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

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
Revised Table of Integrals
Substitution for Definite Integrals

Area Between Curves

The Slice and Dice Principle
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
Volumes Rotating About the $y$-axis

Integration by Parts

Behind IBP
Going in Circles
Tricks of the Trade

Integrals of Trig Functions

Basic Trig Functions
Product of Sines and Cosines (1)
Product of Sines and Cosines (2)
Product of Secants and Tangents
Other Cases

Trig Substitutions

How it works
Completing the Square

Partial Fractions

Linear Factors
Quadratic Factors
Improper Rational Functions and Long Division

Strategies of Integration

Integration by Parts
Trig Integrals
Trig Substitutions
Partial Fractions

Improper Integrals

Type I Integrals
Type II Integrals
Comparison Tests for Convergence

Differential Equations

Separable Equations
Mixing and Dilution

Models of Growth

Exponential Growth and Decay
Logistic Growth

Infinite Sequences

Close is Good Enough (revisited)
Limit Laws for Sequences
Monotonic Convergence

Infinite Series

Geometric Series
Limit Laws for Series
Telescoping Sums and the FTC

Integral Test

Road Map
The Integral Test
When the Integral Diverges
When the Integral Converges

Comparison Tests

The Basic Comparison Test
The Limit Comparison Test

Convergence of Series with Negative Terms

Alternating Series and the AS Test
Absolute Convergence

The Ratio and Root Tests

The Ratio Test
The Root Test

Strategies for testing Series

List of Major Convergence Tests

Power Series

Radius and Interval of Convergence
Finding the Interval of Convergence
Other Power Series

Representing Functions as Power Series

Functions as Power Series
Derivatives and Integrals of Power Series
Applications and Examples

Taylor and Maclaurin Series

The Formula for Taylor Series
Taylor Series for Common Functions
Adding, Multiplying, and Dividing Power Series
Miscellaneous Useful Facts

Applications of Taylor Polynomials

What are Taylor Polynomials?
How Accurate are Taylor Polynomials?
What can go Wrong?
Other Uses of Taylor Polynomials

Partial Derivatives

Definitions and Rules
The Geometry of Partial Derivatives
Higher Order Derivatives
Differentials and Taylor Expansions

Multiple Integrals

What is a Double Integral?
Volumes as Double Integrals

Iterated Integrals over Rectangles

One Variable at the Time
Fubini's Theorem
Notation and Order

Double Integrals over General Regions

Type I and Type II regions
Order of Integration
Area and Volume Revisited

Definitions and Rules

Partial derivatives help us track the change of multi-variable functions by dealing with one variable at the time.

DNA forms a double helix, but the curvature of this helix depends on the temperature and on the salinity (concentration of salt). If we wanted to understand the curvature, we would do experiments by varying the conditions and measuring the curvature each time. If we did our experiments well, we wouldn't try changing both the temperature and the salinity. We would first hold the salinity fixed and change the temperature. Once we understood how temperature affects curvature, we would run a second set of experiments, holding the temperature fixed and varying the salinity. Combining the results, we would understand how both temperature and salinity affect curvature.

Mathematically, the curvature is a function $f(x,y)$, where $x$ is the temperature and $y$ is the salinity. Varying the temperature means comparing $f(x,y)$ to $f(x+h,y)$, and we can ask for the rate of change. Varying the salinity means comparing $f(x,y)$ to $f(x,y+h)$. By taking limits, we can compute two kinds of derivatives:

Definitions of partial derivatives:
The partial derivative of $f$ with respect to $x$ is $$\displaystyle{ \lim_{h \rightarrow 0} \frac{f(x+h,y)-f(x,y)}{h}}.$$
The partial derivative of $f$ with respect to $y$ is $$\displaystyle{ \lim_{h \rightarrow 0} \frac{f(x,y+h)-f(x,y)}{h}}$$

There are many notations for partial derivatives. If $z = f(x,y)$, then $$\displaystyle f_x(x,y) = f_x = \frac{\partial f}{\partial x}= \frac{\partial}{\partial x}f(x,y) = \frac{\partial z}{\partial x}= f_1 = D_1 f= D_x f$$ and $$\displaystyle f_y(x,y) = f_y = \frac{\partial f}{\partial y}= \frac{\partial}{\partial y}f(x,y) = \frac{\partial z}{\partial y}= f_2 = D_2 f= D_y f$$

The rough and precise definitions of limits of functions of two (or more) variables work the same way:

Rules for finding partial derivatives:
  1. To find $f_x$, hold $y$ constant and differentiate with respect to $x$.
  2. To find $f_y$, hold $x$ constant and differentiate with respect to $y$.

When computing $f_x$, we treat $y$ as a constant because it is a constant. After all, we are doing today's experiments at fixed salinity. This means that we can apply all of our familiar differentiation rules, pretending that the only variable is $x$.

Example: Compute $f_x$ and $f_y$ when $f(x,y) = \sin(x+y^2)$.

Solution: Since the derivative of $\sin(x + \hbox{ constant })$ with respect to $x$ is $\cos(x + \hbox{ constant })$, $$f_x = \cos(x+y^2).$$ Since the derivative of $\sin(\hbox{constant} + y^2)$ with respect to $y$ is $2y \cos(\hbox{constant }+y^2)$, $$f_y = 2y \cos(x+y^2).$$