An Essential Guide to Calculus and Linear Algebra: A Must Have For University Students

An Essential Guide to Calculus and Linear Algebra This is a must have guide. Calculus and linear algebra are fundamental to virtually all of higher mathematics and its applications in the natural, social and management sciences. These topics therefore form the core of the basic requirements in mathematics, both for Mathematics majors, and for students in science and engineering. Calculus and linear algebra are fundamental to virtually all of higher mathematics and its applications in the natural, social and management sciences. These topics therefore form the core of the basic requirements in mathematics, both for Mathematics majors, and for students in science and engineering. This guide will help you master these concepts. 1 - Introduction Sets and notation Operations on Sets Functions Operations on Functions Symmetries Roots Piecewise Functions Inverse Functions Polynomials and Rational Functions Absolute Values The Absolute Value Relation to Intervals Algebra with Inequalities Exponential Functions Roots Logarithms The Exponential and Logarithmic Functions Sigma Notation Geometric Series 2 - Financial Mathematics Compounding Interest Present Value Continuous Compounding Interest Annuities Amortization Perpetuities 3 - Linear Algebra Linear Equations and Systems Parameterizations of Solutions Matrix Representations of Linear Systems Gaussian Elimination The Rank of a Matrix Matrix Operations Column Vectors The Transpose of a Matrix Matrix Multiplication Matrix Inversion Determinants Definition Properties of the Determinant Eigenvalues and Eigenvectors Exercises 4 - Probability and Counting Counting First Principles of Probability Conditional Probability Applications Expected Value Markov Chains Binomial Evolution Options Pricing Exercises 5 - Limits Some Motivation Intuition One Sided Limits Limit Laws Infinite Limits Vertical Asymptotes Horizontal Asymptotes Continuity One-Sided Continuity and Failures of Continuity 6 - Derivatives First Principles The Geometry of the Derivative A Different Parameterization Relating Variables and Leibniz Notation Some Derivative Results Linearity and the Power Rule The Natural Exponent The Product and Quotient Rule Higher Order Derivatives Smoothness of Differentiable Functions Differentiable implies Continuous Failures of Differentiability Chains and Inverses The Chain Rule Derivatives of Inverse Functions Logarithmic Differentiation 7 - Applications of Derivatives Implicit Differentiation The Idea of Implicit Functions How Implicit Differentiation Works Rates of Change Economics Exponential Growth Derivatives and the Shape of a Graph First Derivative Information Second Derivative Information Maxima and Minima Optimization Curve Sketching Approximation a Function Quadratic and Higher 8 - Integration The Definite Integral The Intuition Estimating Areas Defining the Definite Integral Anti-Deri