 Kaisa_2012_3_photo by Veikko Somerpuro

### Description

Programme: Bachelor's programme in Science.

Module: Study module in Science or Basics studies in Mathematics.

The course is compulsory for all students of the programme.

The course is available to the students of other programmes upon agreement with the teacher.

astering precalculus such as in high school mathematics curriculum is necessary. Students are expected already be able to

• Solve polynomial, rational, trigonometric, exponential equations among others
• Perform algebraic and trigonometric manipulations and rewritings to mathematical expressions
• Be familiar with basic functions and their properties
• Draw graphs of functions, be able to explain where in the graph a function is increasing or decreasing, where does it attain its minimum or maximum values, etc.

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After successfully completing this course students will

• Understand basic concepts of real-valued functions, especially what is meant by limit of a function and continuity.
• Be familiar with consequences of continuity, especially Bolzano's Theorem, and apply them for example in finding solutions to equations.
• Be able to find limits of sequences of real numbers and real valued functions with various techniques involving algebraic manipulations or with the squeeze theorem.
• Understand concept of the derivative from number of perspectives, e.g. as instantaneous rate of change, limit of so called difference quotient, slope of the tangent line to the graph of a continuous function at a point, and through differentiability.
• Find derivatives of basic functions, derive differentiation rules and apply them.
• Be familiar with Mean Value Theorem for differentiation, be able to justify why it holds, as well as use it in applications.
• Use derivative as a tool for studying behavior of functions and in other applications.

First year of studies

Course covers the following main topics

1. Functions
2. Limits of Functions
3. Continuity
4. Derivatives & Differentiation Rules
5. The Mean Value Theorem
6. Applications of Derivatives

Either by final exam or combination of class starters, exercises and final exam.

All necessary study material can be found in the course area. Standard Calculus text books, such as Adams' or Stewart's Calculus books can be used as side material among others.

Scale 1-5 of grades will be used. Points consist of either final exam 100% or 30% exercises, 20% class starters, 50% final exam. Passing course with grade 1 will require approximately half of the points of the exam and for the best grade approximately 5/6 of the exam points are required.

The course is offered every year in the autumn (period II).

Prof. Tuomo Kuusi