Essential Ideas Archives - The Nature of Mathematics

Section 18.5: Review for Chapter 18

August 17, 2015July 23, 2015 by elaine

Studying for a chapter examination is a personal process, one which nobody else can do for you. Simply take the time to review what you have done.

Here are the new terms in Chapter 18.

Antiderivative [18.4]
Area function [18.4]… See the whole entry

Section 18.4: Integrals

September 27, 2015July 23, 2015 by elaine

18.4 Outline

Area function
1. definition
2. antiderivative
3. antiderivative of; ex.
4. properties of antiderivative
  1. constant multiple
  2. antiderivative of a sum
Antiderivative’s as area
The definite integral
1. definition
2. integral
3. limits of integration
4. dummy variable
5. indefinite integral
Areas as antiderivatives

18.4 Essential Ideas

… See the whole entry

Section 18.3: Derivatives

September 27, 2015July 23, 2015 by elaine

18.3 Outline

Average rate of change
Instantaneous rate of change
Derivative
1. derivative of f at x
2. differentiable function
3. derivative of e^x
Tangent line
1. tangent-line
  formula
2. equation of tangent line at a point (h, k ).
Velocity

18.3

… See the whole entry

Section 18.2: Limits

September 27, 2015July 23, 2015 by elaine

18.2 Outline

Introduction
1. sequence
  1. general term
  2. successor
  3. predecessor
2. notation
The limit of a sequence
Limits to infinity
1. converges
2. limit of the sequence
3. diverges
4. limit of 1/n to infinity
Infinite limits

18.2 Essential Ideas

18.2.1 … See the whole entry

Section 18.1: What is Calculus?

September 28, 2015July 23, 2015 by elaine

18.1 Outline

What is Calculus?
1. Calculus is the mathematics of motion and change.
2. The three main ideas of calculus are limit, derivative, and integral
The Limit: Zeno’s Paradox
1. Achilles and the tortoise
2. limit of a sequence
The derivative: the tangent

… See the whole entry

Problems 1-10
These problems are designed to help you understand the material of this section. Read each criteria and then paraphrase it into your own words as requested in each of these problems.

Problems 11-14
These problems are based on … See the whole entry

Section 17.5: Review for Chapter 17

August 17, 2015July 23, 2015 by elaine

Studying for a chapter examination is a personal process, one which nobody else can do for you. Simply take the time to review what you have done.

Here are the new terms in Chapter 17.

Adams’ plan [17.3]
Alabama paradox … See the whole entry

Section 17.4: Apportionment Paradoxes

September 26, 2015July 23, 2015 by elaine

17.4 Outline

Introduction
Alabama paradox
Population paradox
New states paradox
Balinski and Young’s impossibility theorem

17.4 Essential Ideas

Alabama Paradox

A reapportionment in which an increase in the total number of seats results in a loss of seats for … See the whole entry

Section 17.3: Apportionment

September 26, 2015July 23, 2015 by elaine

17.3 Outline

Definition of apportionment
The apportionment process
1. upper quota
2. lower quota
3. standard divisor
4. standard quota
5. quota rule
Adams’ plan
1. definition
2. modified divisor
3. modified quota
Jefferson’s plan
Hamilton’s plan
Webster’s plan
Hungtington-Hill plan
1. arithmetic mean
2. geometric mean
Summary of apportionment

… See the whole entry

Section 17.2: Voting Dilemmas

September 26, 2015July 23, 2015 by elaine

17.2 Outline

Fair voting principles
Majority criterion
Condorcet criterion
1. Condorcet candidate
2. definition
Monotonicity criterion
1. straw vote
2. definition
Irrelevant alternative criterion
Arrow’s impossibility theorem
1. fairness criterion
2. comparison of voting methods
3. insincere voting
4. transitive law
5. Codorcet’s paradox
6. impossibility theorem

17.2 Essential

… See the whole entry