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 1: A Library of Functions
1.1 Functions & change
1.2 Exponential functions
Handout from Prof. Hwang clarifying difference between forms of exponential functions.
1.3 New functions from old
1.4 Logarithmic functions
1.5 Trigonometric functions
1.6 Powers, polynomials, and rational functions
1.7 Introduction to continuity

Chapter 2: Key Concept: The Derivative
2.1 How do we measure speed?
2.2 Limits
Activity for 2.2
2.3 The derivative at a point
Activity for 2.3
2.4 The derivative function
Activity for 2.4
2.5 Interpretations of the derivative
Activity for 2.5
2.6 The second derivative
2.7 Continuity & differentiability

Water Tank Activity

Chapter 3: Short-cuts to Differentiation
3.1 Powers and polynomials
3.2 The exponential function
3.3 The product and quotient rules
3.4 The chain rule
3.5 The trigonometric functions
3.6 Applications of the chain rule
3.7 Implicit functions
<3.8 skipped>
3.9 Local linearity
3.10 L'Hopital's Rule
Prof. Hwang's Worksheet on L'Hopital's Rule

Chapter 4: Using the Derivative
4.1 Using first and second derivatives
4.2 Families of curves
4.3 Optimization
Prof. Hwang's worksheet on Optimization
4.5 Optimization and modeling
4.6 Hyperbolic functions
4.7 Theorems about continuous and differentiable functions
Template for practice examining y=f(x)

Chapter 5: Key Concept: The Definite Integral
5.1 How do we measure distance traveled?
5.2 The definite integral
5.3 Interpretations of the definite integral
5.4 Theorems about the definite integral
Professor Hwang's worksheet on the definite integral

Professor Hwang's review questions for the final exam
Solutions to Professor Hwang's review questions

Chapter 6: Constructing Antiderivatives
6.1 Antiderivatives Graphically and Numerically
Using your Graphing Calculator to estimate definite integrals
6.2 Constructing Antiderivatives Analytically
6.3 Differential Equations and 6.5 Equations of Motion
6.4 Second Fundamental Theorem of Calculus

Chapter 7: Integration
7.1 Substitution
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, 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 , 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.

11.3 Euler's Method

Graphing Calculators can use Euler's Method!
Go to a search engine, such as, 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 , 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.

11.4 Separation of Variables
11.5 Growth and Decay

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