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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

Implicit Differentiation

Derivatives aren't just for those functions whose formulas we already know.  We can meaningfully ask for the rate of change of anything.

In particular, if a certain equation holds for all values of $x$, then the derivative of the left hand side must equal the derivative of the right hand side.  If the expressions involve $y$, then the derivatives will involve $\displaystyle y'=\frac{dy}{dx}$ by the chain rule.  Then solve for $\displaystyle \frac{dy}{dx}$ by putting all the terms that include $\displaystyle \frac{dy}{dx}$ on one side of the equation, and all the terms that don't on the other side.

Example: Find $y'$ where $$x^2y + y^3 =\sin(x).$$
DO:  Try to work this problem before reading the solution.

Solution: Differentiating on both sides, using the product rule on the first term and the chain rule on the second, we get $$2xy + x^2 y' + 3y^2y' = \cos(x).$$ After grouping, we have $$\left(x^2+3y^2\right)y' = \cos(x)-2xy,$$ or $$\displaystyle{y' = \frac{\cos(x)-2xy}{x^2+3y^2}}.$$Note that the answer is typically an expression involving both $x$ and $y$.  Occasionally we can simplify this into something that just involves $x$, but usually we can't.

There are 2 main uses of implicit differentiation:

The first is to get information about curves where $x$ and $y$ are related in a way that's more complicated than just the function $y=f(x)$.  Many curves are not functions (such as a spiral, or a circle, etc.), but we still might want to know the rate of change of the curve at a point.  At each point $(x,y)$ on the curve, we can figure out the derivative, plot the tangent line, and estimate what the curve is doing nearby.

The second use is to compute the derivatives of inverse functions.   If $y = f^{-1}(x)$, then $x = f(y)$, so $1 = f'(y) y'$, so $y' = 1/f'(y)$.  Often that can be expressed in terms of $x$.  We used this to compute the inverse trig functions, which we review here:

Example:  Compute the derivative of $\tan^{-1}(x)$.

DO:  Try to work this problem before reading the solution.

Solution:  We write \begin{eqnarray*}y &=& \tan^{-1}(x) \cr\cr x &=& \tan(y) \cr \cr1 &=& \sec^2(y)\, y' \cr\cr y' &=& \frac{1}{\sec^2(y)} \cr\cr y' &=& \frac{1}{1+\tan^2(y)}\cr \cr y' &=& \frac{1}{1+x^2}\end{eqnarray*}The derivatives of $\ln(x)$, $\sin^{-1}(x)$ and $\sec^{-1}(x)$ can be derived similarly.

Another way to say this is that, since $x=f(y)$, $\displaystyle \frac{dx}{dy} = f'(y)$.  However, $$\displaystyle \frac{dy}{dx} = \frac{\quad 1\quad }{\frac{dx}{dy}} = \frac{1}{f'(y)}.$$ In other words, $\displaystyle \frac{dx}{dy}$ is the reciprocal of $\displaystyle \frac{dy}{dx}$.