Section 2.3: Infinite sets and cardinality
... Suppose that A and B are sets (finite or infinite). We say that A and B have the same cardinality (written |A| = |B|) if a bijective correspondence exists between A and B. In other words, A and B have the same cardinality if it’s possible to match each element of A to a different element of B in such a ...
... Suppose that A and B are sets (finite or infinite). We say that A and B have the same cardinality (written |A| = |B|) if a bijective correspondence exists between A and B. In other words, A and B have the same cardinality if it’s possible to match each element of A to a different element of B in such a ...
1.5 M - Thierry Karsenti
... In this activity we formulate the Riemann integral which depends explicitly on the order structure of the real line. Accordingly we begin by discussing the concept of a partition of an interval and show that formulation of the Riemann integral is essentially one of the methods of estimating area und ...
... In this activity we formulate the Riemann integral which depends explicitly on the order structure of the real line. Accordingly we begin by discussing the concept of a partition of an interval and show that formulation of the Riemann integral is essentially one of the methods of estimating area und ...
Millionaire - WOWmath.org
... Explanation The concavity of f(x) at any value of x is determined by the sign ( + or - ) of f’’(x). If the sign is + then the concavity is positive and negative if the sign is -. Points of infection divide intervals of different concavity. P of I occur where f’’(x) = 0 and f’’(x) = 0 at x = ±0.408 ...
... Explanation The concavity of f(x) at any value of x is determined by the sign ( + or - ) of f’’(x). If the sign is + then the concavity is positive and negative if the sign is -. Points of infection divide intervals of different concavity. P of I occur where f’’(x) = 0 and f’’(x) = 0 at x = ±0.408 ...
1.4 Limits and Continuity
... • We learned about left and right limits. • We learned about continuity and the properties of continuity. ...
... • We learned about left and right limits. • We learned about continuity and the properties of continuity. ...
Calculus Math 1710.200 Fall 2012 (Cohen) Lecture Notes
... may write 2p/q = q 2p = ( q 2)p , which is the definition we all know and understand. So how does one extend√the definition to irrational numbers? Just what is meant by the expressions 2π or 2 2 ? Question 4 (Zeno’s Paradox). Does Achilles ever catch the tortoise? Here is a variation of one of Zeno ...
... may write 2p/q = q 2p = ( q 2)p , which is the definition we all know and understand. So how does one extend√the definition to irrational numbers? Just what is meant by the expressions 2π or 2 2 ? Question 4 (Zeno’s Paradox). Does Achilles ever catch the tortoise? Here is a variation of one of Zeno ...
3 The Introductory Course on Higher Mathematics\ V.B.Zhivetin. The
... And lastly, there exists one-to-one correspondence both between the set of triplets of real numbers (x,y,z) and the set of points in space, i. e. to each triplet of numbers there corresponds a point in space with coordinates x, y, z, and, vice versa, to each point in space there corresponds a triple ...
... And lastly, there exists one-to-one correspondence both between the set of triplets of real numbers (x,y,z) and the set of points in space, i. e. to each triplet of numbers there corresponds a point in space with coordinates x, y, z, and, vice versa, to each point in space there corresponds a triple ...
class exam II review
... (b) Find all horizontal asymptotes of the graph of y = f (x). (c) Find all relative maxima and all relative minima of the graph of y = f (x). (d) Where is f increasing? Decreasing? (e) Where is the graph of y = f (x) concave up? Where is it concave down? (f) Sketch the graph of y = f (x). ...
... (b) Find all horizontal asymptotes of the graph of y = f (x). (c) Find all relative maxima and all relative minima of the graph of y = f (x). (d) Where is f increasing? Decreasing? (e) Where is the graph of y = f (x) concave up? Where is it concave down? (f) Sketch the graph of y = f (x). ...
Analysis 1.pdf
... In this activity we formulate the Riemann integral which depends explicitly on the order structure of the real line. Accordingly we begin by discussing the concept of a partition of an interval and show that formulation of the Riemann integral is essentially one of the methods of estimating area und ...
... In this activity we formulate the Riemann integral which depends explicitly on the order structure of the real line. Accordingly we begin by discussing the concept of a partition of an interval and show that formulation of the Riemann integral is essentially one of the methods of estimating area und ...
looking at graphs through infinitesimal microscopes
... increment along the tangent, not along the graph. Of course, if dx is extremely small, then Fig. 2 approximates to Fig. 1. Leibniz imagined dx to be an infinitesimal, and that Fig. 1 was accurate within infinitesimals of higher FIGURE 2. order. In the nineteenth century the arrival of the analysis o ...
... increment along the tangent, not along the graph. Of course, if dx is extremely small, then Fig. 2 approximates to Fig. 1. Leibniz imagined dx to be an infinitesimal, and that Fig. 1 was accurate within infinitesimals of higher FIGURE 2. order. In the nineteenth century the arrival of the analysis o ...
MAT 1613 TEST #1 Name
... 12. (a) Use the limit definition to find the slope of the tangent line to the graph of f ( x) ...
... 12. (a) Use the limit definition to find the slope of the tangent line to the graph of f ( x) ...
Chapter 5 Sect. 1,2,3 - Columbus State University
... Finding absolute extrema on [a , b] 1. Find all critical numbers for f (x) in (a,b). 2. Evaluate f (x) for all critical numbers in (a,b). 3. Evaluate f (x) for the endpoints a and b of the interval [a,b]. 4. The largest value found in steps 2 and 3 is the absolute maximum for f on the interval [a , ...
... Finding absolute extrema on [a , b] 1. Find all critical numbers for f (x) in (a,b). 2. Evaluate f (x) for all critical numbers in (a,b). 3. Evaluate f (x) for the endpoints a and b of the interval [a,b]. 4. The largest value found in steps 2 and 3 is the absolute maximum for f on the interval [a , ...
Leibniz`s Harmonic Triangle Paper
... ultimate definition of the Calculus. We will delve into the harmonic triangle shortly, but first, a brief history of Leibniz leading up to this point. Leibniz was born in Leipzig, Germany, in 1646. His father was a professor of Moral Philosophy at the University of Leipzig, but died when G.W. Leibni ...
... ultimate definition of the Calculus. We will delve into the harmonic triangle shortly, but first, a brief history of Leibniz leading up to this point. Leibniz was born in Leipzig, Germany, in 1646. His father was a professor of Moral Philosophy at the University of Leipzig, but died when G.W. Leibni ...
M129-Tutorial_1
... Graph of a Function: The set of all points (x, f (x)) where x is the domain of f (x). Generally, this forms a curve in the xyplane. ...
... Graph of a Function: The set of all points (x, f (x)) where x is the domain of f (x). Generally, this forms a curve in the xyplane. ...
a review sheet for test #03
... an L for every n > N. If there is no such number L, then we say the sequence diverges. (This definition is the starting point for proving Theorems 1.1, 1.2, and 1.3…) Theorem 1.1: Limits of Combinations of Sequences. ...
... an L for every n > N. If there is no such number L, then we say the sequence diverges. (This definition is the starting point for proving Theorems 1.1, 1.2, and 1.3…) Theorem 1.1: Limits of Combinations of Sequences. ...
The untyped Lambda Calculus
... In the early 20th century, several mathematicians tried to define basic, abstract systems upon which one can build all of mathematics. In contrast to systems such as Zermelo-Fraenkel, which is based on sets as the most basic structure, some systems based un functions were developed. One of them is t ...
... In the early 20th century, several mathematicians tried to define basic, abstract systems upon which one can build all of mathematics. In contrast to systems such as Zermelo-Fraenkel, which is based on sets as the most basic structure, some systems based un functions were developed. One of them is t ...
Polygonal Numbers and Finite Calculus
... itself. As the name suggests, finite calculus is similar to conventional calculus. In fact, the two are analogous, yet while in calculus we needed to compute the area under a function, in finite calculus we want to compute the area under a sequence. Hence the name: sums in finite calculus are compos ...
... itself. As the name suggests, finite calculus is similar to conventional calculus. In fact, the two are analogous, yet while in calculus we needed to compute the area under a function, in finite calculus we want to compute the area under a sequence. Hence the name: sums in finite calculus are compos ...
Integral identities and constructions of approximations to
... 3.2. Hypergeometric integrals. Here we will find integral representations for sums defined in Proposition 1. These representations are useful in applications for computation of asimptotics for constructed linear forms. Let r > 2 be integer and be complex number. Denote ...
... 3.2. Hypergeometric integrals. Here we will find integral representations for sums defined in Proposition 1. These representations are useful in applications for computation of asimptotics for constructed linear forms. Let r > 2 be integer and be complex number. Denote ...
Limits at Infinity
... Two additional topics of interest with limits are limits as x → ±∞ and limits where f (x) → ±∞. Before we can properly discuss the notion of infinite limits, we will need to begin with a discussion on the concept of infinity. We begin by emphasizing that ∞ and −∞ are not a numbers; they are symbols ...
... Two additional topics of interest with limits are limits as x → ±∞ and limits where f (x) → ±∞. Before we can properly discuss the notion of infinite limits, we will need to begin with a discussion on the concept of infinity. We begin by emphasizing that ∞ and −∞ are not a numbers; they are symbols ...
Finding increasing and decreasing intervals
... 1. Increasing function - graph moves up as x moves to the right. 2. Decreasing function - graph moves down as x moves to the right. What do you know about the slope of an increasing function? a decreasing function? a constant function? ...
... 1. Increasing function - graph moves up as x moves to the right. 2. Decreasing function - graph moves down as x moves to the right. What do you know about the slope of an increasing function? a decreasing function? a constant function? ...