Final Review Sheet 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.*

*The Final Exam wil
lnot cover complex numbers,
centroids of figures in more than two dimensions, L'Hopital's Rule, or
improper integrals.*

*Larger Bold Italic for
items beyond the Midterms that could be on the Final Exam. (Dec. 10, 11:30 am – 1:18 pm in our
classroom)*

*There are three items
of theory marked *** and there will be a question about one of them on the
final exam.*

*The only material that you are newly responsible
for on the final exam is hyperbolic and inverse hyperbolic functions.*

*You may bring
calculators to the Final Exam, 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 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
how to use reduction formulas 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]**

** **

*Hyperbolic and inverse hyperbolic functions
[195.0-201.1]*

** **

** ****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. [27]

A nonempty set S
of real numbers which has a least upper bound for 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, 124, 128]**

**Calculations**

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

Indefinite integrals
and definite integrals by using 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 e**^{x}** [101
– 104]**

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

**Derivatives and
integrals involving inverse tangent
[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 [180-194.3]*

**Notation**

Notation used for 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 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]**