After this course, the student is able to
- Reflect critically on existing theoretical knowledge and ideas.
- Analyze algebraic properties of one variable functions, including critical points, limits, asymptotic behavior and singularities.
- Define the concept of Riemann integration and apply this to appropriate problems.
- Apply partial integration to find antiderivatives.
- Check if a solution of an ordinary differential equation (ODE) exists, and whether it is unique;
- Analyze improper integrals.
- Study different categories of infinite series and discuss their convergence.
- Study the concept of vectors in 2-D and 3-D.
- Classify ODEs and solve ODEs analytically using separation of variables, integrating factors or variation of constants
- Discuss the stability of solutions of differential equations.
- Explain ideas, demonstrate personal autonomy, develop critical thinking and problem-solving skills and apply the principles of teamwork and collaboration.
- Demonstrate that has reached all underlying ideas at the expected level.
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In this 6 EC course you will develop a conceptual understanding on Differential and Integral Calculus of 1-D functions, and on Ordinary Differential Equations. You will become competent in applying the concepts to solve concrete problems, increase your computational skills, and practice in communicating your results.
The course will cover the concepts of Derivatives, Integrals, Infinite Series, and Ordinary Differential Equations. In order to solve certain types of differential equations, the topic of complex numbers will be covered as well. Emphasis will be given on applications of Differentiation, Integration and Modelling.
External students who are interested in this elective: please contact electives-uct@utwente.nl
Prior knowledge:
Knowledge presumed as a basis:
First year students registered for the course will have to master the topics provided on the online Math Platform Grasple.
The aim of this preparatory set of units is to provide a link between the topics already taught in high school and the more advanced topics to be covered in the course Introduction to Calculus and Linear Algebra in Semester 1.
Topics to be covered and Learning Objectives:
Fundamentals (Learning Objectives):
Recall and apply basic algebra skills.
Sketch graphs of various expressions.
Find intersection points.
Solve equations and inequalities.
Unit 1: Functions (Learning Objectives)
Define a function.
Recognize different categories of functions.
Analyse the behaviour for different categories of functions.
Use the properties of these functions to solve equations and application problems.
Unit 2: Limits (Learning Objectives)
Learning Objectives:
Demonstrate, describe, and recognize ways in which limits exist (or not) .
Evaluate limits given analytic, graphical, numerical function information .
Explain indeterminate form of limits.
Evaluate one sided limits.
Describe relationship between Limits and Continuity.
Unit 3: Derivatives (Learning Objectives)
State the limit definition of derivative of a function.
Calculate derivatives of functions by applying differentiation rules.
Interpret and make use of notation for higher-order derivatives.
Unit 4: Integration (Learning Objectives)
Define the First and Second Fundamental Theorem of Calculus.
Relate the Definite integral with the Area under a curve.
Solve Indefinite and Definite Integrals using appropriate Integration Rules.
Use Substitution Rule to solve integrals.
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