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The Six Pillars of Calculus

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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
One-sided 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 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 $f'$
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

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

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

How to take derivatives
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
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 Wrap-up

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

Strategies
Another Example

Newton's Method

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

Anti-derivatives

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 Anti-derivatives
A Table of Common Anti-derivatives
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

Limits necessary to determine the derivatives of $\sin x$ and $\cos x$

On this page, we set the stage to derive the values of $(\sin x)'$ and $(\cos x)'$.  This is particularly interesting to those of you who are curious about how we determine such derivatives for which we do not (yet) have a derivative rule.

You will need to know the values of the following limits later in the semester, not only for this application.

Unless you are told otherwise, you will not be expected to be able to reproduce the following proofs, but it is certainly true that working through the rest of this page, carefully, will increase your mathematical understanding in many ways.

The two important limits (derived below): $$\lim_{x \to 0} \frac{\sin(x)}{x} = 1 \qquad \mbox{and} \qquad \lim_{x \to 0}\frac{1-\cos(x)}{x}=0.$$ Notice that both of these limits have the $\frac{0}{0}$ indeterminate form, so we need to do more work, but what work?

In the following video, we use geometry and the squeeze theorem to derive the first of the two limits:

 $$\lim_{x \to 0} \frac{\sin(x)}{x} = 1$$

Once we know the first limit, the second limit

 $$\lim_{x \to 0}\frac{\cos(x)-1}{x}=0$$

is fairly easy: \begin{eqnarray*} \displaystyle{\lim_{x \to 0} \frac{\cos(x)-1}{x}} & = & \displaystyle{\lim_{x \to 0} \left ( \frac{\cos(x)-1}{x}\cdot \frac{\cos(x)+1}{\cos(x)+1} \right ) } \cr & = & \displaystyle{\lim_{x \to 0} \left ( \frac{\cos^2(x)-1}{x(1+\cos(x))}\right )} \cr &=&\displaystyle{\lim_{x \to 0} \left (\frac{-\sin^2(x)}{x(1+\cos(x))}\right )} \cr &=& \displaystyle{-\lim_{x \to 0} \left ( \frac{\sin(x)}{x}\cdot\frac{\sin(x)}{(1+\cos(x))}\right )}\cr &=& \displaystyle{-\left ( \lim_{x \to 0} \frac{\sin(x)}{x} \right )} \cdot \displaystyle{\left ( \lim_{x \to 0} \frac{\sin(x)}{1+\cos(x)}\right )}. \end{eqnarray*}

We just showed that the first of the limits on the last line is 1, and the second limit is $\displaystyle{\frac{\sin(0)}{1+\cos(0)}=\frac{0}{2}=0}$, so the product is 0.