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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

Determining Intervals of Concavity and Inflection Points

The intervals of concavity can be found in the same way used to determine the intervals of increase/decrease, except that we use the second derivative instead of the first. In particular, since $(f')'=f''$, the intervals of increase/decrease for the first derivative will determine the concavity of $f$.

The process to find intervals of concavity

  1. If possible, factor $f''$.  If $f''$ is a quotient, factor the numerator and denominator (separately).

  2. Find all critical numbers $x=s$ of $f'$.   These are the points where $(f')'=0$ or $(f')'$ doesn't exist (i.e., the points where $f''=0$ or where $f''$ doesn't exist).  You might want to call these $s$ subcritical numbers.

  3. Draw a number line with tick marks at each subcritical number $s$.

  4. For each interval between subcritical numbers in which the function $f$ is defined, pick a number $b$, and then find the sign of the second derivative $f''(b)$.

  5. If $f''(b) \gt 0$, then $f'$ is increasing on the interval containing $b$.  This means that the slopes are increasing, so $f$ is concave up.  Draw a right-side-up bowl over that interval on your number line.  Similarly, if $f''(b) \lt 0$, draw an upside-down bowl.

  6. That's it! You can now see the intervals where $f$ is concave up or down.

If this box above looks familiar, it is because we use that exact same process to find intervals of increase/decrease of $f$, by looking at $f'$ instead of $f''$.  Keep in mind:  the sign of $f'$ indicates the direction (up or down) of $f$, while the sign of $f''$ indicates the concavity of $f$.

Inflection points

An inflection point is a point where concavity changes.  In each of the graphs below, the point of inflection lies between the location of the two tangent lines; the tangent lines show that the concavity has changed. 
DO:  The inflection point is not marked -- can you find it?


The process to find inflection points

Take the number line showing subcritical numbers and intervals of concavity from the process above.  The points $(s,f(s))$ where the concavity changes are inflection points.  Note:  not all subcritical numbers will yield inflection points (just like not all critical numbers yield local extrema).

Example: Find the intervals of concavity and any inflection points of f(x)=x33x2 .

DO:  Try to work this problem, using the process above, before reading the solution.

Solution: Since f(x)=3x26x=3x(x2), our two critical points for f are at x=0 and x=2.  We used these critical numbers to find intervals of increase/decrease as well as local extrema on previous slides.  Meanwhile, f(x)=6x6 , so the only subcritical number is at x=1.  It's easy to see that f is negative for x<1 and positive for x>1 , so our curve is concave down for x<1 and concave up for x>1 , and thus there is a point of inflection at x=1.