CLP-1 Differential Calculus

The Basics
Numbers
Sets
Other Important Sets
Functions
Parsing Formulas
Inverse Functions
Limits
Drawing Tangents and a First Limit
Another Limit and Computing Velocity
The Limit of a Function
Calculating Limits with Limit Laws
Limits at Infinity
Continuity
(Optional) — Making the Informal a Little More Formal
(Optional) — Making Infinite Limits a Little More Formal
(Optional) — Proving the Arithmetic of Limits
Derivatives
Revisiting Tangent Lines
Definition of the Derivative
Interpretations of the Derivative
Arithmetic of Derivatives - a Differentiation Toolbox
Proofs of the Arithmetic of Derivatives
Using the Arithmetic of Derivatives – Examples
Derivatives of Exponential Functions
Derivatives of Trigonometric Functions
One More Tool – the Chain Rule
The Natural Logarithm
Implicit Differentiation
Inverse Trigonometric Functions
The Mean Value Theorem
Higher Order Derivatives
(Optional) — Is limxcf'(x) Equal to f'(c)?
Applications of Derivatives
Velocity and Acceleration
Related Rates
Exponential Growth and Decay — a First Look at Differential Equations
Carbon Dating
Newton's Law of Cooling
Population Growth
Approximating Functions Near a Specified Point — Taylor Polynomials
Zeroth Approximation — the Constant Approximation
First Approximation — the Linear approximation
Second Approximation — the Quadratic Approximation
Still Better Approximations — Taylor Polynomials
Some Examples
Estimating Change and x, y Notation
Further Examples
The Error in the Taylor Polynomial Approximations
(Optional) — Derivation of the Error Formulae
Optimisation
Local and Global Maxima and Minima
Finding Global Maxima and Minima
Max/Min Examples
Sketching Graphs
Domain, Intercepts and Asymptotes
First Derivative — Increasing or Decreasing
Second Derivative — Concavity
Symmetries
A Checklist for Sketching
Sketching Examples
L'Hôpital's Rule and Indeterminate Forms
Standard Examples
Variations
Towards Integral Calculus
Introduction to Antiderivatives
High School Material
Similar Triangles
Pythagoras
Trigonometry — Definitions
Radians, Arcs and Sectors
Trigonometry — Graphs
Trigonometry — Special Triangles
Trigonometry — Simple Identities
Trigonometry — Add and Subtract Angles
Inverse Trigonometric Functions
Areas
Volumes
Powers
Logarithms
Highschool Material You Should be Able to Derive
Origin of Trig, Area and Volume Formulas
Theorems about Triangles
Thales' Theorem
Pythagoras
Trigonometry
Angles — Radians vs Degrees
Trig Function Definitions
Important Triangles
Some More Simple Identities
Identities — Adding Angles
Identities — Double-angle Formulas
Identities — Extras
Inverse Trigonometric Functions
Cosine and Sine Laws
Cosine Law or Law of Cosines
Sine Law or Law of Sines
Circles, cones and spheres
Where Does the Formula for the Area of a Circle Come From?
Where Do These Volume Formulas Come From?
Root Finding
Newton's Method
The Error Behaviour of Newton's Method
The false position (regula falsi) method
The secant method
The Error Behaviour of the Secant Method