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-Derivatives
Initial Value Problems
The Fundamental Theorem of Calculus
Properties of the Definite Integral
Indefinite Integrals
Integral Notation
Integration Techniques
Integration by Substitution
Integration by Parts

9 - Applications of Integration
Area Computations
What We Already Know
More Complicated Shapes
Unsigned (Absolute) Area
Integrating along the y-axis
The Area Between Curves
Improper Integrals
Infinite Intervals
Unbounded Functions
The Basic Comparison Test
The Limit Comparison Test
Applications in Economics and Finance

10 - Differential Equations
Basic Differential Equations
Separable Differential Equations
Linear Differential Equations
Second Order Differential Equation
Exercises

11 - Multivariable Calculus
Partial Derivatives
Applications of Partial Derivatives
Higher-Order Partial Derivatives
The Chain Rule
Optimization
Critical Points
Constrained Optimization
Iterated Integrals
Change of Variables
Coordinates
Integration