Home

The Six Pillars of Calculus

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
Definition
One-sided Limits
How can a limit fail to exist?
Infinite Limits and Vertical Asymptotes
Summary

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 $f'$
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

Two important Limits
Sine and Cosine
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 and 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

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
Fermat's Theorem
How-to

The Mean Value and other Theorems

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

$f$ vs. $f'$

Increasing/Decreasing Test and Critical Numbers
How-to
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
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

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

The Area under a curve

The Area Problem and Examples
Riemann Sums Notation
Summary

Definite Integrals

Definition
Properties
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

In Physics

Derivatives with respect to time

In physics, we are often looking at how things change over time:

1. Velocity is the derivative of position with respect to time: $v(t) = \displaystyle{\frac{d}{dt}\big(x(t)\big)}$.
2. Acceleration is the derivative of velocity with respect to time: $\displaystyle{a(t) = \frac{d}{dt}\big(v(t)\big)= \frac{d^2 }{dt^2}}\big(x(t)\big)$.
3. Momentum (usually denoted $p$) is mass times velocity, and force ($F$) is mass times acceleration, so the derivative of momentum is $\displaystyle{\frac{dp}{dt} = \frac{d}{dt}\big(mv\big)=m \frac{dv}{dt} = ma = F}$.

One of Newton's laws says that for every action there is an equal and opposite reaction, meaning that if particle 2 puts force $F$ on particle 1, then particle 1 must put force $-F$ on particle 2. But this means that the total momentum is constant, since $$\frac{d}{dt}\big(p_1 + p_2\big) = \frac{dp_1}{dt} + \frac{dp_2}{dt} = F - F = 0.$$ This is the law of conservation of momentum.

Derivatives with respect to position

In physics, we also take derivatives with respect to $x$.

1. For so-called "conservative" forces, there is a function $V(x)$ such that the force depends only on position and is minus the derivative of $V$, namely $F(x) = - \frac{dV(x)}{dx}$. The function $V(x)$ is called the potential energy. For instance, for a mass on a spring the potential energy is $\frac{1}{2}kx^2$, where $k$ is a constant, and the force is $- k x$.
2. The kinetic energy is $\frac{1}{2} m v^2$. Using the chain rule we find that the total energy is $$\frac{d }{dt} \Big(\frac{1}{2} m v^2 + V(x)\Big)= m v \frac{dv}{dt} + V'(x) \frac{dx}{dt} = m v a - F v = (m a - F) v = 0$$ since $F = m a$. This means that the total energy never changes.

These are just a few of the examples of how derivatives come up in physics. In fact, most of physics, and especially electromagnetism and quantum mechanics, is governed by differential equations in several variables. We will learn about partial derivatives in M408L/S and M408M.