Review Sheet1   Math H161   Autumn 04

[n] means the topic is found on acetate n.

Theory that might be on Midterm  I  is marked by asterisks  ***.  Five items are marked.  One of them will be on the exam.

Tools

Number line  [1-3]

Illustrating individual numbers  

Illustrating sets  [2, 3]

Picturing the inequality  |x – x0| <  e  on the number line [11.0-11.2]

Motivation for the definition of definite integral [17-18, 20.0-21]

Differentiation formulas [28.0-28.1]

Simple integration formulas 

Integration formula for  1/x  

Trigonometric functions and identitie  [textbook pp. 37-45]

Axioms  (know them and be able to write them correctly)

***The Number Line Axiom  [4.0 -4.2]

***Least upper bound axiom  

Definitions (know them and be able to write them correctly)

Upper bound and least upper bound for a nonempty set  [5-7]

***Riemann sum for a continuous function on a closed interval and its motivation from finding areas [20.0-21]

Definition of definite integral as the limit of Riemann sums  [18, 27]

Antiderivative 

Indefinite integral 

Natural logarithm 

The number  e  

Calculation of a definite integral by taking a limit [23-24.1]

Theorems--with proofs

Properties of logarithms [54-57]

***Fundamental Theorem of Calculus [38-43]

Theorems--no proofs (know what they mean and be able to state them correctly)

***Archimedean property of real numbers  [12-13]

Existence of the definite integral for continuous functions on a closed interval.  (First main result about integration.)  

If  S  is a nonempty set of real numbers and there is a least upper bound for  S, then  S  cannot have two different least upper bounds.  

Elementary properties of definite integrals [34-37]

If  F(x)  is an antiderivative of  f(x), then all of the other antiderivatives of  f(x)  are the functions  F(x) + C, where  C  is a constant  [38, 40-41.1]

Mean Value Theorem (for derivatives)  [41.0-41.1]

Calculations

Riemann sums (without taking limits) [22, 25, 26]

Calculation of indefinite integrals and definite integrals through the use of antiderivatives  [45-49, 51-52, 58-59]

Areas by using definite integrals [31-33, 37, 60-62]

Volumes by using definite integrals  [63.0-64.1]

Disk (and washer) method  [63.0-64.1, 67]

Shell method  [65.0-66]

Substitution in integration [46-49, 51-52, 58-59]

Arc length  [68.0 - 72]

Surface area for surface of revolution  [73.0-78]

Notation

Notation used in writing down Riemann sums, including the summation notation using S  (capital Greek sigma)  [20.1-20.4]

Definite integrals  [18, 27]

Indefinite integrals  

Facts

More on the natural logarithm [54 - 57]