Mathematics A

Basic information

Support materials

• Mathematical rigor
• Numbers
• Real functions
• Elementary functions - graphs
• Continuity and limits of functions
• Derivatives
• Parametric equations of planar curves
• Integrals
• Differential equations

Slides from lecture

• Introduction
• Real functions
• Continuity and limits of functions
• Derivatives
• Graphing functions
• Taylor polynomial, differential
• Parametric equations of planar curves
• Integral calculus
• Definite integral
• Applications of integrals
• Differential equations of the first order
• Linear differential equations with constant coefficients
• Applications of differential equations

Syllabus – here

1. Functions of a single real variable. Domain and range. Graphs of elementary functions. Basic properties. Composition of functions.
2. Inverse functions. Exponential and logarithmic functions. Trigonometric and inverse trigonometric functions.
3. Continuity of a function. Properties of continuous functions. Limits of sequences and functions.
4. Derivatives. Geometrical and physical meaning of derivatives. Rules for computing derivatives. Differential of a function.
5. Physical and geometrical applications of derivatives. L’Hospital’s rule. Approximation of a function value using Taylor polynomial. Analysis and graphing of a function.
6. Numerical solution of an equation of a single uknown variable - Newton’s method. Parametric curves, tangent vector to a curve.
7. Antiderivatives and their properties. Newton definite integral, its properties and geometrical meaning.
8. Methods for computing indefinite and definite integrals – integration by parts and substitution method.
9. Integration of rational functions. Improper integrals. Numerical integration – trapezoidal method.
10. Riemann definite integral. Selected geometrical and physical applications of the integral.
11. Differential equations. Terminology, general and particular solution. Separation of variables.
12. First order linear differential equations. Variation of constants. Numerical solution of a first order differential equations – Euler’s method.
13. Second order linear differential equations with constant coefficients and a special right-hand. Estimation method.
14. Application of differential equations in Physics, Chemistry, and Biochemistry. Revision and discussion.

Consultations – by agreement personal or email