We have already seen that recognizing the product rule can be useful, when we noticed that
[int sec^3u+sec u an^2u,du=sec u an u.]
As with substitution, we do not have to rely on insight or cleverness to discover such antiderivatives; there is a technique that will often help to uncover the product rule.
Start with the product rule:
[{dover dx}f(x)g(x)=f'(x)g(x)+f(x)g'(x).]
We can rewrite this as
[f(x)g(x)=int f'(x)g(x),dx +int f(x)g'(x),dx,]
and then
[int f(x)g'(x),dx=f(x)g(x)int f'(x)g(x),dx.]
This may not seem particularly useful at first glance, but it turns out that in many cases we have an integral of the form
[int f(x)g'(x),dx]
but that
[int f'(x)g(x),dx]
is easier. This technique for turning one integral into another is called integration by parts, and is usually written in more compact form. If we let (u=f(x)) and (v=g(x)) then (du=f'(x),dx) and (dv=g'(x),dx) and
[int u,dv = uvint v,du.]
To use this technique we need to identify likely candidates for (u=f(x)) and (dv=g'(x),dx).
Example (PageIndex{1})
Evaluate (displaystyle int xln x,dx).
Solution
Let (u=ln x) so (du=1/x,dx). Then we must let (dv=x,dx) so ( v=x^2/2) and
[ int xln x,dx={x^2ln xover 2}int {x^2over2}{1over x},dx= {x^2ln xover 2}int {xover2},dx={x^2ln xover 2}{x^2over4}+C. ]
Example (PageIndex{2})
Evaluate (displaystyle int xsin x,dx).
Solution
Let (u=x) so (du=dx). Then we must let (dv=sin x,dx) so (v=cos x) and
[int xsin x,dx=xcos xint cos x,dx= xcos x+int cos x,dx=xcos x+sin x+C.]
Example (PageIndex{3})
Evaluate (displaystyle intsec^3 x,dx).
Solution
Of course we already know the answer to this, but we needed to be clever to discover it. Here we'll use the new technique to discover the antiderivative. Let (u=sec x) and ( dv=sec^2 x,dx). Then (du=sec x an x,dx) and (v= an x) and
[eqalign{ intsec^3 x,dx&=sec x an xint an^2xsec x,dxcr &=sec x an xint (sec^2x1)sec x,dxcr &=sec x an xint sec^3x,dx +intsec x,dx.cr }]
At first this looks uselesswe're right back to ( intsec^3x,dx). But looking more closely:
[eqalign{ intsec^3x,dx&=sec x an xint sec^3x,dx +intsec x,dxcr intsec^3x,dx+int sec^3x,dx&=sec x an x +intsec x,dxcr 2intsec^3x,dx&=sec x an x +intsec x,dxcr intsec^3x,dx&={sec x an xover2} +{1over2}intsec x,dxcr &={sec x an xover2} +{lnsec x+ an xover2}+C.cr }]
Example (PageIndex{4})
Evaluate ( displaystyle int x^2sin x,dx).
Solution
Let (u=x^2), (dv=sin x,dx); then (du=2x,dx) and (v=cos x). Now ( int x^2sin x,dx=x^2cos x+int 2xcos x,dx). This is better than the original integral, but we need to do integration by parts again. Let (u=2x), (dv=cos x,dx); then (du=2) and (v=sin x), and
[eqalign{ int x^2sin x,dx&=x^2cos x+int 2xcos x,dxcr &=x^2cos x+ 2xsin x  int 2sin x,dxcr &=x^2cos x+ 2xsin x + 2cos x + C.cr }]
Such repeated use of integration by parts is fairly common, but it can be a bit tedious to accomplish, and it is easy to make errors, especially sign errors involving the subtraction in the formula. There is a nice tabular method to accomplish the calculation that minimizes the chance for error and speeds up the whole process. We illustrate with the previous example. Here is the table:
 or 

To form the first table, we start with (u) at the top of the second column and repeatedly compute the derivative; starting with (dv) at the top of the third column, we repeatedly compute the antiderivative. In the first column, we place a "()'' in every second row. To form the second table we combine the first and second columns by ignoring the boundary; if you do this by hand, you may simply start with two columns and add a "()'' to every second row.
To compute with this second table we begin at the top. Multiply the first entry in column (u) by the second entry in column (dv) to get ( x^2cos x), and add this to the integral of the product of the second entry in column (u) and second entry in column (dv). This gives:
[x^2cos x+int 2xcos x,dx,]
or exactly the result of the first application of integration by parts. Since this integral is not yet easy, we return to the table. Now we multiply twice on the diagonal, ( (x^2)(cos x)) and ((2x)(sin x)) and then once straight across, ((2)(sin x)), and combine these as
[x^2cos x+2xsin xint 2sin x,dx,]
giving the same result as the second application of integration by parts. While this integral is easy, we may return yet once more to the table. Now multiply three times on the diagonal to get ( (x^2)(cos x)), ((2x)(sin x)), and ((2)(cos x)), and once straight across, ((0)(cos x)). We combine these as before to get
[ x^2cos x+2xsin x +2cos x+int 0,dx= x^2cos x+2xsin x +2cos x+C. ]
Typically we would fill in the table one line at a time, until the "straight across'' multiplication gives an easy integral. If we can see that the (u) column will eventually become zero, we can instead fill in the whole table; computing the products as indicated will then give the entire integral, including the "(+C,)'', as above.
Integration by parts
Figure 5.4 shows a technique called integration by parts. If the integral is easier than the integral , then we can calculate the easier one, and then by simple geometry determine the one we wanted. Identifying the large rectangle that surrounds both shaded areas, and the small white rectangle on the lower left, we have
In the case of an indefinite integral, we have a similar relationship de rived from the product rule:
Integrating both sides, we have the following relation.
Integration by parts
Since a definite integral can always be done by evaluating an indefinite integral at its upper and lower limits, one usually uses this form. Integrals don't usually come prepackaged in a form that makes it obvious that you should use integration by parts. What the equation for integration by parts tells us is that if we can split up the integrand into two factors, one of which (the ) we know how to integrate, we have the option of changing the integral into a new form in which that factor becomes its integral, and the other factor becomes its derivative. If we choose the right way of splitting up the integrand into parts, the result can be a simplification.
How do you evaluate the integral of #(ln x)^2 dx#?
bp has one great solution Method 1. There are other solutions:
Both of the solution presented below use Integration by Parts.
I use the form:
Both of the solution presented below use #int lnx dx = xlnx  x +C# , which can be done by integration by parts. (And, of course, verified by differentiating the answer.)
Then #du = (2lnx)/x dx# and #v = x#
Integration by parts gives us:
#int (lnx)^2 dx = x(lnx)^2  2int lnx dx# #
#color(white)"sssssss"# # =x(lnx)^22(xlnx  x) +C#
#color(white)"sssssss"# # =x(lnx)^22xlnx + 2x +C#
#int (lnx)^2 dx = int (lnx)(lnx)dx#
So, #du = 1/x dx# and #v= xlnx x#
The parts formula gives us:
#int (lnx)^2 dx = (lnx)(xlnx x)int(xlnxx)/x dx#
#color(white)"sssssss"# # =x(lnx)^2xlnx int (color(red)(lnx)  color(green)(1))dx#
#color(white)"sssssss"# # =x(lnx)^2xlnx (color(red)(xlnxx)  color(green)(x)) +C#
AP Calculus BC  A. Tennyson
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Calculus BC
Unit 1 Limits and Continuity
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 You belong to 2nd Period
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Unit 6 Area and Integrals
Unit 7 usubstitution
Unit 8 Differential Equations
Unit 9 Area and Volume
Unit 10 Integration by Parts, L'Hopital's Rule, Improper Integrals
Unit 11 Series Tests
Unit 12 Taylor Polynomials
Unit 13 Vectors and Polar
AP Test Reviews
Mrs. Tennyson
Amie Tennyson
[email protected]
8176985659
Calculus BC Tutorials:
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AP Calculus BC  A. Tennyson
Students , click here for a list of all of the BNHS teacher's advisory Zoom links! Advisory meets every Monday and Friday 9:3010!
Open all Close all
Instructions: Clicking on the section name will show / hide the section.
Intro to Course/Calendars/Unit Reviews
Northwest ISD is not responsible for content on external websites or services.
Calculus BC
Unit 1 Limits and Continuity
Class Notes (you may choose the video or mp4 version of each topic)
 You belong to 2nd Period
 It is before September 22 2020, 6:15 PM
Unit 2 Derivatives
Class Notes (you may choose the video or mp4 version of each topic)
Unit 3 Chain Rule, Implicit, Related Rates
Class Notes (you may choose the video or mp4 version of each topic)
Unit 4 Derivatives of Transcendental Functions
Unit 5 Applications of Derivatives
Class Notes (you may choose the video or mp4 version of each topic)
Unit 6 Area and Integrals
Unit 7 usubstitution
Unit 8 Differential Equations
Unit 9 Area and Volume
Unit 10 Integration by Parts, L'Hopital's Rule, Improper Integrals
Unit 11 Series Tests
Unit 12 Taylor Polynomials
Unit 13 Vectors and Polar
AP Test Reviews
Mrs. Tennyson
Amie Tennyson
[email protected]
8176985659
Calculus BC Tutorials:
Tuesdays & Thursdays 44:30
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 Glossary of more than 450 mathematical terms
 Engaging content to help students advance their mathematical knowledge:
 Limits and derivatives
 Computational techniques such as the power rule, product rule, quotient rule, and chain rule
 Differentiation, optimization, and related rates
 Antiderivatives, integration, and the fundamental theorem of calculus
 Indeterminate forms and L'Hopital's rule
 Sequences and series
 Differential equations
 Parametric equations and polar coordinates
 Vector calculus
Table of Contents
1. The Basics
 1.1 Overview
 1.1.1 An Introduction to Thinkwell Calculus
 1.1.2 The Two Questions of Calculus
 1.1.3 Average Rates of Change
 1.1.4 How to Do Math
 1.2 Precalculus Review
 1.2.1 Functions
 1.2.2 Graphing Lines
 1.2.3 Parabolas
 1.2.4 Some NonEuclidean Geometry
2. Limits
 2.1 The Concept of the Limit
 2.1.1 Finding Rate of Change over an Interval
 2.1.2 Finding Limits Graphically
 2.1.3 The Formal Definition of a Limit
 2.1.4 The Limit Laws, Part I
 2.1.5 The Limit Laws, Part II
 2.1.6 OneSided Limits
 2.1.7 The Squeeze Theorem
 2.1.8 Continuity and Discontinuity
 2.2 Evaluating Limits
 2.2.1 Evaluating Limits
 2.2.2 Limits and Indeterminate Forms
 2.2.3 Two Techniques for Evaluating Limits
 2.2.4 An Overview of Limits
3. An Introduction to Derivatives
 3.1 Understanding the Derivative
 3.1.1 Rates of Change, Secants, and Tangents
 3.1.2 Finding Instantaneous Velocity
 3.1.3 The Derivative
 3.1.4 Differentiability
 3.2 Using the Derivative
 3.2.1 The Slope of a Tangent Line
 3.2.2 Instantaneous Rate
 3.2.3 The Equation of a Tangent Line
 3.2.4 More on Instantaneous Rate
 3.3 Some Special Derivatives
 3.3.1 The Derivative of the Reciprocal Function
 3.3.2 The Derivative of the Square Root Function
4. Computational Techniques
 4.1 The Power Rule
 4.1.1 A Shortcut for Finding Derivatives
 4.1.2 A Quick Proof of the Power Rule
 4.1.3 Uses of the Power Rule
 4.2 The Product and Quotient Rules
 4.2.1 The Product Rule
 4.2.2 The Quotient Rule
 4.3 The Chain Rule
 4.3.1 An Introduction to the Chain Rule
 4.3.2 Using the Chain Rule
 4.3.3 Combining Computational Techniques
5. Special Functions
 5.1 Trigonometric Functions
 5.1.1 A Review of Trigonometry
 5.1.2 Graphing Trigonometric Functions
 5.1.3 The Derivatives of Trigonometric Functions
 5.1.4 The Number Pi
 5.2 Exponential Functions
 5.2.1 Graphing Exponential Functions
 5.2.2 Derivatives of Exponential Functions
 5.2.3 The Music of Math
 5.3 Logarithmic Functions
 5.3.1 Evaluating Logarithmic Functions
 5.3.2 The Derivative of the Natural Log Function
 5.3.3 Using the Derivative Rules with Transcendental Functions
6. Implicit Differentiation
 6.1 Implicit Differentiation Basics
 6.1.1 An Introduction to Implicit Differentiation
 6.1.2 Finding the Derivative Implicitly
 6.2 Applying Implicit Differentiation
 6.2.1 Using Implicit Differentiation
 6.2.2 Applying Implicit Differentiation
7. Applications of Differentiation
 7.1 Position and Velocity
 7.1.1 Acceleration and the Derivative
 7.1.2 Solving Word Problems Involving Distance and Velocity
 7.2 Linear Approximation
 7.2.1 HigherOrder Derivatives and Linear Approximation
 7.2.2 Using the Tangent Line Approximation Formula
 7.2.3 Newton's Method
 7.3 Related Rates
 7.3.1 The Pebble Problem
 7.3.2 The Ladder Problem
 7.3.3 The Baseball Problem
 7.3.4 The Blimp Problem
 7.3.5 Math Anxiety
 7.4 Optimization
 7.4.1 The Connection Between Slope and Optimization
 7.4.2 The Fence Problem
 7.4.3 The Box Problem
 7.4.4 The Can Problem
 7.4.5 The WireCutting Problem
8. Curve Sketching
 8.1 Introduction
 8.1.1 An Introduction to Curve Sketching
 8.1.2 Three Big Theorems
 8.1.3 Morale Moment
 8.2 Critical Points
 8.2.1 Critical Points
 8.2.2 Maximum and Minimum
 8.2.3 Regions Where a Function Increases or Decreases
 8.2.4 The First Derivative Test
 8.2.5 Math Magic
 8.3 Concavity
 8.3.1 Concavity and Inflection Points
 8.3.2 Using the Second Derivative to Examine Concavity
 8.3.3 The Möbius Band
 8.4 Graphing Using the Derivative
 8.4.1 Graphs of Polynomial Functions
 8.4.2 Cusp Points and the Derivative
 8.4.3 DomainRestricted Functions and the Derivative
 8.4.4 The Second Derivative Test
 8.5 Asymptotes
 8.5.1 Vertical Asymptotes
 8.5.2 Horizontal Asymptotes and Infinite Limits
 8.5.3 Graphing Functions with Asymptotes
 8.5.4 Functions with Asymptotes and Holes
 8.5.5 Functions with Asymptotes and Critical Points
9. The Basics of Integration
 9.1 Antiderivatives
 9.1.1 Antidifferentiation
 9.1.2 Antiderivatives of Powers of x
 9.1.3 Antiderivatives of Trigonometric and Exponential Functions
 9.2 Integration by Substitution
 9.2.1 Undoing the Chain Rule
 9.2.2 Integrating Polynomials by Substitution
 9.3 Illustrating Integration by Substitution
 9.3.1 Integrating Composite Trigonometric Functions by Substitution
 9.3.2 Integrating Composite Exponential and Rational Functions by Substitution
 9.3.3 More Integrating Trigonometric Functions by Substitution
 9.3.4 Choosing Effective Function Decompositions
 9.4 The Fundamental Theorem of Calculus
 9.4.1 Approximating Areas of Plane Regions
 9.4.2 Areas, Riemann Sums, and Definite Integrals
 9.4.3 The Fundamental Theorem of Calculus, Part I
 9.4.4 The Fundamental Theorem of Calculus, Part II
 9.4.5 Illustrating the Fundamental Theorem of Calculus
 9.4.6 Evaluating Definite Integrals
10. Applications of Integration
 10.1 Motion
 10.1.1 Antiderivatives and Motion
 10.1.2 Gravity and Vertical Motion
 10.1.3 Solving Vertical Motion Problems
 10.2 Finding the Area between Two Curves
 10.2.1 The Area between Two Curves
 10.2.2 Limits of Integration and Area
 10.2.3 Common Mistakes to Avoid When Finding Areas
 10.2.4 Regions Bound by Several Curves
 10.3 Integrating with Respect to y
 10.3.1 Finding Areas by Integrating with Respect to y: Part One
 10.3.2 Finding Areas by Integrating with Respect to y: Part Two
 10.3.3 Area, Integration by Substitution, and Trigonometry
11. Calculus I Review
12. Math Fun
 12.1 Paradoxes
 12.1.1 An Introduction to Paradoxes
 12.1.2 Paradoxes and Air Safety
 12.1.3 Newcomb's Paradox
 12.1.4 Zeno's Paradox
 12.2 Sequences
 12.2.1 Fibonacci Numbers
 12.2.2 The Golden Ratio
13. An Introduction to Calculus II
 13.1 Introduction
 13.1.1 Welcome to Calculus II
 13.1.2 Review: Calculus I in 20 Minutes
14. L'Hôpital's Rule
 14.1 Indeterminate Quotients
 14.1.1 Indeterminate Forms
 14.1.2 An Introduction to L'Hôpital's Rule
 14.1.3 Basic Uses of L'Hôpital's Rule
 14.1.4 More Exotic Examples of Indeterminate Forms
 14.2 Other Indeterminate Forms
 14.2.1 L'Hôpital's Rule and Indeterminate Products
 14.2.2 L'Hôpital's Rule and Indeterminate Differences
 14.2.3 L'Hôpital's Rule and One to the Infinite Power
 14.2.4 Another Example of One to the Infinite Power
15. Elementary Functions and Their Inverses
 15.1 Inverse Functions
 15.1.1 The Exponential and Natural Log Functions
 15.1.2 Differentiating Logarithmic Functions
 15.1.3 Logarithmic Differentiation
 15.1.4 The Basics of Inverse Functions
 15.1.5 Finding the Inverse of a Function
 15.2 The Calculus of Inverse Functions
 15.2.1 Derivatives of Inverse Functions
 15.3 Inverse Trigonometric Functions
 15.3.1 The Inverse Sine, Cosine, and Tangent Functions
 15.3.2 The Inverse Secant, Cosecant, and Cotangent Functions
 15.3.3 Evaluating Inverse Trigonometric Functions
 15.4 The Calculus of Inverse Trigonometric Functions
 15.4.1 Derivatives of Inverse Trigonometric Functions
 15.4.2 More Calculus of Inverse Trigonometric Functions
 15.5 The Hyperbolic Functions
 15.5.1 Defining the Hyperbolic Functions
 15.5.2 Hyperbolic Identities
 15.5.3 Derivatives of Hyperbolic Functions
16. Techniques of Integration
 16.1 Integration Using Tables
 16.1.1 An Introduction to the Integral Table
 16.1.2 Making uSubstitutions
 16.2 Integrals Involving Powers of Sine and Cosine
 16.2.1 An Introduction to Integrals with Powers of Sine and Cosine
 16.2.2 Integrals with Powers of Sine and Cosine
 16.2.3 Integrals with Even and Odd Powers of Sine and Cosine
 16.3 Integrals Involving Powers of Other Trigonometric Functions
 16.3.1 Integrals of Other Trigonometric Functions
 16.3.2 Integrals with Odd Powers of Tangent and Any Power of Secant
 16.3.3 Integrals with Even Powers of Secant and Any Power of Tangent
 16.4 An Introduction to Integration by Partial Fractions
 16.4.1 Finding Partial Fraction Decompositions
 16.4.2 Partial Fractions
 16.4.3 Long Division
 16.5 Integration by Partial Fractions with Repeated Factors
 16.5.1 Repeated Linear Factors: Part One
 16.5.2 Repeated Linear Factors: Part Two
 16.5.3 Distinct and Repeated Quadratic Factors
 16.5.4 Partial Fractions of Transcendental Functions
 16.6 Integration by Parts
 16.6.1 An Introduction to Integration by Parts
 16.6.2 Applying Integration by Parts to the Natural Log Function
 16.6.3 Inspirational Examples of Integration by Parts
 16.6.4 Repeated Application of Integration by Parts
 16.6.5 Algebraic Manipulation and Integration by Parts
 16.7 An Introduction to Trigonometric Substitution
 16.7.1 Converting Radicals into Trigonometric Expressions
 16.7.2 Using Trigonometric Substitution to Integrate Radicals
 16.7.3 Trigonometric Substitutions on Rational Powers
 16.8 Trigonometric Substitution Strategy
 16.8.1 An Overview of Trigonometric Substitution Strategy
 16.8.2 Trigonometric Substitution Involving a Definite Integral: Part One
 16.8.3 Trigonometric Substitution Involving a Definite Integral: Part Two
 16.9 Numerical Integration
 16.9.1 Deriving the Trapezoidal Rule
 16.9.2 An Example of the Trapezoidal Rule
17. Improper Integrals
 17.1 Improper Integrals
 17.1.1 The First Type of Improper Integral
 17.1.2 The Second Type of Improper Integral
 17.1.3 Infinite Limits of Integration, Convergence, and Divergence
18. Applications of Integral Calculus
 18.1 The Average Value of a Function
 18.1.1 Finding the Average Value of a Function
 18.2 Finding Volumes Using CrossSections
 18.2.1 Finding Volumes Using CrossSectional Slices
 18.2.2 An Example of Finding CrossSectional Volumes
 18.3 Disks and Washers
 18.3.1 Solids of Revolution
 18.3.2 The Disk Method along the yAxis
 18.3.3 A Transcendental Example of the Disk Method
 18.3.4 The Washer Method across the xAxis
 18.3.5 The Washer Method across the yAxis
 18.4 Shells
 18.4.1 Introducing the Shell Method
 18.4.2 Why Shells Can Be Better Than Washers
 18.4.3 The Shell Method: Integrating with Respect to y
 18.5 Arc Lengths and Functions
 18.5.1 An Introduction to Arc Length
 18.5.2 Finding Arc Lengths of Curves Given by Functions
 18.6 Work
 18.6.1 An Introduction to Work
 18.6.2 Calculating Work
 18.6.3 Hooke's Law
 18.7 Moments and Centers of Mass
 18.7.1 Center of Mass
 18.7.2 The Center of Mass of a Thin Plate
19. Sequences and Series
 19.1 Sequences
 19.1.1 The Limit of a Sequence
 19.1.2 Determining the Limit of a Sequence
 19.1.3 The Squeeze and Absolute Value Theorems
 19.2 Monotonic and Bounded Sequences
 19.2.1 Monotonic and Bounded Sequences
 19.3 Infinite Series
 19.3.1 An Introduction to Infinite Series
 19.3.2 The Summation of Infinite Series
 19.3.3 Geometric Series
 19.3.4 Telescoping Series
 19.4 Convergence and Divergence
 19.4.1 Properties of Convergent Series
 19.4.2 The nthTerm Test for Divergence
 19.5 The Integral Test
 19.5.1 An Introduction to the Integral Test
 19.5.2 Examples of the Integral Test
 19.5.3 Using the Integral Test
 19.5.4 Defining pSeries
 19.6 The Direct Comparison Test
 19.6.1 An Introduction to the Direct Comparison Test
 19.6.2 Using the Direct Comparison Test
 19.7 The Limit Comparison Test
 19.7.1 An Introduction to the Limit Comparison Test
 19.7.2 Using the Limit Comparison Test
 19.7.3 Inverting the Series in the Limit Comparison Test
 19.8 The Alternating Series
 19.8.1 Alternating Series
 19.8.2 The Alternating Series Test
 19.8.3 Estimating the Sum of an Alternating Series
 19.9 Absolute and Conditional Convergences
 19.9.1 Absolute and Conditional Convergence
 19.10 The Ratio and Root Tests
 19.10.1 The Ratio Test
 19.10.2 Examples of the Ratio Test
 19.10.3 The Root Test
 19.11 Polynomial Approximations of Elementary Functions
 19.11.1 Polynomial Approximation of Elementary Functions
 19.11.2 HigherDegree Approximations
 19.12 Taylor and Maclaurin Polynomials
 19.12.1 Taylor Polynomials
 19.12.2 Maclaurin Polynomials
 19.12.3 The Remainder of a Taylor Polynomial
 19.12.4 Approximating the Value of a Function
 19.13 Taylor and Maclaurin Series
 19.13.1 Taylor Series
 19.13.2 Examples of the Taylor and Maclaurin Series
 19.13.3 New Taylor Series
 19.13.4 The Convergence of Taylor Series
 19.14 Power Series
 19.14.1 The Definition of Power Series
 19.14.2 The Interval and Radius of Convergence
 19.14.3 Finding the Interval and Radius of Convergence: Part One
 19.14.4 Finding the Interval and Radius of Convergence: Part Two
 19.14.5 Finding the Interval and Radius of Convergence: Part Three
 19.15 Power Series Representations of Functions
 19.15.1 Differentiation and Integration of Power Series
 19.15.2 Finding Power Series Representations by Differentiation
 19.15.3 Finding Power Series Representations by Integration
 19.15.4 Integrating Functions Using Power Series
20. Differential Equations
 20.1 Separable Differential Equations
 20.1.1 An Introduction to Differential Equations
 20.1.2 Solving Separable Differential Equations
 20.1.3 Finding a Particular Solution
 20.1.4 Direction Fields
 20.2 Solving a Homogeneous Differential Equation
 20.2.1 Separating Homogeneous Differential Equations
 20.2.2 Change of Variables
 20.3 Growth and Decay Problems
 20.3.1 Exponential Growth
 20.3.2 Radioactive Decay
 20.4 Solving FirstOrder Linear Differential Equations
 20.4.1 FirstOrder Linear Differential Equations
 20.4.2 Using Integrating Factors
21. Parametric Equations and Polar Coordinates
 21.1 Understanding Parametric Equations
 21.1.1 An Introduction to Parametric Equations
 21.1.2 The Cycloid
 21.1.3 Eliminating Parameters
 21.2 Calculus and Parametric Equations
 21.2.1 Derivatives of Parametric Equations
 21.2.2 Graphing the Elliptic Curve
 21.2.3 The Arc Length of a Parameterized Curve
 21.2.4 Finding Arc Lengths of Curves Given by Parametric Equations
 21.3 Understanding Polar Coordinates
 21.3.1 The Polar Coordinate System
 21.3.2 Converting between Polar and Cartesian Forms
 21.3.3 Spirals and Circles
 21.3.4 Graphing Some Special Polar Functions
 21.4 Polar Functions and Slope
 21.4.1 Calculus and the Rose Curve
 21.4.2 Finding the Slopes of Tangent Lines in Polar Form
 21.5 Polar Functions and Area
 21.5.1 Heading toward the Area of a Polar Region
 21.5.2 Finding the Area of a Polar Region: Part One
 21.5.3 Finding the Area of a Polar Region: Part Two
 21.5.4 The Area of a Region Bounded by Two Polar Curves: Part One
 21.5.5 The Area of a Region Bounded by Two Polar Curves: Part Two
22. Vector Calculus and the Geometry of R 2 and R 3
 22.1 Vectors and the Geometry of R 2 and R 3
 22.1.1 Coordinate Geometry in ThreeDimensional Space
 22.1.2 Introduction to Vectors
 22.1.3 Vectors in R 2 and R 3
 22.1.4 An Introduction to the Dot Product
 22.1.5 Orthogonal Projections
 22.1.6 An Introduction to the Cross Product
 22.1.7 Geometry of the Cross Product
 22.1.8 Equations of Lines and Planes in R 3
 22.2 Vector Functions
 22.2.1 Introduction to Vector Functions
 22.2.2 Derivatives of Vector Functions
 22.2.3 Vector Functions: Smooth Curves
 22.2.4 Vector Functions: Velocity and Acceleration
About the Author
Edward Burger is an awardwinning professor with a passion for teaching mathematics.
Since 2013, Edward Burger has been President of Southwestern University, a topranked liberal arts college in Georgetown, Texas. Previously, he was Professor of Mathematics at Williams College. Dr. Burger earned his Ph.D. at the University of Texas at Austin, having graduated summa cum laude with distinction in mathematics from Connecticut College.
Prof. Burger's unique sense of humor and his teaching expertise combine to make him the ideal presenter of Thinkwell's entertaining and informative video lectures.
Keywords
This work was supported by the Natural Sciences and Engineering Research Council (NSERC) , the Canada Research Chairs program , Bombardier Aerospace , Mathematics of Information Technology and Complex Systems (MITACS) , and the University of Toronto .
Rensselaer Polytechnic Institute, Department of Mechanical, Aerospace, and Nuclear Engineering.
Tier 1 Canada Research Chair in Computational Aerodynamics, J. Armand Bombardier Foundation Chair in Aerospace Flight.
Math 231/249, Honors Calculus II
A second course in calculus, focusing on sequences and series, but also covering techniques of integration, parametric equations, polar coordinates, and complex numbers. While covering the same basic material as the standard sections, this honors class does so in more detail, including some additional topics. As such, it is for those students who, regardless of their major, are particularly interested in, and excited by, mathematics. In addition, a score of 5 on the AP Calculus AB exam, or a grade of "A" in Math 220 or 221 is required for enrollment. Students who enroll in this course must also register for Math 249 Q1H, the "honors supplement", with CRN 32044.
Grading:
 Weekly Homework (15%). Homework will be assigned during each lecture and due at the beginning of class each Wednesday. No late homework will be accepted however, your lowest homework grade will be dropped so you are effectively allowed one infinitely late assignment. Collaboration on homework is permitted, nay encouraged. However, you must write up your solutions individually and understand them completely. You may use a computer or calculator on the HW for experimentation and to check your answers, but may not refer to it directly in the solution, e.g. "by Mathematica" is not an acceptable justification for deriving one equation from another. (Also, computers and calculators will not be allowed on the exams, so it's best not to get too dependent on them.)
 Three inclass exams (20% each). These will be closedbook, calculatorfree exams, though you will be allowed to bring one piece of paper with handwritten formulas. They will be on Fridays, in particular, September 19, October 17, and November 14.
 A final exam (25%) Our final exam is scheduled for Friday, December 12 from 1:304:30 pm.
Textbook
 Smith and Minton, Calculus: Early Transcendental Functions, 3rd edition, McGraw Hill, 2006 or 2007.
We will be covering Chapters 69, so either the "Single Variable" or "Full" version of this book is fine. As to the future value of the longer version for those planning on taking Math 241 (Calculus III), the honors sections of that course do not use this text, though some, but not all, of the standard sections do.
8.5: Integration by Parts
Consider a discrete grid consisting of points and uniform spacing on some, possibly unbounded, domain .
Definition 17. A difference operator approximating is said to satisfy SBP on the domain with respect to a positive definite scalar product ,
This is the discrete counterpart of integration by parts for the operator,
If the interval is infinite, say or , certain falloff conditions are required and Eq. (8.22 ) replaced by dropping the corresponding boundary term(s).
Example 50. Standard centered differences as defined by Eq. (8.19 ) in the domain or for periodic domains and functions satisfy SBP with respect to the trivial scalar product (),
The scalar product or associated norm are said to be diagonal if
Accuracy and Efficiency.
When constructing SBP operators, the discrete scalar product cannot be arbitrarily fixed and afterward the difference operator solved for so that it satisfies the SBP property (8.22 ) &ndash in general this leads to no solutions. The coefficients of and those of have to be simultaneously solved for. The resulting systems of equations lead to SBP operators being in general not unique, with increasing freedom with the accuracy order. In the diagonal case the resulting norm is automatically positive definite but not so in the fullrestricted case.
We label the operators by their order of accuracy in the interior and near boundary points. For diagonal norms and restricted full ones this would be and , respectively.
Example 51. : For the simplest case, , the SBP operator and scalar product are unique:
The operator and its associated scalar product are also unique in the diagonal norm case:
On the other hand, the operators have one, three and ten free parameters, respectively. Up to their associated scalar products are unique, while for one of the free parameters enters in . For the fullrestricted case, have three, four and five free parameters, respectively, all of which appear in the corresponding scalar products.
A possibility [396 ] is to use the nonuniqueness of SBP operators to minimize the boundary stencil s ize . If the difference operator in the interior is a standard centered difference with accuracyorder then there are points at and near each boundary, where the accuracy is of order (with in the diagonal case and in the full restricted one). The integer can be referred to as the b oundary width. The boundary stencil size is the number of gridpoints that the difference operator uses to evaluate its approximation at those boundary points.
However, minimizing such size, as well as any naive or arbitrary choice of the free parameters, easily leads to a large spectral radius and as a consequence restrictive CFL (see Section 7 ) limit in the case of explicit evolutions. Sometimes it also leads to rather large boundary truncation errors. Thus, an alternative is to numerically compute the spectral radius for these multiparameter families of SBP operators and find in each case the parameter choice that leads to a minimum [399 , 281 ] . It turns out that in this way the order of accuracy can be increased from the very low one of to higherorder ones such as or with a very small change in the CFL limit. It involves some work, but since the SBP property (8.22 ) is independent of the system of equations one wants to solve, it only needs to be done once. In the fullrestricted case, when marching through parameter space and minimizing the spectral radius, this minimization has to be constrained with the condition that the resulting norm is actually positive definite.
The nonuniqueness of highorder SBP operators can be further used to minimize a combination of the average of the boundary truncation error (ABTE), defined below, without a significant increase in the spectral radius. For definiteness consider a left boundary. If a Taylor expansion of the FD operator is written as
Watch the video: How To Integrate Using USubstitution (December 2021).