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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

Integration by Parts

Integration by Parts
Examples
Integration by Parts with a definite integral
Going in Circles

Integrals of Trig Functions

Antiderivatives of Basic Trigonometric Functions
Product of Sines and Cosines (mixed even and odd powers or only odd powers)
Product of Sines and Cosines (only even powers)
Product of Secants and Tangents
Other Cases

Trig Substitutions

How Trig Substitution Works
Summary of trig substitution options
Examples
Completing the Square

Partial Fractions

Introduction
Linear Factors
Improper Rational Functions and Long Division
Summary

Strategies of Integration

Substitution
Integration by Parts
Trig Integrals
Trig Substitutions
Partial Fractions

Improper Integrals

Type 1 - Improper Integrals with Infinite Intervals of Integration
Type 2 - Improper Integrals with Discontinuous Integrands
Comparison Tests for Convergence

Differential Equations

Introduction
Separable Equations
Mixing and Dilution

Models of Growth

Exponential Growth and Decay
Logistic Growth

Infinite Sequences

Examples of Infinite Sequences
Limit Laws for Sequences
Theorems for and Examples of Computing Limits of Sequences
Monotonic Covergence

Infinite Series

Introduction
Geometric Series
Limit Laws for Series
Test for Divergence and Other Theorems
Telescoping Sums and the FTC

Integral Test

The Integral Test
Estimates of Value of the Series

Comparison Tests

The Basic Comparison Test
The Limit Comparison Test

Convergence of Series with Negative Terms

Introduction, Alternating Series,and the AS Test
Absolute Convergence
Rearrangements

The Ratio Test
The Root Test
Examples

Strategies for testing Series

Strategy to Test Series and a Review of Tests
Examples, Part 1
Examples, Part 2

Power Series

Finding the Interval of Convergence
Power Series Centered at $x=a$

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

Taylor Polynomials
When Functions Are Equal to Their Taylor Series
When a Function Does Not Equal Its Taylor Series
Other Uses of Taylor Polynomials

Partial Derivatives

Visualizing Functions in 3 Dimensions
Definitions and Examples
An Example from DNA
Geometry of Partial Derivatives
Higher Order Derivatives
Differentials and Taylor Expansions

Multiple Integrals

Background
What is a Double Integral?
Volumes as Double Integrals

Iterated Integrals over Rectangles

How To Compute Iterated Integrals
Examples of Iterated Integrals
Cavalieri's Principle
Fubini's Theorem
Summary and an Important Example

Double Integrals over General Regions

Type I and Type II regions
Examples 1-4
Examples 5-7
Order of Integration

Definitions and Examples

Partial derivatives help us track the change of multivariable functions by dealing with one variable at a time.  If we think of $z=f(x,y)$ as being in 3-space, we can discuss movement in the $x$-direction, or $y$-direction, and see how this movement affects $z$. We do this by holding $y$ fixed, and varying $x$, or vice versa.  Holding one variable fixed has the effect of slicing a cross section of 3-space, which is then 2-space and we can use our knowledge to understand it.

Definitions and notation

 The definition 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 $\displaystyle{ \lim_{h \rightarrow 0} \frac{f(x,y+h)-f(x,y)}{h}}.$ In general we do not use these definitions to compute partial derivatives.

There are many notations for partial derivatives. If $z = f(x,y)$, then some, but not all, of the notations:

The partial derivative of $f$ with respect to $x$:   $\displaystyle f_x(x,y) = f_x = \frac{\partial f}{\partial x}= \frac{\partial}{\partial x}f(x,y) = \frac{\partial z}{\partial x}= f_1$.

The partial derivative of $f$ with respect to $y$:   $\displaystyle f_y(x,y) = f_y = \frac{\partial f}{\partial y}= \frac{\partial}{\partial y}f(x,y) = \frac{\partial z}{\partial y}= f_2$.

We can evaluate these partial derivatives at particular values of $x$ and $y$, e.g. $f_x(2,7)=\frac{\partial f}{\partial x}\big|_{(2,7)}$ or $f_y(1,-10)$.

Computing partial derivatives

To find $f_x$, hold $y$ constant and differentiate with respect to $x$.  To find $f_y$, hold $x$ constant and differentiate with respect to $y$.  Literally, when computing $f_y$ we treat $x$ as a constant because it is a constant.  This appears as a slice of 3-space.  We can slice 3 dimensions at a particular $x$-value, say we slice at $x=1$ (see the previous page for a graphic example); such a slice is parallel to the $yz$-plane. The $x$-value is the same everywhere in this slice -- it's constant.  Then we observe what happens to $z$ as $y$ changes.  This procedure makes computing partial derivatives very simple.

Example 1:  Compute both partial derivatives of $f$, where $f(x,y)=3x^2-4y^3-7x^2y^3$.
Solution 1:
When we look at $f_x$, we fix $y$.  You can imagine that $y=5$, for example.  Then when we differentiate with respect to $x$, we get $\displaystyle f_x=6x-0-14xy^3=6x-14xy^3$.  Now fix $x$ (imagine that $x=5$) and differentiate with respect to $y$: $\displaystyle f_y=0-12y^2-21x^2y^2=-12y^2-21x^2y^2$.

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Example 2:   Compute $f_x$ and $f_y$ when $f(x,y) = \sin(x+y^2)$.
Solution 2:  We must use the chain rule here.  Since the derivative with respect to $x$ of $\sin(x + \hbox{ constant })$ is $\cos(x + \hbox{ constant })\cdot(1+0)$, and since the derivative with respect to $y$ of $\sin(\hbox{constant} + y^2)$ is $\cos(\hbox{constant }+y^2)\cdot(0+2y)$, we get
$\displaystyle f_x = \cos(x+y^2)$ and $\displaystyle f_y = 2y \cos(x+y^2).$

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Example 3:  Compute both partial derivatives of $\tan{\left(xy^2+7\right)}$

Example 4:  Find $\displaystyle\frac{\partial f}{\partial x},\frac{\partial f}{\partial y}$ for $f(x,y)=\sqrt{x^2-5y}\left(\ln{xy}\right)$.

Example 5:  Find $f_x(-1,2)$ and $f_y(-1,2)$ for $f(x,y)=3x^2-4y^3-7x^2y^3$ from Example 1 above.  What do these numbers mean?

DO:  Try to compute these derivatives before looking ahead.

Solution 3:  We must use the chain rule.
$\displaystyle\frac{\partial}{\partial x}(\tan(xy^2+7))=\sec^2(xy^2+7)\frac{\partial}{\partial x}(xy^2+7)=\sec^2(xy^2+7)(y^2)$.

$\displaystyle\frac{\partial}{\partial y}(\tan(xy^2+7))=\sec^2(xy^2+7)\frac{\partial}{\partial y}(xy^2+7)=\sec^2(xy^2+7)(2xy)$.

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Solution 4:  We must use the product rule and the chain rule:
$\displaystyle\frac{\partial f}{\partial x}=\frac12(x^2-5y)^{-1/2}(2x-0)\ln(xy)+\sqrt{x^2-5y}\ \frac{y}{xy}=\frac{x}{\sqrt{x^2-5y}}\ln(xy)+\frac{\sqrt{x^2-5y}}{x}$.

$\displaystyle\frac{\partial f}{\partial y}=\frac12(x^2-5y)^{-1/2}(0-5)\ln(xy)+\sqrt{x^2-5y}\ \frac{x}{xy}=-\frac{5\ln(xy)}{2\sqrt{x^2-5y}}+\frac{\sqrt{x^2-5y}}{y}$.

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Solution 5:  Since $f_x(x,y)=6x-14xy^3$, $f_x(-1,2)=6(-1)-14(-1)2^3=-6+112=106$.  And since $f_y(x,y)=12y^2-21x^2y^2$,  $f_y(-1,2)=12\cdot 2^2-21(-1)^2\cdot 2^2=48-84=-48$.  This means that if we stand at the point $(-1,2)$ and look in the positive $x$ direction, $z=f(x,y)$ is heading upward, but if we look in the positive $y$ direction $z$ is heading downward.