain (lin. homog. 1st order, lin. 1st order constant coefficient, lin. 1st order variable coefficient, lin. 2nd order constant coefficient),

Laplace transform, numerical propagation methods: Euler, Runge-Kutta.

Series, arithmetic, geometric and power series (Taylor and Maclaurin), Fourier series, integration with series, finite difference discretization of the derivative.

Statistical and probability(?)

01) Limit and continuity of a function.

02) Derivative of a function and its geometric interpretation.

03) Differentiation rules.

04) Partial derivative and the differential.

05) Extreme values of a function.

06) Integral function.

07) Definite integral and its geometric interpretation.

08) Integration methods: partial integration, substitution, fractioning.

09) The volume of the rotational body.

10) Calculating the center of gravity of a disc.

11) First order differential equations.

12) Second-order linear differential equation with constant coefficients.

16) Series: arithmetic and geometric series.

17) Power series: Maclaurin's series.

18) Binomial series.

19) Calculating definite integrals using power series.

## Recommended or Required Reading and Other Learning Resources/Tools

Material compiled by the teacher.

## Mode of Delivery / Planned Learning Activities and Teaching Methods

Lectures, homework exercises.

## Assessment Criteria

Grade 5: The student knows all the concepts and resulting procedures/algorithms discussed on the course, and understands how they are relatated with each other. The student is able to independently apply the tools discussed on the course while solving complicated problems related with the contents of the course.

Grade 3: The student knows most of the concepts and resulting procedures/algorithms discussed on the course, and understands a significant amount of the relationships between them. The student is able to apply the tools discussed on the course while solving medium-level problems related with the contents of the course.

Grade 1: The student knows the most important concepts and resulting procedures/algorithms discussed on the course, and understand the most important relationships between them. The student is able to apply the tools discussed on the course while solving basic problems related with the contents of the course.

## Assessment Methods

Examinations (2), homework exercise activity.