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#### The Six Pillars of Calculus

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

In Physics
In Economics
In Biology

#### Related Rates

Overview
How to tackle the problems

#### Linear Approximation and Differentials

Overview
Examples
An example with negative $dx$

#### Differentiation Review

Basic 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

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

### Three Versions of L'Hospital's Rule

Here we present three versions of L'Hospital's rule with proof, which we will call Baby L'Hospital's rule, Macho L'Hospital's rule and the Extended L'Hospital's rule. The material here is somewhat more advanced.

• Baby L'Hospital's Rule

 Theorem: Let $f(x)$ and $g(x)$ be continuous functions on an interval containing $x=a$, with $f(a)=g(a)=0$. Suppose that $f$ and $g$ are differentiable, and that $f'$ and $g'$ are continuous. Finally, suppose that $g'(a) \ne 0$. Then $$\lim_{x\to a}\, \frac{f(x)}{g(x)} = \lim_{x \to a} \,\frac{f'(x)}{g'(x)} = \frac{f'(a)}{g'(a)}.$$ Also, $$\lim_{x \to a^+}\, \frac{f(x)}{g(x)} = \lim_{x\to a^+} \,\frac{f'(x)}{g'(x)}$$ and $$\lim_{x \to a^-}\, \frac{f(x)}{g(x)} = \lim_{x \to a^-}\,\frac{f'(x)}{g'(x)}.$$

(Proof.)

The baby version is good enough to compute limits like $$\lim_{x \to 0}\, \frac{\sin(2x)}{x+x^2}.$$

However, it isn't good enough to compute limits like $$\lim_{x \to 0} \,\frac{1 - \cos(2x)}{x^2},$$ since in that case $g'(0)=0$.  To compute limits like that, we need the macho version.

• Macho L'Hospital's Rule:

 Theorem: Suppose that $f$ and $g$ are continuous on a closed interval $[a,b]$, and are differentiable on the open interval $(a,b)$. Suppose that $g'(x)$ is never zero on $(a,b)$, and that $\displaystyle\lim_{x \to a^+} \frac{f'(x)}{g'(x)}$ exists, and that $\displaystyle\lim_{x \to a^+} f(x) = \lim_{x\to a^+}g(x)=0$. Then $$\lim_{x \to a^+}\, \frac{f(x)}{g(x)} = \lim_{x\to a^+} \,\frac{f'(x)}{g'(x)}.$$

(Proof.)

Note that this theorem doesn't require anything about $g'(a)$, just about how $g'$ behaves to the right of $a$. An analogous theorem applies to the limit as $x \to a^-$ (and requires $f$ and $g$ and $f'$ and $g'$ to be defined on an interval that ends at $a$, rather than one that starts at $a$). You can combine the two to get a theorem about an overall limit as $x \to a$.

The conclusion of Macho L'Hospital's Rule relates one limit (of $f/g$) to another limit (of $f'/g'$), and not to the value of $f'(a)/g'(a)$. This is what allows the theorem to be used recursively to solve problems like $\displaystyle\lim_{x \to 0} \frac{1-\cos(2x)}{x^2}$.

• Extended L'Hospital's Rule:

 Theorem: The previous versions apply to indefinite forms of type "$\infty/\infty$" as well as "$0/0$", and apply to limits as $x\to \pm \infty$ as well as to limits $x \to a^\pm$. In all of these cases, the rule is: $$\lim \frac{f(x)}{g(x)} = \lim \frac{f'(x)}{g'(x)}.$$

(Proof.)