Introduction to Calculus

... Relation between Sequences and Functions, Establish Bounds for Sequences, Additional Terminology Associated with Sequences, Stolz -Cesàro Theorem, Examples of Sequences, Infinite Series, Sequence of Partial Sums, Convergence and Divergence of a Series, Comparison of Two Series, Test For Divergence, ...

... Relation between Sequences and Functions, Establish Bounds for Sequences, Additional Terminology Associated with Sequences, Stolz -Cesàro Theorem, Examples of Sequences, Infinite Series, Sequence of Partial Sums, Convergence and Divergence of a Series, Comparison of Two Series, Test For Divergence, ...

a la Finance University Paris 1 Stochastic Calculus 2 Annie Millet

... The aim of this chapter is to extend the notions of Brownian motion, Itô process and the Itô formula from dimension 1 to any finite dimension d. 1.1 Reminders In order to make these notes as self-contained as possible, let us start by recalling some notions already introduced in the notes of Stoch ...

... The aim of this chapter is to extend the notions of Brownian motion, Itô process and the Itô formula from dimension 1 to any finite dimension d. 1.1 Reminders In order to make these notes as self-contained as possible, let us start by recalling some notions already introduced in the notes of Stoch ...

Introduction to Homogenization and Gamma

... We have at our disposal now a powerful tool to obtain the desired convergence of minima in (1.2) and (1.3). The next, crucial point now is to understand what the right choice for the space (X; d) is, and how to dene the functionals Fh . At this point, we have to specify the conditions we require on ...

... We have at our disposal now a powerful tool to obtain the desired convergence of minima in (1.2) and (1.3). The next, crucial point now is to understand what the right choice for the space (X; d) is, and how to dene the functionals Fh . At this point, we have to specify the conditions we require on ...

limits and derivatives conditions for the existence of a limit

... The gradient of a line is constant and does not change at different values of x . As the derivative represents the gradient to a curve, the derivative of a linear expressions will result in a constant (numeric) value. The gradient to a curve modelled by a non linear equation changes, and is dependen ...

... The gradient of a line is constant and does not change at different values of x . As the derivative represents the gradient to a curve, the derivative of a linear expressions will result in a constant (numeric) value. The gradient to a curve modelled by a non linear equation changes, and is dependen ...

convex functions on symmetric spaces, side lengths of polygons and

... When studying asymptotic properties of the spectra of certain linear partial differential equations in mathematical physics, Hermann Weyl was led in [We] to the question how the spectra of two compact self adjoint operators are related to the spectrum of their sum. The restrictions turn out to be ho ...

... When studying asymptotic properties of the spectra of certain linear partial differential equations in mathematical physics, Hermann Weyl was led in [We] to the question how the spectra of two compact self adjoint operators are related to the spectrum of their sum. The restrictions turn out to be ho ...

Pdf - Text of NPTEL IIT Video Lectures

... there is no fluid produced anywhere in the flow or there is no volume of fluid which disappears during the flow. So, if you make this assumptions then we get that the velocity field of a fluid flow is the conjugate of an analytic function. So, let us first start by interpreting what the line integr ...

... there is no fluid produced anywhere in the flow or there is no volume of fluid which disappears during the flow. So, if you make this assumptions then we get that the velocity field of a fluid flow is the conjugate of an analytic function. So, let us first start by interpreting what the line integr ...

Notes on Calculus II Integral Calculus Miguel A. Lerma

... The symbols at the left historically were intended to mean an infinite R sum, represented by a long “S” (the integral symbol ), of infinitely small amounts f (x) dx. The symbol dx was interpreted as the length of an “infinitesimal” interval, sort of what ∆x becomes for infinite n. This interpretatio ...

... The symbols at the left historically were intended to mean an infinite R sum, represented by a long “S” (the integral symbol ), of infinitely small amounts f (x) dx. The symbol dx was interpreted as the length of an “infinitesimal” interval, sort of what ∆x becomes for infinite n. This interpretatio ...

DEFINITE INTEGRALS

... shaded region in Fig.27.3. Now the problem is to find the area of the shaded region. In order to solve this problem, we consider three special cases of f (x) as rectangular region , triangular region and trapezoidal region. The area of these regions = base × average height In general for any functio ...

... shaded region in Fig.27.3. Now the problem is to find the area of the shaded region. In order to solve this problem, we consider three special cases of f (x) as rectangular region , triangular region and trapezoidal region. The area of these regions = base × average height In general for any functio ...

Part 1 - CSUN.edu

... Proof: As in many uniqueness proofs we do this one by contradiction. Assume that M1 and M2 are to least upper bounds if M1 6= m2 then either M1 > M2 which would make M1 not the least uper bound, or M2 > M1 which makes M2 not the least upper bound. Therefore, M1 = M2 . In analysis you will come to lo ...

... Proof: As in many uniqueness proofs we do this one by contradiction. Assume that M1 and M2 are to least upper bounds if M1 6= m2 then either M1 > M2 which would make M1 not the least uper bound, or M2 > M1 which makes M2 not the least upper bound. Therefore, M1 = M2 . In analysis you will come to lo ...

Math 55b Lecture Notes Contents

... distance between two vertices to be the graph-theoretic distance between them. (The discrete metric is the special case when G is the complete graph on S.) ...

... distance between two vertices to be the graph-theoretic distance between them. (The discrete metric is the special case when G is the complete graph on S.) ...

MATH_125_online_L5__..

... part two of the fundamental theorem of calculus, which, recall is in Section 5.3 of your textbook, and on a related result from Section 5.4, which is called the net change theorem. Let's start by recalling part one of the fundamental theorem of calculus, which we discussed last time. [00:00:26.25] T ...

... part two of the fundamental theorem of calculus, which, recall is in Section 5.3 of your textbook, and on a related result from Section 5.4, which is called the net change theorem. Let's start by recalling part one of the fundamental theorem of calculus, which we discussed last time. [00:00:26.25] T ...

3 Lipschitz condition and Lipschitz continuity

... A fact that we have used in the proof of uniqueness is the following: 3.17 Lemma (Characterization of local Lipschitz-continuity) A continuous function f , such as in (25), is locally Lipschitz continuous with respect to its first argument if and only if for each closed and bounded subset K of Rm × ...

... A fact that we have used in the proof of uniqueness is the following: 3.17 Lemma (Characterization of local Lipschitz-continuity) A continuous function f , such as in (25), is locally Lipschitz continuous with respect to its first argument if and only if for each closed and bounded subset K of Rm × ...

Multidimensional Calculus. Lectures content. Week 10 22. Tests for

... if ∀ε > 0∃N0 ∈ IN such that ∀k ≥ N0 ∀x ∈ M : |f (x) − fk (x)| < ε. Theorem. Let fk → →f on M . (i) If all fk are continuous on M , then also f is continuous there. (ii) If all fk have a derivative on M , then also f has it there and f 0 = lim (fk0 ) on M . ...

... if ∀ε > 0∃N0 ∈ IN such that ∀k ≥ N0 ∀x ∈ M : |f (x) − fk (x)| < ε. Theorem. Let fk → →f on M . (i) If all fk are continuous on M , then also f is continuous there. (ii) If all fk have a derivative on M , then also f has it there and f 0 = lim (fk0 ) on M . ...

educative commentary on jee 2014 advanced mathematics papers

... that now that the number of candidates appearing for the JEE (Advanced) is manageable enough to permit evaluation by humans, the classic practice of requiring the candidates to give justifications for their answers would be revived at least from 2014, if not from 2013 (when there might not have been ...

... that now that the number of candidates appearing for the JEE (Advanced) is manageable enough to permit evaluation by humans, the classic practice of requiring the candidates to give justifications for their answers would be revived at least from 2014, if not from 2013 (when there might not have been ...

Antiderivatives - John Abbott College

... What is the Antiderivative? In a derivative problem, a function f (x) is given and you find the derivative f ′ (x) using the formulas and rules of derivatives shown in a previous tutorial. In an antiderivative problem, the derivative f ′ (x) is given and you find a function f (x) using the formulas ...

... What is the Antiderivative? In a derivative problem, a function f (x) is given and you find the derivative f ′ (x) using the formulas and rules of derivatives shown in a previous tutorial. In an antiderivative problem, the derivative f ′ (x) is given and you find a function f (x) using the formulas ...

Example 1: Solution: f P

... From what we have learned already about differentiability, it will not be difficult to show that continuity is an important condition for differentiability. The following theorem is one of the most important theorems in calculus: Differentiability and Continuity If f is differentiable at x 0, then ...

... From what we have learned already about differentiability, it will not be difficult to show that continuity is an important condition for differentiability. The following theorem is one of the most important theorems in calculus: Differentiability and Continuity If f is differentiable at x 0, then ...

Integral Calculus - Exercises

... Therefore, the desired function is f (x) = 14 x4 + x2 + 2x − 54 . 9. It is estimated that t years from now the population of a certain lakeside community will be changing at the rate of 0.6t2 + 0.2t + 0.5 thousand people per year. Environmentalists have found that the level of pollution in the lake ...

... Therefore, the desired function is f (x) = 14 x4 + x2 + 2x − 54 . 9. It is estimated that t years from now the population of a certain lakeside community will be changing at the rate of 0.6t2 + 0.2t + 0.5 thousand people per year. Environmentalists have found that the level of pollution in the lake ...

Chapter 7

... contradicting the assumption that M = sup S. Proof of Proposition 7D. We may clearly suppose that f (x1 ) < y < f (x2 ). By considering the function −f (x) if necessary, we may further assume, without loss of generality, that x1 < x2 . The idea of the proof is then to follow the graph of the funct ...

... contradicting the assumption that M = sup S. Proof of Proposition 7D. We may clearly suppose that f (x1 ) < y < f (x2 ). By considering the function −f (x) if necessary, we may further assume, without loss of generality, that x1 < x2 . The idea of the proof is then to follow the graph of the funct ...

... infinite hyper-reals with negative sign, and non-constant hyper-reals. 8. The hyper-reals are totally ordered, and aligned along a line: the Hyper-real Line. 9. That line includes the real numbers separated by the nonconstant hyper-reals. Each real number is the center of an interval of hyper-reals, ...

INTRODUCTION TO POLYNOMIAL CALCULUS 1. Straight Lines

... What about the slope of a curve that is not a straight line? Does this make sense? If so, how do we calculate it? Let's look at an example, say the curve y = x2 The graph of this curve (Figure 2.1) makes it clear that, if its slope makes sense, it cannot be a xed number. To the left of x = 0 the cu ...

... What about the slope of a curve that is not a straight line? Does this make sense? If so, how do we calculate it? Let's look at an example, say the curve y = x2 The graph of this curve (Figure 2.1) makes it clear that, if its slope makes sense, it cannot be a xed number. To the left of x = 0 the cu ...

The mathematics of PDEs and the wave equation

... D2 = ∇2 . Good luck solving this one.5 The operator D is called the Dirac operator; finding particular Dirac operators is a major intellectual achievement of modern mathematics and physics. The Atiyah-Singer index theorem is a deep result connecting the Dirac operator with the geometry of manifolds. ...

... D2 = ∇2 . Good luck solving this one.5 The operator D is called the Dirac operator; finding particular Dirac operators is a major intellectual achievement of modern mathematics and physics. The Atiyah-Singer index theorem is a deep result connecting the Dirac operator with the geometry of manifolds. ...

REVIEW FOR FINAL EXAM April 08, 2014 • Final Exam Review Session:

... I. Local maximum/minimal values: a. A function f has a local maximum value at point (a, b) if f (x, y) ≤ f (a, b) for all (x, y) in some small open disk centered at (a, b). b. A function f has a local minimum value at point (a, b) if f (x, y) ≥ f (a, b) for all (x, y) in some small open disk centere ...

... I. Local maximum/minimal values: a. A function f has a local maximum value at point (a, b) if f (x, y) ≤ f (a, b) for all (x, y) in some small open disk centered at (a, b). b. A function f has a local minimum value at point (a, b) if f (x, y) ≥ f (a, b) for all (x, y) in some small open disk centere ...

Calculus II

... Similarly to Cartesian coordinates on the Euclidean plane we could introduce rectangular coordinate system or xyz-coordinate system in three dimensions. The origin is usually denoted by O and three axises are OX, OY , OZ. The positive directions are selected in the way to form the right-handed coord ...

... Similarly to Cartesian coordinates on the Euclidean plane we could introduce rectangular coordinate system or xyz-coordinate system in three dimensions. The origin is usually denoted by O and three axises are OX, OY , OZ. The positive directions are selected in the way to form the right-handed coord ...

Economic Techniques 102 Week 3 Lecture DCDM BUSINESS

... FACULTY OF MANAGEMENT ECONOMIC TECHNIQUES 102 LECTURE 3 NON-LINEAR FUNCTIONS 0. Preliminaries The following functions will be discussed briefly first: ...

... FACULTY OF MANAGEMENT ECONOMIC TECHNIQUES 102 LECTURE 3 NON-LINEAR FUNCTIONS 0. Preliminaries The following functions will be discussed briefly first: ...