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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

One-sided Limits
When limits don't exist
Infinite Limits

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 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
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 $f'$
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

Necessary Limits
Derivatives of Sine and Cosine
Derivatives of Tangent, Cotangent, Secant, and Cosecant

The Chain Rule

Two Forms of the Chain Rule
Version 1
Version 2
Why does it work?
A hybrid chain rule

Implicit Differentiation

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

How to tackle the problems
Example (ladder)
Example (shadow)

Linear Approximation and Differentials

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

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, Points of Inflection, and the Second Derivative Test
The Second Derivative Test
Visual Wrap-up

Indeterminate Forms and L'Hospital's Rule

What does $\frac{0}{0}$ equal?
Indeterminate Differences
Indeterminate Powers
Three Versions of L'Hospital's Rule


Another Example

Newton's Method

The Idea of Newton's Method
An Example
Solving Transcendental Equations
When NM doesn't work


Common antiderivatives
Initial value problems
Antiderivatives are not Integrals

The Area under a curve

The Area Problem and Examples
Riemann Sum Notation

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 Anti-derivatives
A Table of Common Anti-derivatives
The Net Change Theorem
The NCT and Public Policy


Substitution for Indefinite Integrals
Examples to Try
Revised Table of Integrals
Substitution for Definite Integrals

Area Between Curves

Computation Using Integration
To Compute a Bulk Quantity
The Area Between Two Curves
Horizontal Slicing


Slicing and Dicing Solids
Solids of Revolution 1: Disks
Solids of Revolution 2: Washers
More Practice


Intuitively, the absolute maximum value is the largest of the possible values of $f(x)$, and similarly for minimum.  Notice that a function may reach its maximum (or minimum) at more than one $x$-value. 
DO:  Read the following definition carefully, and slowly, with an example $f$ graphed, and think of what this precise language is saying.

Definition (Absolute Extrema)
  • If $c$ is a number in the domain of $f$, then $f(c)$ is the absolute maximum value of $f$ if $f(c)\geq f(x)$ for all $x$ in the domain of $f$.
  • If $c$ is a number in the domain of $f$, then $f(c)$ is the absolute minimum value of $f$ if $f(c) \leq f(x)$ for all $x$ in the domain of $f$.

A value of $f(x)$ may not be the largest (or smallest) of all, but it might be the largest (or smallest) compared to nearby values.  We call these local extrema, or local extreme values
:  Read the following definition carefully and look at an example or two.

Definition (Local Extrema)
  • If $c$ is a number in the domain of $f$, then $f(c)$ is a local maximum value of $f$ if $f(c) > f(x)$ when $x$ is "near" $c$.
  • Likewise, $f(c)$ is local minimum value of $f$ if $f(c) < f(x)$ when $x$ is "near" $c$.

Notice that when a function is defined on a closed interval, an absolute extreme value may occur at the endpoint of that interval of domain, since the endpoint of the interval may yield the largest value of $f$ on that closed interval. 

This situation is different with local extrema.  When we say $x$ is near $c$, we mean on an open interval containing $c$, so on either side of $c$.  When $f$ is defined on a closed interval, there is no open interval containing an endpoint of the closed interval on which $f$ is defined.  Hence, a local extreme value cannot occur at the endpoint of an interval of domain.  This is a definition, and it could be defined differently. 

This video will help clarify these concepts.

Notice that the absolute and local maxima and minima are $y$-values.  Graphically, this means that the max/min value is the maximum/minimum height of the graph at some $x=c$.  Then $x=c$ is where the max/min occurs.

Example: Suppose you are driving westward across Colorado on IH 70 (to get to Vail to go snowskiing). As you head west, the road will sometimes be going up, and other times going down, but overall your altitude is increasing.

Suppose that, along the way, you go up, then at mile marker 324 (where your altitude is 6,200 feet) you start going down.  We would say that you reached a local maximum altitude of 6,200 feet at $x=324$.  You keep driving, some up, some down (attaining local maximum and maximum altitudes) until you reach Vail Pass (the Continental Divide) at mile marker 278, where the altitude is 10,662 feet.  We say you reached an absolute maximum altitude of 10,662 feet at $x=278$.  If you are traveling from Denver at 5280 feet high, to Grand Junction at 4560 feet high, what is your guess about the absolute minimum altitude you attain, and where it occurs?