Review Sheet1 Math H161 Autumn 04

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

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

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

**There will be nothing involving complex
numbers or partial fractions on Midterm II.**

**You may bring calculators to Midterm II,
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-x_{0}| < e (Greek
epsilon) 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 d**^{b}** [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 [63-64.1]

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 e**^{x}** [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]

**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 [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 e**^{x}** [97]**

** ***Properties
of d**^{b}** [98-100]**

** Inverse
functions and their properties
[86-96]**

** **

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

** **

** **

** **

** **

** **