Assignment 1 Exercise 1 - Chinese University of Hong Kong

PROOF OF TAYLOR’S THEOREM

ection on the proof(s) of Taylor’s theorem. First we recall the (derivative form) of the theorem: Theorem 1 (Taylor’s theorem). Suppose f: (a;b) !R is a function on (a;b), where a;b 2R with a < b. Assume that for some positive integer n, f is n-times di erentiable on the open interval

  Notice

Chapter 3 The Space of Continuous Functions

Baire category theorem is proved and, as an application, it is shown that continuous, nowhere di erentiable functions form a set of second category in the space of continuous functions. 3.1 Spaces of Continuous Functions We studied continuous functions on an interval in MATH2050/60 and in a domain bounded

  Category, Theorem, Rabies, Baire category theorem

Black-Scholes Equations - CUHK Mathematics

Black-Scholes formulation establishes the equilibrium condition between the expected return on the option, the expected return on the stock, and the riskless interest rate. We will derive the formula in this chapter. Since the publication of Black-Scholes’ and Merton’s papers, the growth of the field of derivative securities has been ...

  Black, Scholes, Black scholes

Chapter 3 The Contraction Mapping Principle

3.2. THE CONTRACTION MAPPING PRINCIPLE 3 This theorem will be further explained and proved in the appendix. 3.2 The Contraction Mapping Principle Solving an equation f(x) = 0;where f is a function from Rn to itself frequently comes up in application. This problem can be turned into a problem for xed points. Literally,

  Principles

3.1 Uniform Convergence of Functions - CUHK Mathematics

proven. They are reformulated in the context of in nite series of functions in Section 3. The last two important sections demonstrate the power of uniform convergence. In Sections 4 and 5 we introduce the exponential function, sine and cosine functions based on di erential equations. Although various de nitions of

  Functions, Sine, Isceon, Sine and cosine functions

1 Derivatives of Piecewise Defined Functions

1 Derivatives of Piecewise Defined Functions For piecewise defined functions, we often have to be very careful in com-puting the derivatives. The di↵erentiation rules (product, quotient, chain rules) can only be applied if the function is defined by ONE formula in a neighborhood of the point where we evaluate the derivative. If we want

  Functions, Derivatives

MATH2060A Solution to Assignment 1

examine the di erentiability of j at 0 using de niton. Indeed, using the fact the cosine function is even, lim h!0 cosjhj cos0 h 0 = lim h!0 cosh 1 h = 0; from which we conclude that jis also di erentiable at x= 0. Hence jis di erentiable everywhere. A shortcut is to realize that the cosine is an even function, so j(x) = cosxis dif-ferentiable ...

  Niton

Chapter 4 Inverse Function Theorem

Indeed, in MATH2060 we learned that if f is continuously di erentiable on (a;b) with non-vanishing f0, it is either strictly increasing or decreasing so that its global inverse exists and is again continuously di erentiable. Example 4.3. Consider the map F: R2!R2 given by F(x;y) = (x2;y). Its Jacobian matrix is singular at (0;0).

Two-dimensional modeling of steady state heat transfer in ...

The aim of the thesis is offering to the universities an alternative way of calculating heat transfer by numerical method. Nowadays programs like ANSYS, which costs use to be quite ... 2.2 HOW TO PERFORM NUMERICAL METHOD USING MS EXCEL. ... thermal resistance r [ - ] common ratio r [m] radius [K/W] thermal conduction resistance T [K] temperature

  Resistance, Calculating

Voltage Drop - Adams Electric

formulas. The table also shows another method of calculating — called the “resistance per 1,000 ft” method — which we won’t get into in this discussion. Example 1 — A single-phase motor is located 250 feet from its power source and is supplied with 10 AWG copper. The motor has a full load current draw of 24 amps. What is

  Resistance, Calculating, Voltage, Drop, Voltage drop

Portal Hypertension: Introduction - Hopkins Medicine

This resistance causes increased pressure, resulting in varices or dilations of the veins and tributaries. ... and l ow platelet counts may be the earliest abnormal laboratory finding. Hepatomegaly is variable and dependent upon the cause and stage of liver disease. ... The most commonly used and preferred method for measuring the portal ...

  Resistance, Medicine, Hopkins, Hopkins medicine

HE “TURNING-OUT OF BOYS IN A MAN S PRISON WHY AND …

820 INDIANA LAW REVIEW [Vol. 44:819 2. T.J. PARSELL, FISH: A MEMOIR OF A BOY IN A MAN’S PRISON 86-94 (2006). 3. Id. at xi. 4. See infra notes 13-14 and accompanying text (delineating the number of boy prisoners). The term “boy prisoners” designates men under age eighteen at the commission of their offenses

  Prison

GLOBAL HISTORY AND GEOGRAPHY II (GRADE 10)

Global Hist. & Geo. II Rating Guide – June ’19 [6] 31 Identify and explain a cause-and-effect relationship associated with the historical developments in documents 1 and 2. Be sure to use evidence from both documents 1 and 2 in your response. Score of 1:

K5 External Wall Board

The heat fl ow at these junctions and opening locations, over and above that through the adjoining plane elements is the linear thermal transmittance of the thermal bridge: measured in W/mK; referred to as a ‘psi-value’; and expressed as a ψ-value. The lower the ψ-value, the better the performance. -values ψ