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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
Overview
Definition
Onesided Limits
When limits don't exist
Infinite Limits
Summary
Limit Laws and Computations
Limit Laws
Intuitive idea of why these laws work
Two limit theorems
How to algebraically manipulate a 0/0?
Indeterminate forms involving fractions
Limits with Absolute Values
Limits involving indeterminate forms with square roots
Limits of Piecewise Functions
The Squeeze Theorem
Continuity and the Intermediate Value Theorem
Definition of continuity
Continuity and piecewise functions
Continuity properties
Types of discontinuities
The Intermediate Value Theorem
Summary of using continuity to evaluate limits
Limits at Infinity
Limits at infinity and horizontal asymptotes
Limits at infinity of rational functions
Which functions grow the fastest?
Vertical asymptotes (Redux)
Summary and selected graphs
Rates of Change
Average velocity
Instantaneous velocity
Computing an instantaneous rate of change of any function
The equation of a tangent line
The Derivative of a Function at a Point
The Derivative Function
The derivative function
Sketching the graph of $f'$
Differentiability
Notation and higherorder 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
Necessary Limits
Derivatives of Sine and Cosine
Derivatives of 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
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
How to take derivatives
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
Critical Numbers
Steps to Find Absolute Extrema
The Mean Value and other Theorems
Rolle's Theorems
The Mean Value Theorem
Finding $c$
$f$ vs. $f'$
Increasing/Decreasing Test and Critical Numbers
Process for finding intervals of increase/decrease
The First Derivative Test
Concavity
Concavity, Points of Inflection, and the Second Derivative Test
The Second Derivative Test
Visual Wrapup
Indeterminate Forms and L'Hospital's Rule
What does $\frac{0}{0}$ equal?
Examples
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
Antiderivatives
Antiderivatives
Common antiderivatives
Initial value problems
Antiderivatives are not Integrals
The Area under a curve
The Area Problem and Examples
Riemann Sum Notation
Summary
Definite Integrals
Definition of the Integral
Properties of Definite Integrals
What is integration good for?
More Applications of Integrals
The Fundamental Theorem of Calculus
Three Different Concepts
The Fundamental Theorem of Calculus (Part 2)
The Fundamental Theorem of Calculus (Part 1)
More FTC 1
The Indefinite Integral and the Net Change
Indefinite Integrals and Antiderivatives
A Table of Common Antiderivatives
The Net Change Theorem
The NCT and Public Policy
Substitution
Substitution for Indefinite Integrals
Examples to Try
Revised Table of Integrals
Substitution for Definite Integrals
Examples
Area Between Curves
Computation Using Integration
To Compute a Bulk Quantity
The Area Between Two Curves
Horizontal Slicing
Summary
Volumes
Slicing and Dicing Solids
Solids of Revolution 1: Disks
Solids of Revolution 2: Washers
More Practice


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 \begin{equation}\lim_{x \to 0}\, \frac{\sin(2x)}{x+x^2}.\end{equation}
However, it isn't good enough to compute limits like
\begin{equation} \lim_{x \to 0} \,\frac{1  \cos(2x)}{x^2},
\end{equation} 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.)
