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

Close is good enough
One-sided Limits
How can a limit fail to exist?
Infinite Limits and Vertical Asymptotes

Limit Laws and Computations

A summary of Limit Laws
Why do these laws work?
Two limit theorems
How to algebraically manipulate a 0/0?
Limits with fractions
Limits with Absolute Values
Limits involving Rationalization
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
Examples of continuous functions

Limits at Infinity

Limits at infinity and horizontal asymptotes
Limits at infinity of rational functions
Which functions grow the fastest?
Vertical asymptotes (Redux)
Toolbox of graphs

Rates of Change

Tracking change
Average and instantaneous velocity
Instantaneous rate of change of any function
Finding tangent line equations
Definition of derivative

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

Two important Limits
Sine and Cosine
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

Introduction and Examples
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

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
Fermat's Theorem

The Mean Value and other Theorems

Rolle's Theorems
The Mean Value Theorem
Finding $c$

$f$ vs. $f'$

Increasing/Decreasing Test and Critical Numbers
The First Derivative Test
Concavity, Points of Inflection, and the Second Derivative Test

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


Anti-derivatives and Physics
Some formulas
Anti-derivatives are not Integrals

The Area under a curve

The Area Problem and Examples
Riemann Sums Notation

Definite Integrals

What is integration good for?
More Examples

The Fundamental Theorem of Calculus

Three Different Quantities
The Whole as Sum of Partial Changes
The Indefinite Integral as Antiderivative
The FTC and the Chain Rule

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 $\displaystyle y'=\frac{dy}{dx}$, thanks to 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 $$x^2y + y^3 =\sin(x).$$

Solution: Differentiating on both sides we get $$2xy + x^2 y' + 3y^2y' = \cos(x).$$ After grouping, we'd 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 $y=f(x)$. 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 $x$.

    Example: To compute the derivative of $\tan^{-1}(x)$, 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 $\displaystyle \frac{dy}{dx}$.