Foundational Mathematics for Actuaries / CFA / FRM

Master the chain rule for nested functions and compute the derivative of F(X)=G(H(X)) with inner H(X)=3X^2+2X−1 and outer G(u)=u^5.

Learn how to locate stationary points on surfaces using partial derivatives, classify them as minima, maxima, or saddle points with second partial derivatives, and preview constrained optimization via Lagrange multipliers.

Use Lagrange multipliers to find constrained stationary points of a bivariate function under the constraint 4x^2 + y^2 = 9, identifying the maximum at (-3/2,0) and the minimum at (0,±3).

Utilize technique 3: substitution with u = 2x + 3 to simplify the integral, adjusting limits from x to u and replacing dx with du/2.

Master double integrals to compute volume under a surface using rectangle and triangle regions. Apply inner and outer integrals, variable constants, and region limits for actuarial contexts.

Learn to solve differential equations with two functions by separating variables, integrating dy/y and the x-part, and obtaining the general solution y = K(1+x^2).

Explore integrating factors to solve Bernoulli differential equations, transforming to linear form and solving examples such as dy/dx + 4y = 2 e^{3x}.

Define a series as the sum of a sequence and use sigma notation to set start and end indices; derive arithmetic, geometric, and infinite geometric series formulas for |r|<1.

Discover power series, Taylor and Maclaurin expansions, and radius of convergence, including geometric series cases, then differentiate and integrate term-by-term to represent functions.

Explore linear algebra by representing linear equations and maps with vectors and matrices, and apply these tools to computations in actuarial science, engineering, and computer science.

Explore matrices and their operations, including transpose, zero, diagonal, and identity matrices, for actuarial applications. Learn addition, subtraction, scalar multiplication, and both pre- and post-multiplication with dimension rules.