Mathematics: Analysis in One Variable

The course treats:

• The real numbers: axioms, examples of proofs of basic arithmetical rules.
• The elementary functions, polynomials, rational functions, the exponential function and the natural logarithm, the trigonometric functions and the inverse trigonometric functions; definitions, basic properties, and quantitative approximations using representations in terms of areas and arclengths.
• Sequences of numbers and their limits: formal definition of the limit, examples of proofs of their computational rules, visual representation of convergence of recursive sequences, quantitative approximations.
• Infinite series: applications and proofs of convergence tests, absolute convergence, quantitative approximations using partial sums and tail estimates.
• Functions and their limits: formal definition of the limit, proofs and applications of their computational rules, indeterminate forms and asymptote
• Continuity: continuity of elementary functions, the intermediate value theorem and the min-max theorem.
• Derivatives: definition, proofs and applications of computational rules, differentiation formulas for elementary functions, Rolle’s lemma, the mean value theorem and L’Hopital’s rule.
• Applications of the derivative: optimisation and graph sketching, techniques for establishing identities and inequalities.
• Indefinite integrals: proofs and applications of basic computational rules and integration methods, such as change of variables, partial integration and use of partial fraction decomposition.
• Definite integrals: Darboux integrability of monotone functions and functions with bounded derivative with related error estimates, the fundamental theorem of calculus, applications to arclength, rotational volumes and surfaces, numerical approximations of definite integrals.
• Improper integrals: convergence criteria for improper integrals for positive functions, absolute convergence, comparison to infinite series.
• Differential equations: direction fields, analytic solution methods for separable and linear first order differential equations, solution method for linear higher- order differential equations with constant coefficients, numerical approximations of solutions of initial value problems using Euler’s method.
• Taylor expansions: Taylor's formula with Lagrange’s formula for the error term, uniqueness theorem for Taylor polynomials, numerical approximations of function values and integrals using Taylor polynomials.
• In addition, materials on sets, functions and relations, induction, the binomial theorem, as well as variables, for-loops and if-statements in Python are covered at the beginning of the course.