After successfully finishing this study unit a student is able to:
1. work with elementary properties of integrals and calculate integrals using different techniques, for functions of 1 variable
2. work with power series and Taylor series, for functions of 1 variable
- formulate Riemann sums
- formulate and use the Fundamental Theorem of Calculus
- calculate integrals using anti-derivates
- calculate integrals using the substitution method
- calculate integrals using the technique of integration by parts
- calculate improper integrals using limits
3. solve first- and second-order ordinary differential equations
- calculate the radius of convergence of a power series
- calculate Taylor series and Taylor polynomials
4. work with complex numbers
- solve first-order separable differential equations using separation of variables
- solve first-order linear differential equations using the integrating factor
- solve first- and second-order linear homogeneous differential equations with constant coefficients using the characteristic equation
- solve first- and second-order linear inhomogeneous differential equations with constant coefficients using the method of undetermined coefficients
- solve initial / boundary value problems
- plot (sets of) complex numbers in the plane
- convert complex numbers from Cartesian form to polar form and vice versa
- apply complex arithmetic operations
- find roots of a complex number and solve binomial equations
Just like Calculus 1A, Calculus 1B is a course in the basic mathematics programme of the UT, called the Mathematics Line.
Calculus 1B starts with a topic many students are already familiar with: integration theory. The integral of a function of one variable is introduced. Integration techniques such as substitution and integration by parts are discussed, as well as the concept of improper integrals. The follow-up course Calculus 2 will continue with integral calculus for functions of two or three variables.
Next, a short introduction to power series is given, with Taylor series and Taylor polynomials as its main application.
Another new concept that is introduced is that of a differential equation. The idea behind a differential equation is discussed. Methods for solving first-order separable differential equations and first-order linear differential equations are presented in detail. A method for solving second-order linear differential equations with constant coefficients is also presented, but several details are left out.
Before second-order linear differential equations are discussed, the set of complex numbers is introduced. Different representations of complex numbers are presented, as well as algebraic operations on complex numbers and solving equations involving complex numbers.