Section 1.1 Calculus: Areas And Tangents

... Example Suppose we wish to find the area inside a circle of radius one centered at the origin. Of course, we have all learned that the answer is π. But why? Indeed, what does it mean to find the area of a disk? Area is best defined for polygons, regions in the plane with line segments for sides. One ...

... Example Suppose we wish to find the area inside a circle of radius one centered at the origin. Of course, we have all learned that the answer is π. But why? Indeed, what does it mean to find the area of a disk? Area is best defined for polygons, regions in the plane with line segments for sides. One ...

A Direct Proof of the Prime Number Theorem

... We have used this technique to prove the prime number theorem directly, without recourse to ψ(x), derived both the Riemann and “exact” forms of π(x), and found a correction to the classical ω, Ω average order ...

... We have used this technique to prove the prime number theorem directly, without recourse to ψ(x), derived both the Riemann and “exact” forms of π(x), and found a correction to the classical ω, Ω average order ...

Sequences, Series and Taylor Approximation

... on special sequences defined by real functions such that an = f (n), for all n ∈ N, where f : [1, ∞) → R. This will allow us to use what you did with limits of functions at Level 1. This is not a significant restriction because, in practice, we shall use sequences of this type. In lecture six, the l ...

... on special sequences defined by real functions such that an = f (n), for all n ∈ N, where f : [1, ∞) → R. This will allow us to use what you did with limits of functions at Level 1. This is not a significant restriction because, in practice, we shall use sequences of this type. In lecture six, the l ...

Understanding Calculus II: Problems, Solutions, and Tips

... The goal of this course is to further your understanding and appreciation of calculus. Just as in Understanding Calculus: Problems, Solutions, and Tips, you will see how calculus plays a fundamental role in all of science and engineering. In the first third of the course, you’ll use the tools of der ...

... The goal of this course is to further your understanding and appreciation of calculus. Just as in Understanding Calculus: Problems, Solutions, and Tips, you will see how calculus plays a fundamental role in all of science and engineering. In the first third of the course, you’ll use the tools of der ...

(pdf)

... (3) E S X 2 (t, w)dt < ∞. Just like the Riemann integral is built using step functions, we will first define stochastic integrals directly on a very well-behaved subset of V , which we will call elementary processes. Then, for the remaining processes in V , we will define the stochastic integral by ...

... (3) E S X 2 (t, w)dt < ∞. Just like the Riemann integral is built using step functions, we will first define stochastic integrals directly on a very well-behaved subset of V , which we will call elementary processes. Then, for the remaining processes in V , we will define the stochastic integral by ...

Integral

... all these anti derivatives is called the indefinite integral of the function and such process of finding anti derivatives is called integration. Such type of problems arise in many practical situations. For instance, if we know the instantaneous velocity of an object at any instant, then there arise ...

... all these anti derivatives is called the indefinite integral of the function and such process of finding anti derivatives is called integration. Such type of problems arise in many practical situations. For instance, if we know the instantaneous velocity of an object at any instant, then there arise ...

Introduction to Calculus

... If you are a beginner in calculus, then be sure that you have had the appropriate background material of algebra and trigonometry. If you don’t understand something then don’t be afraid to ask your instructor a question. Go to the library and check out some other calculus books to get a presentation ...

... If you are a beginner in calculus, then be sure that you have had the appropriate background material of algebra and trigonometry. If you don’t understand something then don’t be afraid to ask your instructor a question. Go to the library and check out some other calculus books to get a presentation ...

Lecture Slides on Maple - Department of Statistics

... This website has a number of useful-looking tutorials: http://www.gmi.edu/acad/scimath/appmath/maple/ ...

... This website has a number of useful-looking tutorials: http://www.gmi.edu/acad/scimath/appmath/maple/ ...

Functional Limit theorems for the quadratic variation of a continuous

... The sequence xn (t) ∈ D converges to x0 (t) ∈ D in the J1 −topology if there exists a sequence of increasing homeomorphisms λn : [0, T ] → [0, T ] such that sup |λn (t) − t| → 0, sup |xn (λn (t)) − x0 (t)| → 0, t∈[0,T ] ...

... The sequence xn (t) ∈ D converges to x0 (t) ∈ D in the J1 −topology if there exists a sequence of increasing homeomorphisms λn : [0, T ] → [0, T ] such that sup |λn (t) − t| → 0, sup |xn (λn (t)) − x0 (t)| → 0, t∈[0,T ] ...

Section 4.1: The Definite Integral

... error. This error is removed by taking the standard part to form the integral. It is often difficult to compute an infinite Riemann sum, since it is a sum of infinitely many infinitesimal rectangles. We shall first study finite Riemann sums, which can easily be computed on a hand calculator. Suppose ...

... error. This error is removed by taking the standard part to form the integral. It is often difficult to compute an infinite Riemann sum, since it is a sum of infinitely many infinitesimal rectangles. We shall first study finite Riemann sums, which can easily be computed on a hand calculator. Suppose ...

MAT 1015 Calculus I 2010/2011 John F. Rayman

... We can describe lines and curves in the complex plane by deﬁning conditions on z: if these conditions are inequalities we have a description of a region of the complex plane. • The distance between two points in the complex plane, z and w is |w − z|, thus the set {z : |z| = a} is a circle centre at ...

... We can describe lines and curves in the complex plane by deﬁning conditions on z: if these conditions are inequalities we have a description of a region of the complex plane. • The distance between two points in the complex plane, z and w is |w − z|, thus the set {z : |z| = a} is a circle centre at ...

Pdf - Text of NPTEL IIT Video Lectures

... This inequality is right from here 3. I am using 3 and then I can conclude this thing here. So, what I get is the contour integration of f of z d z minus the contour integration on the boundary of R n f of z star d z minus the contour integration of f prime of z star times z d z plus the contour int ...

... This inequality is right from here 3. I am using 3 and then I can conclude this thing here. So, what I get is the contour integration of f of z d z minus the contour integration on the boundary of R n f of z star d z minus the contour integration of f prime of z star times z d z plus the contour int ...

Mean Square Calculus for Random Processes

... (such as means and variances) are of interest, not individual sample functions. Many applications are served well by a calculus based on mean-square convergence concepts similar to those introduced in Chapter 11. Such a mean-square calculus discards the difficulties of having to deal with all sample ...

... (such as means and variances) are of interest, not individual sample functions. Many applications are served well by a calculus based on mean-square convergence concepts similar to those introduced in Chapter 11. Such a mean-square calculus discards the difficulties of having to deal with all sample ...

Course Title:

... This Calculus course aims at preparing students for success in college-level Math. At the foundation of mathematics is the idea of a function. Functions express the way one variable quantity is related to another quantity. Calculus was invented to deal with the rate at which a quantity varies, parti ...

... This Calculus course aims at preparing students for success in college-level Math. At the foundation of mathematics is the idea of a function. Functions express the way one variable quantity is related to another quantity. Calculus was invented to deal with the rate at which a quantity varies, parti ...

Primitive and General Recursive Functions

... Given functions gi : Nm → N for i = 1, . . . , n , h : Nn → N , we define a new function f : Nm → N by composition as follows: f (x) = h(g1 (x), . . . , gn (x)) ...

... Given functions gi : Nm → N for i = 1, . . . , n , h : Nn → N , we define a new function f : Nm → N by composition as follows: f (x) = h(g1 (x), . . . , gn (x)) ...

Part 1 - CSUN.edu

... 2. M is an upper bound of A and for every ε > 0 there is an a ∈ A such that M − ε < a ≤ M . Proof: Suppose M = sup A, and suppose that there is a number ε0 > 0 such that a ≤ M − ε0 for all a ∈ A. Then M − ε0 is an upper bound of A which is smaller than M . Thus for every ε > 0 there is an a ∈ A such ...

... 2. M is an upper bound of A and for every ε > 0 there is an a ∈ A such that M − ε < a ≤ M . Proof: Suppose M = sup A, and suppose that there is a number ε0 > 0 such that a ≤ M − ε0 for all a ∈ A. Then M − ε0 is an upper bound of A which is smaller than M . Thus for every ε > 0 there is an a ∈ A such ...

Conditional expectation

... The final claim holds since every random variable is independent of the trivial σ-algebra. l Theorem 3.4. If G1 Ă G2 Ă F, then ErErX | G1 s | G2 s “ ErErX | G2 s | G1 s “ ErX | G1 s. Remark 3.5. In words, this says that in a battle between nested σ-algebras, the smallest ...

... The final claim holds since every random variable is independent of the trivial σ-algebra. l Theorem 3.4. If G1 Ă G2 Ă F, then ErErX | G1 s | G2 s “ ErErX | G2 s | G1 s “ ErX | G1 s. Remark 3.5. In words, this says that in a battle between nested σ-algebras, the smallest ...

Rationalizing Substitutions

... this way we may be able to integrate the original functions by referring to the method of Partial Fractions from Chapter 8. Before we start though we need to remind the reader of a notion called the least common multiple of two given positive integers. As the phrase suggests, the least common multip ...

... this way we may be able to integrate the original functions by referring to the method of Partial Fractions from Chapter 8. Before we start though we need to remind the reader of a notion called the least common multiple of two given positive integers. As the phrase suggests, the least common multip ...

CLEP® Precalculus - The College Board

... Most textbooks used in college-level precalculus courses cover the topics in the outline above, but the approaches to certain topics and the emphases given to them may differ. To prepare for the CLEP Precalculus exam, it is advisable to study one or more college textbooks, which can be found for sal ...

... Most textbooks used in college-level precalculus courses cover the topics in the outline above, but the approaches to certain topics and the emphases given to them may differ. To prepare for the CLEP Precalculus exam, it is advisable to study one or more college textbooks, which can be found for sal ...

Derivative of General Exponential and Logarithmic

... J. Garvin — Derivatives of General Exponential andLogarithmic Functions Slide 11/11 ...

... J. Garvin — Derivatives of General Exponential andLogarithmic Functions Slide 11/11 ...

Weak Complementarity, Path Independence, and the Intuition of the

... uncompensated welfare measure in the environmental good space is equivalent to the uncompensated welfare measure in the private good space. The uncompensated measure derived in private good space can provide a bound for the compensated measure of the value of a quality improvement. In addition to d ...

... uncompensated welfare measure in the environmental good space is equivalent to the uncompensated welfare measure in the private good space. The uncompensated measure derived in private good space can provide a bound for the compensated measure of the value of a quality improvement. In addition to d ...

Discrete Mathematics—Introduction

... Theorem: Suppose F: XY is a one-to-one and onto. Then F-1: YX is also one-to-one and onto. Proof: (1) Show that F-1 is one-to-one. Suppose F-1(y1) = F-1(y2). We must show y1= y2. Let x = F-1(y1) = F-1(y2). F-1(y1)= x and F-1(y2)= x so so y1=F(x) and y2=F(x) [ Recall that F-1(y)=x y=F(x)] so y1= ...

... Theorem: Suppose F: XY is a one-to-one and onto. Then F-1: YX is also one-to-one and onto. Proof: (1) Show that F-1 is one-to-one. Suppose F-1(y1) = F-1(y2). We must show y1= y2. Let x = F-1(y1) = F-1(y2). F-1(y1)= x and F-1(y2)= x so so y1=F(x) and y2=F(x) [ Recall that F-1(y)=x y=F(x)] so y1= ...

inverse sine functions

... However, we have a slight difficulty in this task. As the trigonometric functions are not one-to-one, they don’t have inverse functions. The difficulty is overcome by restricting the domains of these functions so that they become one-to-one. ...

... However, we have a slight difficulty in this task. As the trigonometric functions are not one-to-one, they don’t have inverse functions. The difficulty is overcome by restricting the domains of these functions so that they become one-to-one. ...

# Lebesgue integration

In mathematics, the integral of a non-negative function can be regarded, in the simplest case, as the area between the graph of that function and the x-axis. The Lebesgue integral extends the integral to a larger class of functions. It also extends the domains on which these functions can be defined.Mathematicians had long understood that for non-negative functions with a smooth enough graph—such as continuous functions on closed bounded intervals—the area under the curve could be defined as the integral, and computed using approximation techniques on the region by polygons. However, as the need to consider more irregular functions arose—e.g., as a result of the limiting processes of mathematical analysis and the mathematical theory of probability—it became clear that more careful approximation techniques were needed to define a suitable integral. Also, we might wish to integrate on spaces more general than the real line. The Lebesgue integral provides the right abstractions needed to do this important job.The Lebesgue integral plays an important role in the branch of mathematics called real analysis, and in many other mathematical sciences fields. It is named after Henri Lebesgue (1875–1941), who introduced the integral (Lebesgue 1904). It is also a pivotal part of the axiomatic theory of probability.The term Lebesgue integration can mean either the general theory of integration of a function with respect to a general measure, as introduced by Lebesgue—or the specific case of integration of a function defined on a sub-domain of the real line with respect to Lebesgue measure.