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Set theory
Set theory

... generates the Calkin- Wilf sequence ...
Full text
Full text

... integrated using standard formulas, but the results are not particularly simple. Finally, we note that the above ideas may be carried out to extend general second-order recurring sequences to continuous functions, as indicated in Section 2. However, because of increased complexity, we do not state t ...
Chapter 2 Limits and continuity
Chapter 2 Limits and continuity

... Chapter 2 Limits and continuity ...
a b
a b

Ann Khadaran
Ann Khadaran

... In exercises 1 – 4, you’re given an equation and the graph of a quadratic function. Do each of the following: The graph needed is in a separate pdf file sent. #3.1 #2 on page 303 a) Give the domain and range b) Give the coordinates of the vertex c) Give the equation of the axis d) Find the y-interce ...
Natural Numbers to Integers to Rationals to Real Numbers
Natural Numbers to Integers to Rationals to Real Numbers

Math 109. Instructor: Chow Homework #2 Hints Problem 1: Prove
Math 109. Instructor: Chow Homework #2 Hints Problem 1: Prove

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Solutions

The Learning Strands, Standards and Indicators Subject
The Learning Strands, Standards and Indicators Subject

... M.1.4.6. Construct the table which contains the types of intervals of real numbers; M.1.4.7. Change the interval set into the number line and vice-versa. ...
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study guide

... This set of problems is for preparing for the final exam of Math 190. You need to do more problems in order to perform well on the test. 1. Use a graph to find a number δ such that sin x  ...
function - croninmath
function - croninmath

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Full text

Let S be the set of all positive rational numbers x such that x 2 < 3
Let S be the set of all positive rational numbers x such that x 2 < 3

Sums Products and Proofs Contents 1 Introduction 2 ∑ = Sum
Sums Products and Proofs Contents 1 Introduction 2 ∑ = Sum

... Below the Sigma we wrote ”index = first number”. The index is the number we choose to document where we are in the summation. In the first example i, is the index. The first number is the first value the index assumes in the sum. In the first example this value is 0. To the right of the Sigma is an ...
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Let F(x,y)

THE PRIME NUMBER THEOREM AND THE
THE PRIME NUMBER THEOREM AND THE

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Class notes from November 18

Function f Function
Function f Function

... Horizontal Line Test Theorem If every horizontal line intersects the graph of a function f in at most one point, then f is 1-to-1. Below is the graph of y=x2-4 ...
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Lesson 15: Piecewise Functions
Lesson 15: Piecewise Functions

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Rational Functions - Matrix Mathematics

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Ch03 - Faculty

... Note that the output variables are enclosed in square brackets, while the input variables must be enclosed with parentheses. The function name (here, name) should be the same as the file name in which it is saved (with the .m ...
Notes - REVIEW
Notes - REVIEW

... We can use this concept to measure the rate of change for more complex functions, too However, since the function might be a parabola, we can't (yet) figure out the exact rate of change at any given point Instead, we'll simplify – we'll talk about the average rate of change between two points on a f ...
Zeno`s paradoxes
Zeno`s paradoxes

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Non-standard calculus

In mathematics, non-standard calculus is the modern application of infinitesimals, in the sense of non-standard analysis, to differential and integral calculus. It provides a rigorous justification for some arguments in calculus that were previously considered merely heuristic.Calculations with infinitesimals were widely used before Karl Weierstrass sought to replace them with the (ε, δ)-definition of limit starting in the 1870s. (See history of calculus.) For almost one hundred years thereafter, mathematicians like Richard Courant viewed infinitesimals as being naive and vague or meaningless.Contrary to such views, Abraham Robinson showed in 1960 that infinitesimals are precise, clear, and meaningful, building upon work by Edwin Hewitt and Jerzy Łoś. According to Jerome Keisler, ""Robinson solved a three hundred year old problem by giving a precise treatment of infinitesimals. Robinson's achievement will probably rank as one of the major mathematical advances of the twentieth century.""
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