Overview

Explore advanced mathematical concepts essential for finance and actuarial studies in this comprehensive course. Delve into complex integration techniques, differential equations, sequences and series, multivariable calculus, and double integrals. Master integration methods including trigonometric functions, partial fractions, and substitutions. Solve various types of differential equations, from first-order to second-order with constant coefficients. Investigate sequences, limits, and series, including Taylor and power series. Examine multivariable calculus topics such as partial derivatives, gradients, and critical points. Learn optimization techniques using the second derivative test and Lagrange multipliers. Develop proficiency in double integrals, including applications in polar coordinates and volume calculations. Apply these mathematical tools to solve real-world problems in finance and actuarial science through numerous tutorials and examples.

Syllabus

Integrals of trig functions and reduction formulae.
Integration by trig substitution and partial fractions.
Integration + Partial Fractions.
Integration via substitutions..
Separable Differential Equations.
Linear and Exact Differential Equations.
Homogeneous first order ordinary differential equation.
Mixing problems and differential equations..
How to solve 2nd order differential equations.
2nd order ODE with constant coefficients: simple method of solution.
2nd order ODE with constant coeffcients: non-standard method of solution.
Solution to a 2nd order, linear homogeneous ODE with repeated roots.
How to solve second order differential equations.
Integration and differential equations.
Sequences and their limits.
What is a Taylor polynomial?.
Limit of a sequence.
Limit of a sequence: L'Hopital's rule applied to $(\ln n)/n$.
Limit of a sequence: $n--$th root of $n^2$.
Intro to series + the integral test.
Series, comparison + ratio tests.
Alternating series and absolute convergence.
What is a Taylor series?.
What is a power series?.
How to calculate power series: a tutorial.
Partial derivatives and error estimation.
Gradient and directional derivative.
Gradient & directional derivative tutorial.
Tangent plane approximation and error estimation.
Tutorial on gradient and tangent plane.
Multivariable Taylor Polynomials.
Taylor polynomials: functions of two variables.
Critical points of functions.
How to find critical points of functions.
Second derivative test: two variables.
Critical points + 2nd derivative test: Multivariable calculus.
How to find and classify critical points of functions.
Max / min on closed, bounded sets.
Lagrange multipliers.
Lagrange multipliers: 2 constraints.
Lagrange multipliers example.
Method of Lagrange multipliers..
2nd derivative test, max / min and Lagrange multipliers tutorial.
Lagrange multipliers: Extreme values of a function subject to a constraint.
Intro to double integrals.
Double integrals over general regions.
Double integral tutorial.
Double integrals and area.
Double integrals in polar co-ordinates.
Reversing order in double integrals.
Applications of double integrals..
Double integrals and polar co-ordinates.
Double integrals: reversing the order of integration.
Tutorial on double integrals.
Double integrals: Volume between two surfaces.
Centroid + double integral tutorial.
Double integrals: Volume of a tetrahedron.
Center of mass, double integrals and polar co-ordinates tutorial.
Centroid and double integrals.
Polar coordinates and double integrals.
Reversing order in double integrals.