# College of the Holy Cross Math 133 Calculus Lecture Notes

As a courtesy to my students, I have scanned my personal lecture notes. These are meant to complement your own lecture notes, because during class I amplify on the contents of these lecture notes extensively. These lecture notes are not a replacement for attending the course lectures. If you miss class, in addition to reading my own lecture notes, you should also obtain the notes from one or more of your peers.

Chapter 7: Integration
7.1 Substitution
7.2
Integration by Parts
7.3 Tables (handout will be provided in class)
Review Completing the Square
Review Polynomial Division
7.4 Algebraic Identities and Trigonometric Substitution
Professor Little's handout on Partial Fractions
Professor Little's handout on Trigonometric Substitution
7.5 Approximating Definite Integrals (no lecture notes available at this time)
Numerical Integration Program for a TI-81
Numerical Integration Program for a TI-82
Numerical Integration Program for a TI-85
7.6 Approximation Errors and Simpson's Rule
7.7 Improper Integrals
7.8 Comparison of Improper Integrals

Chapter 8: Using the Definite Integral
8.1 Areas and Volumes
8.2 Applications to Geometry
8.3 Density and Center of Mass
8.4 Applications to Physics
8.5 <skipped>
8.6 <skipped>
8.7 <skipped>

Chapter 9: Series
9.1 Geometric Series
9.2 Convergence of Sequences and Series
9.3 <skipped>
9.4 <skipped>
Practice Problems for Chapter 9
Quinine Activity

Chapter 10: Approximating Functions
10.1 Taylor Polynomials
10.2 Taylor Series
10.3 Finding and Using Taylor Series
10.4 The Error in Taylor Polynomial Approximations
10.5 <skipped>
Practice Problems for Chapter 10

Chapter 11: Differential Equations
11.1 What is a Differential Equation?

11.2 Slope Fields (I can email you these notes, for some reason they won't post!)

Graphing Calculators can sketch Slope Fields!
Go onto a search engine, such as google.com, to find instructions to program your graphing calculator. Enter your model number (e.g. TI-81) and the words "slope field" into the search box. You will find multiple versions of programs. Note that the TI-89 already has this capability, so there's no need to program anything (you need to go to MODE and change from FUNCTION to DIFFERENTIAL EQUATIONS. Then, put any function into the y1 slot and graph it). A favorite link from 2003-2004 was http://math.arizona.edu/~krawczyk/calcul.html#83 , but there are many other choices.

Hint: The expression IS> in many programs is a built-in function.

To test your program, enter in y1=sin(x) and then call the program. You should see a
screen filled with little lines that look like y(x) = cos(x) +C for a bunch of values of C. Next,
try a function like y1=x+y. You can enter in the variable "y" by using the alpha-key.

Graphing Calculators can use Euler's Method!
Go to a search engine, such as google.com, to find instructions to program your graphing calculator. Enter in your calculator model, plus the word Euler. You will find multiple versions of the program. It's best to find one that both graphs your answer and also gives you a table of (x,y) points on the numerically estimated solution. A favorite link from 2003-2004 was http://math.arizona.edu/~krawczyk/calcul.html , but there are many other choices.

To test your program, enter in y1=y and call the program. Start at the point (0,1) and use
a step size of 0.1. Your answer should look like Figure 11.25 on page 489 of your text book.

We probably won't make it to these last few sections, although I include lecture notes here for your interest:
11.6 & 11.7 Handout
11.6 Applications and Modeling
11.7 Models of Population Growth