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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

The Second Derivative Test

Previously, we determined whether a critical number $c$ of a function $f$ indicated a local extreme value of $f$ by looking at the sign of $f'$ on either side of $c$.  If $f'$ changed sign at $c$, there was a local extreme value there, since $f'$ changes sign exactly where $f$ changes direction.  This is the first derivative test, where we use the first derivative to determine where we have local extrema. 

On this slide we use another method that might allow us to determine whether there is a local extreme value at $x=c$.  It uses the second derivative as well as the first, so we call it the second derivative test.

First, we formalize the concavity concepts from our previous work:

Concavity Test:

If $f''(x)>0$ for all $x$ in an open interval, then f is concave up on that interval; i.e. $$\large f''>0\Longleftrightarrow f'\uparrow\Longleftrightarrow f \cup$$

If $f ''(x) < 0$ for all $x$ in an open interval, then f is concave down on that interval; i.e. $$\large f''<0\Longleftrightarrow f'\downarrow\Longleftrightarrow f \cap$$

Now, we use the concavity test to get an alternative process to find local extreme values.

Second Derivative Test:
  • If $f'(c)=0$ and $f''(c) \gt 0$, then there is a local minimum at $x=c$.
  • If $f'(c)=0$ and $f''(c) \lt 0$, then there is a local maximum at $x=c$.
  • If $f'(c)=0$ and $f''(c)=0$, or if $f''(c)$ doesn't exist, then the test is inconclusive. There might be a local maximum or minimum, or there might be a point of inflection.

The reasoning behind the test is simple:  If a point is a max, at the top of a hill, it must be on the (concave down) top of the upside-down bowl.  Similarly, if a point is a min, at the bottom of a valley, it must be on the (concave up) bottom of the right-side-up bowl.  So $f$ concave down at a critical number $c$ indicates a max, and $f$ concave up at $c$ indicates a min.

Another perspective:  If $f''(c) \gt 0$, then $f'(x)$ is increasing near $x=c$. Since $f'(c)=0$, this means that $f'(x)$ used to be negative and is about to be positive. So the curve bottoms out at $x=c$ and then heads back up. The critical number $x=c$ is the bottom of the concave-up bowl. Likewise, if $f''(c) \lt 0$ and $f'(c)=0$, then $f'(x)$ is decreasing; it used to be positive and is about to be negative. The point $x=c$ is at the top of an upside-down bowl.

Notice:  As is indicated in the third option of the test, if a critical number $c$ is also a subcritical number, then the second derivative test cannot help determine whether or not there is a max or min at $c$.

  Find the concavity of $f(x) = x^3 - 3x^2$ using the second derivative test. 

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

Solution: Since $f'(x)=3x^2-6x=3x(x-2)$, our two critical points for $f$ are at $x=0$ and $x=2$.  Meanwhile, $f''(x)=6x-6$, so the only subcritical number for $f$ is at $x=1$.  As we saw in our previous work on this problem, $f''$ is negative for $x \lt 1$ and positive for $x \gt 1$, so our curve is concave down for $x \lt 1$ and concave up for $x \gt 1$, and there is a point of inflection at $x=1$.

To use the second derivative test, we check the concavity of $f$ at the critical numbers.  We see that at $x=0$, $x<1$ so $f$ is concave down there.  Thus we have a local maximum at $x=0$.  At $x=2$, since $x>1$ $f$ is concave up there, so we have a local minimum at $x=2$.  These results agree with what we got from the first derivative test.