Review Sheet 3    Math H161    Autumn 04

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

Boldface for items beyond Midterm I that could be on Midterm II.

Italics for items beyond Midterms I and II that could be on Midterm III.

Here is no theory on Midterm III.

The material from classes on November 18 and 19 will be on Midterm III.

On Midterm III there will be nothing involving complex numbers or on centroids of figures in more than two dimensions.

You may bring calculators to Midterm III, and you will be allowed to use them on part of the exam.

Tools

Number line  [1 - 3]

Illustrating individual numbers  [1]

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 [50]

Integration formula for  1/x  [57]

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

More trigonometry [112 - 120]

Relationship between the graph of a function and the graph of its inverse function (if it has one)  [92, 127]

Know generally what reduction formulas do for you when integrating  [149 – 150.0]

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

The Number Line Axiom  [4.0 – 4.2]

Least upper bound axiom  [8]

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 [29]

Indefinite integral [29]

Natural logarithm [53]

The number  e  [55]

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

Functions

Domain [87, 93, 121, 127]

Range [87, 93]

Exponential function [91 - 94]

Inverse functions [86 – 89, 90.1 - 94]

Inverse sine function [121 – 126.1]

Inverse tangent function [121, 127 – 128, 130]

Definition of db  [98]

Theorems--with proofs

Properties of logarithms [54 - 57]

Fundamental Theorem of Calculus [38 -  43]

The natural logarithm function has an inverse function, the exponential function [86 – 89, 90.1 - 94]

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.)  [27]

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.  [9]

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]

If a function is continuous everywhere in its domain and it has an inverse function, then the inverse function is continuous everywhere in the domain of the inverse function [94]

If a function  f(x) has a derivative at  x = a  and  f'(a) is not zero, and  f(x) has an inverse function, then the inverse function  g(x) has a derivative at  r = f(a)  and  g'(r ) = 1/f'(a).  [94 - 96]

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]

Techniques of differentiation and integration

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

Completing the square [126.0 – 126.1, 130. 133.3]

Derivatives and integrals involving ex  [101 – 104]

Derivatives and integrals involving inverse sine function  [124 – 126.1]

Derivatives and integrals involving inverse tangent function  [127 – 129]

Trigonometric substitution  [130.0 – 135]

Integration by parts  [141 – 150.1]

Motion along a straight line [79 - 83]

Motion under the force of gravity [84 – 85]

Population growth and radioactive decay [105 - 111]

Arc length  [68.0 - 72]

Surface area for surface of revolution  [73.0 – 78]

Factoring polynomials  [155, 157, 161-162]

Simple harmonic motion  [163-169]

Springs and Hooke's Law  [163-179]

Centroids and centers of gravity  [180-thru Nov 17]

Notation

Notation used in writing down Riemann sums, including the summation notation using S  [20.1 – 20.4]

Definite integrals  [18, 27]

Indefinite integrals  [29]

Inverse functions  [93 – 96, 124]

Facts (those marked with *** include proofs)

More on the natural logarithm [54 - 57]

Even more on the natural logarithm [89, 90.1]

About the exponential function [91 - 96]

Properties of ex  [97]

Properties of db  [98 - 100]

Inverse functions and their properties  [86 - 96]

The derivative of an inverse function [94-96, 124, 128]