Absolute and Local Extrema

Fermat’s little theorem proof | Number Theory
Fermat’s little theorem proof | Number Theory

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The Six Pillars of Calculus

The Pillars: A Road Map

A picture is worth 1000 words

Trigonometry Review

The basic trig functions

Basic trig identities

The unit circle

Addition of angles, double and half angle formulas

The law of sines and the law of cosines

Graphs of Trig Functions

Exponential Functions

Exponentials with positive integer exponents

Fractional and negative powers

The function $f(x)=a^x$ and its graph

Exponential growth and decay

Logarithms and Inverse functions

Inverse Functions

How to find a formula for an inverse function

Logarithms as Inverse Exponentials

Inverse Trig Functions

Intro to Limits

Close is good enough

Definition

One-sided Limits

How can a limit fail to exist?

Infinite Limits and Vertical Asymptotes

Summary

Limit Laws and Computations

A summary of Limit Laws

Why do these laws work?

Two limit theorems

How to algebraically manipulate a 0/0?

Limits with fractions

Limits with Absolute Values

Limits involving Rationalization

Limits of Piece-wise Functions

The Squeeze Theorem

Continuity and the Intermediate Value Theorem

Definition of continuity

Continuity and piece-wise functions

Continuity properties

Types of discontinuities

The Intermediate Value Theorem

Examples of continuous functions

Limits at Infinity

Limits at infinity and horizontal asymptotes

Limits at infinity of rational functions

Which functions grow the fastest?

Vertical asymptotes (Redux)

Toolbox of graphs

Rates of Change

Tracking change

Average and instantaneous velocity

Instantaneous rate of change of any function

Finding tangent line equations

Definition of derivative

The Derivative Function

The derivative function

Sketching the graph of $f’$

Differentiability

Notation and higher-order derivatives

Basic Differentiation Rules

The Power Rule and other basic rules

The derivative of $e^x$

Product and Quotient Rules

The Product Rule

The Quotient Rule

Derivatives of Trig Functions

Two important Limits

Sine and Cosine

Tangent, Cotangent, Secant, and Cosecant

Summary

The Chain Rule

Two forms of the chain rule

Version 1

Version 2

Why does it work?

A hybrid chain rule

Implicit Differentiation

Introduction and Examples

Derivatives of Inverse Trigs via Implicit Differentiation

A Summary

Derivatives of Logs

Formulas and Examples

Logarithmic Differentiation

Derivatives in Science

In Physics

In Economics

In Biology

Related Rates

Overview

How to tackle the problems

Example (ladder)

Example (shadow)

Linear Approximation and Differentials

Overview

Examples

An example with negative $dx$

Differentiation Review

Basic Building Blocks

Advanced Building Blocks

Product and Quotient Rules

The Chain Rule

Combining Rules

Implicit Differentiation

Logarithmic Differentiation

Conclusions and Tidbits

Absolute and Local Extrema

Definitions

The Extreme Value Theorem

Fermat’s Theorem

How-to

The Mean Value and other Theorems

Rolle’s Theorems

The Mean Value Theorem

Finding $c$

$f$ vs. $f’$

Increasing/Decreasing Test and Critical Numbers

How-to

The First Derivative Test

Concavity, Points of Inflection, and the Second Derivative Test

Indeterminate Forms and L’Hospital’s Rule

What does $\frac{0}{0}$ equal?

Indeterminate Differences

Indeterminate Powers

Three Versions of L’Hospital’s Rule

Proofs

Optimization

Strategies

Another Example

Newton’s Method

The Idea of Newton’s Method

An Example

Solving Transcendental Equations

When NM doesn’t work

Anti-derivatives

Anti-derivatives and Physics

Some formulas

Anti-derivatives are not Integrals

The Area under a curve

The Area Problem and Examples

Riemann Sums Notation

Summary

Definite Integrals

Definition

Properties

What is integration good for?

More Examples

The Fundamental Theorem of Calculus

Three Different Quantities

The Whole as Sum of Partial Changes

The Indefinite Integral as Antiderivative

The FTC and the Chain Rule

Fermat’s Theorem The Third pillar of Calculus

The Extreme Value Theorem tells us that the minimum and maximum of a function have to be somewhere. But where should we look? The answer lies in the third of the Six Pillars of Calculus:

Fermat’s Theorem

Places where the derivative either

This is the idea behind one of Fermat’s theorems:

Equivalently, if $f'(c)$ exists and is not zero, then $f(c)$ is neither a maximum nor a minimum.

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