LEARNING OUTCOMES
Upon successful completion of the course, the students will:
1. Know the basic notions of calculus (continuity, limits, derivatives and integrals) and use them to analyse properties of mathematical models of growth and other natural processes.
2. Set up and solve simple differential equations and produce mathematical models of population growth.
3. Know the basic notions and techniques of linear algebra (vectors, matrices, Gaussian eliminations, eigenvalues) and use them to solve problems described by linear equations.
4. Get to know Leslie's model of population dynamics and apply the acquired methods of linear algebra.
5. Clearly and precisely define mathematical notions, state theorems and make simple conclusions. Set up mathematical models of natural processes using the acquired notions and techniques.
COURSE CONTENTS
LECTURES
1. Real numbers. Sequences. Discrete models of growth
2. Functions. Elementary functions
3. Limits of functions. Continuity
4. Speed, tangents and derivatives
5. Derivatives of elementary functions. Properties of differentiable functions
6. Applications of differential calculus
7. The definitive integral
8. Integration techniques
9. Applications of integral calculus
10. Differential equations
11. Methods of solving and applications of differential equations
12. Matrices and systems of linear equations. Matrix operations
13. Gaussian eliminations
14. Rank and inverse of a matrix. Eigenvalues and eigenvectors
EXERCISES
1. Set operations. Equation solving
2. Computing limits of sequences
3. Finding domains of functions. Recognising properties of a function from its graph
4. Computing limits of functions
5. Computing derivatives
6. Finding intervals of increase and decrease, minima and maxima of functions using derivatives
7. Computing indefinite and definite integrals
8. Determining area and volume using integrals
9. Solving differential equations
10. Mathematical modelling using differential equations
11. Solving systems of linear equations using Gaussian eliminations
12. Computing with matrices and vectors
13. Computing the inverse of a matrix. Leslie's model
14. Computing the rank of a matrix, eigenvalues and eigenvectors
