Fall 2014 MAT 375 Numerical Methods. Numerical Differentiation (Chapter 9)

Size: px
Start display at page:

Download "Fall 2014 MAT 375 Numerical Methods. Numerical Differentiation (Chapter 9)"

Transcription

1 Fall 2014 MAT 375 Numerical Metods (Capter 9)

2 Idea: Definition of te derivative at x Obviuos approximation: f (x) = lim 0 f (x + ) f (x) f (x) f (x + ) f (x) forward-difference formula? ow good is tis approximation wit respect to

3 Taylor s teorem: so f (x + ) = f (x) + f (x) f (ξ), ξ [x, x + ] f (x) = f (x + ) f (x) 2 f (ξ) Te term 2 f (ξ) is truncation/discretization error It is of order, so we ave 1st order approximation.

4 Let s generalize Mes on [a, b]: a = x 0 < x 1 < < x N 1 < x N = b, wit x i = a + i, step of discretization = (b a)/n Forward-difference formula f (x i ) f x,i = f (x i+1) f (x i ) Backward-difference formula f (x i ) f x,i = f (x i) f (x i 1 ) Centered-difference formula f (x i ) f x,i = f (x i+1) f (x i 1 ) 2

5 Truncation errors f x,i = f (x i ) + 2 f (ξ i ); f x,i = f (x i ) + 2 f (ξ i ) similar to f x,i For centered-difference formula: Taylor s teorem: f (x + ) = f (x) + f (x) f (x) f (ξ), ξ [x, x + ] f (x ) = f (x) f (x) f (x) 3 6 f (η), η [x, x] Ten for x = x i So f x,i = f (x i+1) f (x i 1 ) 2 = f (x i ) (f (ξ i ) + f (η i )) f x,i = f (x i ) f (ξ i ), 2nd degree ξ i [x i 1, x i+1 ]

6 Some conclusions Forward-difference formula: f x,i = f (x i) f (x i 1 ) Backward-difference formula: f x,i = f (x i+1) f (x i ) = f (x i ) + O() = f (x i ) + O() Centered-difference formula: f x,i = f (x i+1) f (x i 1 ) 2 = f (x i ) + O( 2 )

7 Examples: f (x) = e x at x = 1 f (x) = e x and f (1) = e at x = 1 Taking = 1/8: f (1) f x = e e = Errors: f (1) f x = e e Err(f x ) = = Err(f x ) =

8 Wat appens if 0? Approximating f (x) for f (x) = e x at x = 1 1 f x Error f x Error e e e e e e e e e e e e e e-04 Conclusion: Error 0 as 0 and Err(f x ) < Err(f x ) (or Err(f x )) So if we take to be small, life is good. Or is it?

9 Wat about roundoff error? Let f (x) be te rounded value of f (x). Ten f (x) = f (x)(1 + ε) fx = f (x + ) f (x) f (x + ) f (x) = Since δ i < ε - macine epsilon Rounding error = f (x + )(1 + δ 1) f (x)(1 + δ 2 ) + δ 1f (x + ) δ 2 f (x) δ 1 f (x + ) δ 2 f (x) ε ( f (x + ) + f (x) ) ε Rounding error ε as 0 Is it te end of te world as we know it?

10 Best results: truncation error rounding error 2 f (ξ) ε ( f (x + ) + f (x) ) ε So ε ten Error ε

11 Example: Find f x for f (x) = sin x at x = π/3 f (π/3) = 0.5 ε 10 8

12 How about centered-difference f x? Let f (x) be te rounded value of f (x). Ten f (x) = f (x)(1 + ε) f x = f (x + ) f (x ) 2 f (x + ) f (x ) = 2 Since δ i < ε - macine epsilon Rounding error = f (x + )(1 + δ 1) f (x )(1 + δ 2 ) 2 + δ 1f (x + ) δ 2 f (x ) 2 δ 1 f (x + ) δ 2 f (x ) 2 ε 2 ( f (x + ) + f (x ) ) ε

13 Best results: truncation error rounding error 2 6 f (ξ) 2 ε ε ( f (x + ) + f (x ) ) 2 So ε 1/3 ten Error ε 2/3

14 Example: Find f x for f (x) = sin x at x = π/3 f (π/3) = 0.5 ε 1/3 10 5

15 Higer-order derivatives f (x i ) f xx,i = f x,i f x,i = f (x i+1) 2f (x i ) + f (x i 1 ) 2 Error: Taylor s expansion of f (x i+1 ) and f (x i 1 ) at point x i : f xx,i = f (x i ) f (4) (ξ i ) Truncation error 2 (2nd order) Rounding error ε 2 (ε is macine epsilon) Best results: ε 2 2 ε 1/4 wit Err ε (if ε 10 6, ten 10 4 )

16 Unappy conclusion: Operation of numerical differentiation is ill-conditioned: Error as 0 Solution: coose so tat truncation error rounding error

17 Using polynomial interpolation for numerical differentiation f (x) = p n (x) f (x) = p n(x)

18 Mes on [a, b]: a = x 0 < x 1 < < x n 1 < x n = b Lagrange form of interpolation polynomial: f (x) = p n (x) + R n (x) n = f (x k )L k (x) + (x x 0)... (x x n ) f (n+1) (ξ(x)) (n + 1)! k=1 wit L k (x) = n i=0 i k x x i x k x i Differentiate: n [ ] (x f (x) = f (x k )L k (x) + D x0 )... (x x n ) x f (n+1) (ξ(x)) (n + 1)! k=1 + (x x 0)... (x x n ) ] D x [f (n+1) (ξ(x)) (n + 1)!

19 f (x) = n [ ] (x f (x k )L k (x) + D x0 )... (x x n ) x f (n+1) (ξ(x)) (n + 1)! k=1 + (x x 0)... (x x n ) ] D x [f (n+1) (ξ(x)) (n + 1)! How to estimate te truncation error? If we take x = x i for i = 0,..., n ten (x x 0)... (x x n ) (n + 1)! = 0 Ten f (x i ) = n k=1 f (x k )L k (x i) + f (n+1) (ξ(x i )) (n + 1)! (n + 1)-point formula to approximate f (x i ) n (x i x j ) j=0 j i

NUMERICAL DIFFERENTIATION

NUMERICAL DIFFERENTIATION NUMERICAL IFFERENTIATION FIRST ERIVATIVES Te simplest difference formulas are based on using a straigt line to interpolate te given data; tey use two data pints to estimate te derivative. We assume tat

More information

4.2 - Richardson Extrapolation

4.2 - Richardson Extrapolation . - Ricardson Extrapolation. Small-O Notation: Recall tat te big-o notation used to define te rate of convergence in Section.: Definition Let x n n converge to a number x. Suppose tat n n is a sequence

More information

Numerical Differentiation

Numerical Differentiation Numerical Differentiation Finite Difference Formulas for te first derivative (Using Taylor Expansion tecnique) (section 8.3.) Suppose tat f() = g() is a function of te variable, and tat as 0 te function

More information

Chapter 8. Numerical Solution of Ordinary Differential Equations. Module No. 2. Predictor-Corrector Methods

Chapter 8. Numerical Solution of Ordinary Differential Equations. Module No. 2. Predictor-Corrector Methods Numerical Analysis by Dr. Anita Pal Assistant Professor Department of Matematics National Institute of Tecnology Durgapur Durgapur-7109 email: anita.buie@gmail.com 1 . Capter 8 Numerical Solution of Ordinary

More information

Consider a function f we ll specify which assumptions we need to make about it in a minute. Let us reformulate the integral. 1 f(x) dx.

Consider a function f we ll specify which assumptions we need to make about it in a minute. Let us reformulate the integral. 1 f(x) dx. Capter 2 Integrals as sums and derivatives as differences We now switc to te simplest metods for integrating or differentiating a function from its function samples. A careful study of Taylor expansions

More information

(4.2) -Richardson Extrapolation

(4.2) -Richardson Extrapolation (.) -Ricardson Extrapolation. Small-O Notation: Recall tat te big-o notation used to define te rate of convergence in Section.: Suppose tat lim G 0 and lim F L. Te function F is said to converge to L as

More information

Hermite Interpolation

Hermite Interpolation Jim Lambers MAT 77 Fall Semester 010-11 Lecture Notes These notes correspond to Sections 4 and 5 in the text Hermite Interpolation Suppose that the interpolation points are perturbed so that two neighboring

More information

MATH745 Fall MATH745 Fall

MATH745 Fall MATH745 Fall MATH745 Fall 5 MATH745 Fall 5 INTRODUCTION WELCOME TO MATH 745 TOPICS IN NUMERICAL ANALYSIS Instructor: Dr Bartosz Protas Department of Matematics & Statistics Email: bprotas@mcmasterca Office HH 36, Ext

More information

Runge-Kutta methods. With orders of Taylor methods yet without derivatives of f (t, y(t))

Runge-Kutta methods. With orders of Taylor methods yet without derivatives of f (t, y(t)) Runge-Kutta metods Wit orders of Taylor metods yet witout derivatives of f (t, y(t)) First order Taylor expansion in two variables Teorem: Suppose tat f (t, y) and all its partial derivatives are continuous

More information

Floating Point Number Systems. Simon Fraser University Surrey Campus MACM 316 Spring 2005 Instructor: Ha Le

Floating Point Number Systems. Simon Fraser University Surrey Campus MACM 316 Spring 2005 Instructor: Ha Le Floating Point Number Systems Simon Fraser University Surrey Campus MACM 316 Spring 2005 Instructor: Ha Le 1 Overview Real number system Examples Absolute and relative errors Floating point numbers Roundoff

More information

NUMERICAL DIFFERENTIATION. James T. Smith San Francisco State University. In calculus classes, you compute derivatives algebraically: for example,

NUMERICAL DIFFERENTIATION. James T. Smith San Francisco State University. In calculus classes, you compute derivatives algebraically: for example, NUMERICAL DIFFERENTIATION James T Smit San Francisco State University In calculus classes, you compute derivatives algebraically: for example, f( x) = x + x f ( x) = x x Tis tecnique requires your knowing

More information

AMS 147 Computational Methods and Applications Lecture 09 Copyright by Hongyun Wang, UCSC. Exact value. Effect of round-off error.

AMS 147 Computational Methods and Applications Lecture 09 Copyright by Hongyun Wang, UCSC. Exact value. Effect of round-off error. Lecture 09 Copyrigt by Hongyun Wang, UCSC Recap: Te total error in numerical differentiation fl( f ( x + fl( f ( x E T ( = f ( x Numerical result from a computer Exact value = e + f x+ Discretization error

More information

Chapter 5 FINITE DIFFERENCE METHOD (FDM)

Chapter 5 FINITE DIFFERENCE METHOD (FDM) MEE7 Computer Modeling Tecniques in Engineering Capter 5 FINITE DIFFERENCE METHOD (FDM) 5. Introduction to FDM Te finite difference tecniques are based upon approximations wic permit replacing differential

More information

MAT 1339-S14 Class 2

MAT 1339-S14 Class 2 MAT 1339-S14 Class 2 July 07, 2014 Contents 1 Rate of Cange 1 1.5 Introduction to Derivatives....................... 1 2 Derivatives 5 2.1 Derivative of Polynomial function.................... 5 2.2 Te

More information

Exercises for numerical differentiation. Øyvind Ryan

Exercises for numerical differentiation. Øyvind Ryan Exercises for numerical differentiation Øyvind Ryan February 25, 2013 1. Mark eac of te following statements as true or false. a. Wen we use te approximation f (a) (f (a +) f (a))/ on a computer, we can

More information

Simpson s 1/3 Rule Simpson s 1/3 rule assumes 3 equispaced data/interpolation/integration points

Simpson s 1/3 Rule Simpson s 1/3 rule assumes 3 equispaced data/interpolation/integration points CE 05 - Lecture 5 LECTURE 5 UMERICAL ITEGRATIO COTIUED Simpson s / Rule Simpson s / rule assumes equispaced data/interpolation/integration points Te integration rule is based on approximating fx using

More information

First we will go over the following derivative rule. Theorem

First we will go over the following derivative rule. Theorem Tuesday, Feb 1 Tese slides will cover te following 1 d [cos(x)] = sin(x) iger-order derivatives 3 tangent line problems 4 basic differential equations First we will go over te following derivative rule

More information

LECTURE 14 NUMERICAL INTEGRATION. Find

LECTURE 14 NUMERICAL INTEGRATION. Find LECTURE 14 NUMERCAL NTEGRATON Find b a fxdx or b a vx ux fx ydy dx Often integration is required. However te form of fx may be suc tat analytical integration would be very difficult or impossible. Use

More information

158 Calculus and Structures

158 Calculus and Structures 58 Calculus and Structures CHAPTER PROPERTIES OF DERIVATIVES AND DIFFERENTIATION BY THE EASY WAY. Calculus and Structures 59 Copyrigt Capter PROPERTIES OF DERIVATIVES. INTRODUCTION In te last capter you

More information

5.1 introduction problem : Given a function f(x), find a polynomial approximation p n (x).

5.1 introduction problem : Given a function f(x), find a polynomial approximation p n (x). capter 5 : polynomial approximation and interpolation 5 introduction problem : Given a function f(x), find a polynomial approximation p n (x) Z b Z application : f(x)dx b p n(x)dx, a a one solution : Te

More information

MATH 155A FALL 13 PRACTICE MIDTERM 1 SOLUTIONS. needs to be non-zero, thus x 1. Also 1 +

MATH 155A FALL 13 PRACTICE MIDTERM 1 SOLUTIONS. needs to be non-zero, thus x 1. Also 1 + MATH 55A FALL 3 PRACTICE MIDTERM SOLUTIONS Question Find te domain of te following functions (a) f(x) = x3 5 x +x 6 (b) g(x) = x+ + x+ (c) f(x) = 5 x + x 0 (a) We need x + x 6 = (x + 3)(x ) 0 Hence Dom(f)

More information

1 Introduction to Optimization

1 Introduction to Optimization Unconstrained Convex Optimization 2 1 Introduction to Optimization Given a general optimization problem of te form min x f(x) (1.1) were f : R n R. Sometimes te problem as constraints (we are only interested

More information

A.P. CALCULUS (AB) Outline Chapter 3 (Derivatives)

A.P. CALCULUS (AB) Outline Chapter 3 (Derivatives) A.P. CALCULUS (AB) Outline Capter 3 (Derivatives) NAME Date Previously in Capter 2 we determined te slope of a tangent line to a curve at a point as te limit of te slopes of secant lines using tat point

More information

Numerical Methods. King Saud University

Numerical Methods. King Saud University Numerical Methods King Saud University Aims In this lecture, we will... find the approximate solutions of derivative (first- and second-order) and antiderivative (definite integral only). Numerical Differentiation

More information

Order of Accuracy. ũ h u Ch p, (1)

Order of Accuracy. ũ h u Ch p, (1) Order of Accuracy 1 Terminology We consider a numerical approximation of an exact value u. Te approximation depends on a small parameter, wic can be for instance te grid size or time step in a numerical

More information

Polynomial Functions. Linear Functions. Precalculus: Linear and Quadratic Functions

Polynomial Functions. Linear Functions. Precalculus: Linear and Quadratic Functions Concepts: definition of polynomial functions, linear functions tree representations), transformation of y = x to get y = mx + b, quadratic functions axis of symmetry, vertex, x-intercepts), transformations

More information

Numerical di erentiation. Quentin Louveaux (ULg - Institut Montefiore) Numerical analysis / 12

Numerical di erentiation. Quentin Louveaux (ULg - Institut Montefiore) Numerical analysis / 12 Numerical di erentiation Quentin Louveaux (ULg Institut Montefiore Numerical analysis 2018 1 / 12 Numerical di erentiation In general, symbolic di erentiation is very e Sometimes : no analytical form available

More information

Polynomial Interpolation

Polynomial Interpolation Capter 4 Polynomial Interpolation In tis capter, we consider te important problem of approximatinga function fx, wose values at a set of distinct points x, x, x,, x n are known, by a polynomial P x suc

More information

Section 2.7 Derivatives and Rates of Change Part II Section 2.8 The Derivative as a Function. at the point a, to be. = at time t = a is

Section 2.7 Derivatives and Rates of Change Part II Section 2.8 The Derivative as a Function. at the point a, to be. = at time t = a is Mat 180 www.timetodare.com Section.7 Derivatives and Rates of Cange Part II Section.8 Te Derivative as a Function Derivatives ( ) In te previous section we defined te slope of te tangent to a curve wit

More information

Lecture 21. Numerical differentiation. f ( x+h) f ( x) h h

Lecture 21. Numerical differentiation. f ( x+h) f ( x) h h Lecture Numerical differentiation Introduction We can analytically calculate te derivative of any elementary function, so tere migt seem to be no motivation for calculating derivatives numerically. However

More information

232 Calculus and Structures

232 Calculus and Structures 3 Calculus and Structures CHAPTER 17 JUSTIFICATION OF THE AREA AND SLOPE METHODS FOR EVALUATING BEAMS Calculus and Structures 33 Copyrigt Capter 17 JUSTIFICATION OF THE AREA AND SLOPE METHODS 17.1 THE

More information

Math 242: Principles of Analysis Fall 2016 Homework 7 Part B Solutions

Math 242: Principles of Analysis Fall 2016 Homework 7 Part B Solutions Mat 22: Principles of Analysis Fall 206 Homework 7 Part B Solutions. Sow tat f(x) = x 2 is not uniformly continuous on R. Solution. Te equation is equivalent to f(x) = 0 were f(x) = x 2 sin(x) 3. Since

More information

Polynomial Interpolation

Polynomial Interpolation Capter 4 Polynomial Interpolation In tis capter, we consider te important problem of approximating a function f(x, wose values at a set of distinct points x, x, x 2,,x n are known, by a polynomial P (x

More information

Convexity and Smoothness

Convexity and Smoothness Capter 4 Convexity and Smootness 4.1 Strict Convexity, Smootness, and Gateaux Differentiablity Definition 4.1.1. Let X be a Banac space wit a norm denoted by. A map f : X \{0} X \{0}, f f x is called a

More information

We name Functions f (x) or g(x) etc.

We name Functions f (x) or g(x) etc. Section 2 1B: Function Notation Bot of te equations y 2x +1 and y 3x 2 are functions. It is common to ave two or more functions in terms of x in te same problem. If I ask you wat is te value for y if x

More information

MATH1901 Differential Calculus (Advanced)

MATH1901 Differential Calculus (Advanced) MATH1901 Dierential Calculus (Advanced) Capter 3: Functions Deinitions : A B A and B are sets assigns to eac element in A eactl one element in B A is te domain o te unction B is te codomain o te unction

More information

Chapter 4: Numerical Methods for Common Mathematical Problems

Chapter 4: Numerical Methods for Common Mathematical Problems 1 Capter 4: Numerical Metods for Common Matematical Problems Interpolation Problem: Suppose we ave data defined at a discrete set of points (x i, y i ), i = 0, 1,..., N. Often it is useful to ave a smoot

More information

Math Spring 2013 Solutions to Assignment # 3 Completion Date: Wednesday May 15, (1/z) 2 (1/z 1) 2 = lim

Math Spring 2013 Solutions to Assignment # 3 Completion Date: Wednesday May 15, (1/z) 2 (1/z 1) 2 = lim Mat 311 - Spring 013 Solutions to Assignment # 3 Completion Date: Wednesday May 15, 013 Question 1. [p 56, #10 (a)] 4z Use te teorem of Sec. 17 to sow tat z (z 1) = 4. We ave z 4z (z 1) = z 0 4 (1/z) (1/z

More information

Parameter Fitted Scheme for Singularly Perturbed Delay Differential Equations

Parameter Fitted Scheme for Singularly Perturbed Delay Differential Equations International Journal of Applied Science and Engineering 2013. 11, 4: 361-373 Parameter Fitted Sceme for Singularly Perturbed Delay Differential Equations Awoke Andargiea* and Y. N. Reddyb a b Department

More information

Math 212-Lecture 9. For a single-variable function z = f(x), the derivative is f (x) = lim h 0

Math 212-Lecture 9. For a single-variable function z = f(x), the derivative is f (x) = lim h 0 3.4: Partial Derivatives Definition Mat 22-Lecture 9 For a single-variable function z = f(x), te derivative is f (x) = lim 0 f(x+) f(x). For a function z = f(x, y) of two variables, to define te derivatives,

More information

FINITE DIFFERENCE APPROXIMATIONS FOR NONLINEAR FIRST ORDER PARTIAL DIFFERENTIAL EQUATIONS

FINITE DIFFERENCE APPROXIMATIONS FOR NONLINEAR FIRST ORDER PARTIAL DIFFERENTIAL EQUATIONS UNIVERSITATIS IAGELLONICAE ACTA MATHEMATICA, FASCICULUS XL 2002 FINITE DIFFERENCE APPROXIMATIONS FOR NONLINEAR FIRST ORDER PARTIAL DIFFERENTIAL EQUATIONS by Anna Baranowska Zdzis law Kamont Abstract. Classical

More information

Convexity and Smoothness

Convexity and Smoothness Capter 4 Convexity and Smootness 4. Strict Convexity, Smootness, and Gateaux Di erentiablity Definition 4... Let X be a Banac space wit a norm denoted by k k. A map f : X \{0}!X \{0}, f 7! f x is called

More information

Function Composition and Chain Rules

Function Composition and Chain Rules Function Composition and s James K. Peterson Department of Biological Sciences and Department of Matematical Sciences Clemson University Marc 8, 2017 Outline 1 Function Composition and Continuity 2 Function

More information

Math 161 (33) - Final exam

Math 161 (33) - Final exam Name: Id #: Mat 161 (33) - Final exam Fall Quarter 2015 Wednesday December 9, 2015-10:30am to 12:30am Instructions: Prob. Points Score possible 1 25 2 25 3 25 4 25 TOTAL 75 (BEST 3) Read eac problem carefully.

More information

Lecture XVII. Abstract We introduce the concept of directional derivative of a scalar function and discuss its relation with the gradient operator.

Lecture XVII. Abstract We introduce the concept of directional derivative of a scalar function and discuss its relation with the gradient operator. Lecture XVII Abstract We introduce te concept of directional derivative of a scalar function and discuss its relation wit te gradient operator. Directional derivative and gradient Te directional derivative

More information

Chapter 1. Density Estimation

Chapter 1. Density Estimation Capter 1 Density Estimation Let X 1, X,..., X n be observations from a density f X x. Te aim is to use only tis data to obtain an estimate ˆf X x of f X x. Properties of f f X x x, Parametric metods f

More information

These error are made from replacing an infinite process by finite one.

These error are made from replacing an infinite process by finite one. Introduction :- Tis course examines problems tat can be solved by metods of approximation, tecniques we call numerical metods. We begin by considering some of te matematical and computational topics tat

More information

Key Concepts. Important Techniques. 1. Average rate of change slope of a secant line. You will need two points ( a, the formula: to find value

Key Concepts. Important Techniques. 1. Average rate of change slope of a secant line. You will need two points ( a, the formula: to find value AB Calculus Unit Review Key Concepts Average and Instantaneous Speed Definition of Limit Properties of Limits One-sided and Two-sided Limits Sandwic Teorem Limits as x ± End Beaviour Models Continuity

More information

Numerical Algorithms. IE 496 Lecture 20

Numerical Algorithms. IE 496 Lecture 20 Numerical Algorithms IE 496 Lecture 20 Reading for This Lecture Primary Miller and Boxer, Pages 124-128 Forsythe and Mohler, Sections 1 and 2 Numerical Algorithms Numerical Analysis So far, we have looked

More information

Copyright c 2008 Kevin Long

Copyright c 2008 Kevin Long Lecture 4 Numerical solution of initial value problems Te metods you ve learned so far ave obtained closed-form solutions to initial value problems. A closedform solution is an explicit algebriac formula

More information

Bob Brown Math 251 Calculus 1 Chapter 3, Section 1 Completed 1 CCBC Dundalk

Bob Brown Math 251 Calculus 1 Chapter 3, Section 1 Completed 1 CCBC Dundalk Bob Brown Mat 251 Calculus 1 Capter 3, Section 1 Completed 1 Te Tangent Line Problem Te idea of a tangent line first arises in geometry in te context of a circle. But before we jump into a discussion of

More information

ACCESS TO SCIENCE, ENGINEERING AND AGRICULTURE: MATHEMATICS 1 MATH00030 SEMESTER /2019

ACCESS TO SCIENCE, ENGINEERING AND AGRICULTURE: MATHEMATICS 1 MATH00030 SEMESTER /2019 ACCESS TO SCIENCE, ENGINEERING AND AGRICULTURE: MATHEMATICS MATH00030 SEMESTER 208/209 DR. ANTHONY BROWN 6. Differential Calculus 6.. Differentiation from First Principles. In tis capter, we will introduce

More information

Applications of the van Trees inequality to non-parametric estimation.

Applications of the van Trees inequality to non-parametric estimation. Brno-06, Lecture 2, 16.05.06 D/Stat/Brno-06/2.tex www.mast.queensu.ca/ blevit/ Applications of te van Trees inequality to non-parametric estimation. Regular non-parametric problems. As an example of suc

More information

Differential Calculus (The basics) Prepared by Mr. C. Hull

Differential Calculus (The basics) Prepared by Mr. C. Hull Differential Calculus Te basics) A : Limits In tis work on limits, we will deal only wit functions i.e. tose relationsips in wic an input variable ) defines a unique output variable y). Wen we work wit

More information

= h. Geometrically this quantity represents the slope of the secant line connecting the points

= h. Geometrically this quantity represents the slope of the secant line connecting the points Section 3.7: Rates of Cange in te Natural and Social Sciences Recall: Average rate of cange: y y y ) ) ), ere Geometrically tis quantity represents te slope of te secant line connecting te points, f (

More information

Using the definition of the derivative of a function is quite tedious. f (x + h) f (x)

Using the definition of the derivative of a function is quite tedious. f (x + h) f (x) Derivative Rules Using te efinition of te erivative of a function is quite teious. Let s prove some sortcuts tat we can use. Recall tat te efinition of erivative is: Given any number x for wic te limit

More information

1 Lecture 13: The derivative as a function.

1 Lecture 13: The derivative as a function. 1 Lecture 13: Te erivative as a function. 1.1 Outline Definition of te erivative as a function. efinitions of ifferentiability. Power rule, erivative te exponential function Derivative of a sum an a multiple

More information

Solutions to the Multivariable Calculus and Linear Algebra problems on the Comprehensive Examination of January 31, 2014

Solutions to the Multivariable Calculus and Linear Algebra problems on the Comprehensive Examination of January 31, 2014 Solutions to te Multivariable Calculus and Linear Algebra problems on te Compreensive Examination of January 3, 24 Tere are 9 problems ( points eac, totaling 9 points) on tis portion of te examination.

More information

HOMEWORK HELP 2 FOR MATH 151

HOMEWORK HELP 2 FOR MATH 151 HOMEWORK HELP 2 FOR MATH 151 Here we go; te second round of omework elp. If tere are oters you would like to see, let me know! 2.4, 43 and 44 At wat points are te functions f(x) and g(x) = xf(x)continuous,

More information

. Compute the following limits.

. Compute the following limits. Today: Tangent Lines and te Derivative at a Point Warmup:. Let f(x) =x. Compute te following limits. f( + ) f() (a) lim f( +) f( ) (b) lim. Let g(x) = x. Compute te following limits. g(3 + ) g(3) (a) lim

More information

LIMITATIONS OF EULER S METHOD FOR NUMERICAL INTEGRATION

LIMITATIONS OF EULER S METHOD FOR NUMERICAL INTEGRATION LIMITATIONS OF EULER S METHOD FOR NUMERICAL INTEGRATION LAURA EVANS.. Introduction Not all differential equations can be explicitly solved for y. Tis can be problematic if we need to know te value of y

More information

Logarithmic functions

Logarithmic functions Roberto s Notes on Differential Calculus Capter 5: Derivatives of transcendental functions Section Derivatives of Logaritmic functions Wat ou need to know alread: Definition of derivative and all basic

More information

Finite Difference Method

Finite Difference Method Capter 8 Finite Difference Metod 81 2nd order linear pde in two variables General 2nd order linear pde in two variables is given in te following form: L[u] = Au xx +2Bu xy +Cu yy +Du x +Eu y +Fu = G According

More information

MAT 145. Type of Calculator Used TI-89 Titanium 100 points Score 100 possible points

MAT 145. Type of Calculator Used TI-89 Titanium 100 points Score 100 possible points MAT 15 Test #2 Name Solution Guide Type of Calculator Used TI-89 Titanium 100 points Score 100 possible points Use te grap of a function sown ere as you respond to questions 1 to 8. 1. lim f (x) 0 2. lim

More information

5 Ordinary Differential Equations: Finite Difference Methods for Boundary Problems

5 Ordinary Differential Equations: Finite Difference Methods for Boundary Problems 5 Ordinary Differential Equations: Finite Difference Metods for Boundary Problems Read sections 10.1, 10.2, 10.4 Review questions 10.1 10.4, 10.8 10.9, 10.13 5.1 Introduction In te previous capters we

More information

Section 3.1: Derivatives of Polynomials and Exponential Functions

Section 3.1: Derivatives of Polynomials and Exponential Functions Section 3.1: Derivatives of Polynomials and Exponential Functions In previous sections we developed te concept of te derivative and derivative function. Te only issue wit our definition owever is tat it

More information

Chapter Seven The Quantum Mechanical Simple Harmonic Oscillator

Chapter Seven The Quantum Mechanical Simple Harmonic Oscillator Capter Seven Te Quantum Mecanical Simple Harmonic Oscillator Introduction Te potential energy function for a classical, simple armonic oscillator is given by ZÐBÑ œ 5B were 5 is te spring constant. Suc

More information

Math 112 Rahman. Week Taylor Series Suppose the function f has the following power series:

Math 112 Rahman. Week Taylor Series Suppose the function f has the following power series: Math Rahman Week 0.8-0.0 Taylor Series Suppose the function f has the following power series: fx) c 0 + c x a) + c x a) + c 3 x a) 3 + c n x a) n. ) Can we figure out what the coefficients are? Yes, yes

More information

Math 1210 Midterm 1 January 31st, 2014

Math 1210 Midterm 1 January 31st, 2014 Mat 110 Midterm 1 January 1st, 01 Tis exam consists of sections, A and B. Section A is conceptual, wereas section B is more computational. Te value of every question is indicated at te beginning of it.

More information

Differential equations. Differential equations

Differential equations. Differential equations Differential equations A differential equation (DE) describes ow a quantity canges (as a function of time, position, ) d - A ball dropped from a building: t gt () dt d S qx - Uniformly loaded beam: wx

More information

Name: Answer Key No calculators. Show your work! 1. (21 points) All answers should either be,, a (finite) real number, or DNE ( does not exist ).

Name: Answer Key No calculators. Show your work! 1. (21 points) All answers should either be,, a (finite) real number, or DNE ( does not exist ). Mat - Final Exam August 3 rd, Name: Answer Key No calculators. Sow your work!. points) All answers sould eiter be,, a finite) real number, or DNE does not exist ). a) Use te grap of te function to evaluate

More information

Jim Lambers MAT 460/560 Fall Semester Practice Final Exam

Jim Lambers MAT 460/560 Fall Semester Practice Final Exam Jim Lambers MAT 460/560 Fall Semester 2009-10 Practice Final Exam 1. Let f(x) = sin 2x + cos 2x. (a) Write down the 2nd Taylor polynomial P 2 (x) of f(x) centered around x 0 = 0. (b) Write down the corresponding

More information

Physically Based Modeling: Principles and Practice Implicit Methods for Differential Equations

Physically Based Modeling: Principles and Practice Implicit Methods for Differential Equations Pysically Based Modeling: Principles and Practice Implicit Metods for Differential Equations David Baraff Robotics Institute Carnegie Mellon University Please note: Tis document is 997 by David Baraff

More information

3.4 Worksheet: Proof of the Chain Rule NAME

3.4 Worksheet: Proof of the Chain Rule NAME Mat 1170 3.4 Workseet: Proof of te Cain Rule NAME Te Cain Rule So far we are able to differentiate all types of functions. For example: polynomials, rational, root, and trigonometric functions. We are

More information

2.8 The Derivative as a Function

2.8 The Derivative as a Function .8 Te Derivative as a Function Typically, we can find te derivative of a function f at many points of its domain: Definition. Suppose tat f is a function wic is differentiable at every point of an open

More information

Chapter 5 Differentiation

Chapter 5 Differentiation Capter 5 Differentiation Course Title: Real Analsis 1 Course Code: MTH31 Course instrutor: Dr Atiq ur Reman Class: MS-II Course URL: wwwmatitorg/atiq/fa15-mt31 Derivative of a funtion: Let f be defined

More information

The total error in numerical differentiation

The total error in numerical differentiation AMS 147 Computational Metods and Applications Lecture 08 Copyrigt by Hongyun Wang, UCSC Recap: Loss of accuracy due to numerical cancellation A B 3, 3 ~10 16 In calculating te difference between A and

More information

Polynomial Interpolation with n + 1 nodes

Polynomial Interpolation with n + 1 nodes Polynomial Interpolation with n + 1 nodes Given n + 1 distinct points (x 0, f (x 0 )), (x 1, f (x 1 )),, (x n, f (x n )), Interpolating polynomial of degree n P(x) = f (x 0 )L 0 (x) + f (x 1 )L 1 (x) +

More information

SECTION 3.2: DERIVATIVE FUNCTIONS and DIFFERENTIABILITY

SECTION 3.2: DERIVATIVE FUNCTIONS and DIFFERENTIABILITY (Section 3.2: Derivative Functions and Differentiability) 3.2.1 SECTION 3.2: DERIVATIVE FUNCTIONS and DIFFERENTIABILITY LEARNING OBJECTIVES Know, understand, and apply te Limit Definition of te Derivative

More information

LIMITS AND DERIVATIVES CONDITIONS FOR THE EXISTENCE OF A LIMIT

LIMITS AND DERIVATIVES CONDITIONS FOR THE EXISTENCE OF A LIMIT LIMITS AND DERIVATIVES Te limit of a function is defined as te value of y tat te curve approaces, as x approaces a particular value. Te limit of f (x) as x approaces a is written as f (x) approaces, as

More information

DEFINITION OF A DERIVATIVE

DEFINITION OF A DERIVATIVE DEFINITION OF A DERIVATIVE Section 2.1 Calculus AP/Dual, Revised 2017 viet.dang@umbleisd.net 2.1: Definition of a Derivative 1 DEFINITION A. Te derivative of a function allows you to find te SLOPE OF THE

More information

Poisson Equation in Sobolev Spaces

Poisson Equation in Sobolev Spaces Poisson Equation in Sobolev Spaces OcMountain Dayligt Time. 6, 011 Today we discuss te Poisson equation in Sobolev spaces. It s existence, uniqueness, and regularity. Weak Solution. u = f in, u = g on

More information

Exponential and logarithmic functions (pp ) () Supplement October 14, / 1. a and b positive real numbers and x and y real numbers.

Exponential and logarithmic functions (pp ) () Supplement October 14, / 1. a and b positive real numbers and x and y real numbers. MA123, Supplement Exponential and logaritmic functions pp. 315-319) Capter s Goal: Review te properties of exponential and logaritmic functions. Learn ow to differentiate exponential and logaritmic functions.

More information

Computer Derivations of Numerical Differentiation Formulae. Int. J. of Math. Education in Sci. and Tech., V 34, No 2 (March-April 2003), pp

Computer Derivations of Numerical Differentiation Formulae. Int. J. of Math. Education in Sci. and Tech., V 34, No 2 (March-April 2003), pp Computer Derivations o Numerical Dierentiation Formulae By Jon H. Matews Department o Matematics Caliornia State University Fullerton USA Int. J. o Mat. Education in Sci. and Tec. V No (Marc-April ) pp.8-87.

More information

Analytic Functions. Differentiable Functions of a Complex Variable

Analytic Functions. Differentiable Functions of a Complex Variable Analytic Functions Differentiable Functions of a Complex Variable In tis capter, we sall generalize te ideas for polynomials power series of a complex variable we developed in te previous capter to general

More information

1 The concept of limits (p.217 p.229, p.242 p.249, p.255 p.256) 1.1 Limits Consider the function determined by the formula 3. x since at this point

1 The concept of limits (p.217 p.229, p.242 p.249, p.255 p.256) 1.1 Limits Consider the function determined by the formula 3. x since at this point MA00 Capter 6 Calculus and Basic Linear Algebra I Limits, Continuity and Differentiability Te concept of its (p.7 p.9, p.4 p.49, p.55 p.56). Limits Consider te function determined by te formula f Note

More information

The Derivative as a Function

The Derivative as a Function Section 2.2 Te Derivative as a Function 200 Kiryl Tsiscanka Te Derivative as a Function DEFINITION: Te derivative of a function f at a number a, denoted by f (a), is if tis limit exists. f (a) f(a + )

More information

0.1 The Slope of Nonlinear Function

0.1 The Slope of Nonlinear Function WEEK Reading [SB],.3-.7, pp. -38 0. Te Slope of Nonlinear Function If we want approimate a nonlinear function y = f() by a linear one around some point 0, te best approimation is te line tangent to te

More information

MVT and Rolle s Theorem

MVT and Rolle s Theorem AP Calculus CHAPTER 4 WORKSHEET APPLICATIONS OF DIFFERENTIATION MVT and Rolle s Teorem Name Seat # Date UNLESS INDICATED, DO NOT USE YOUR CALCULATOR FOR ANY OF THESE QUESTIONS In problems 1 and, state

More information

Notation Nodes are data points at which functional values are available or at which you wish to compute functional values At the nodes fx i

Notation Nodes are data points at which functional values are available or at which you wish to compute functional values At the nodes fx i LECTURE 6 NUMERICAL DIFFERENTIATION To find discrete approximations to differentiation (since computers can only deal with functional values at discrete points) Uses of numerical differentiation To represent

More information

160 Chapter 3: Differentiation

160 Chapter 3: Differentiation 3. Differentiation Rules 159 3. Differentiation Rules Tis section introuces a few rules tat allow us to ifferentiate a great variety of functions. By proving tese rules ere, we can ifferentiate functions

More information

MAT01A1: Differentiation of Polynomials & Exponential Functions + the Product & Quotient Rules

MAT01A1: Differentiation of Polynomials & Exponential Functions + the Product & Quotient Rules MAT01A1: Differentiation of Polynomials & Exponential Functions + te Prouct & Quotient Rules Dr Craig 22 Marc 2017 Semester Test 1 Scripts will be available for collection from Tursay morning. For marking

More information

Differential Equations

Differential Equations Pysics-based simulation xi Differential Equations xi+1 xi xi+1 xi + x x Pysics-based simulation xi Wat is a differential equation? Differential equations describe te relation between an unknown function

More information

Chapter Primer on Differentiation

Chapter Primer on Differentiation Capter 0.01 Primer on Differentiation After reaing tis capter, you soul be able to: 1. unerstan te basics of ifferentiation,. relate te slopes of te secant line an tangent line to te erivative of a function,.

More information

DELFT UNIVERSITY OF TECHNOLOGY Faculty of Electrical Engineering, Mathematics and Computer Science

DELFT UNIVERSITY OF TECHNOLOGY Faculty of Electrical Engineering, Mathematics and Computer Science DELFT UNIVERSITY OF TECHNOLOGY Faculty of Electrical Engineering, Matematics and Computer Science. ANSWERS OF THE TEST NUMERICAL METHODS FOR DIFFERENTIAL EQUATIONS (WI3097 TU) Tuesday January 9 008, 9:00-:00

More information

Section 15.6 Directional Derivatives and the Gradient Vector

Section 15.6 Directional Derivatives and the Gradient Vector Section 15.6 Directional Derivatives and te Gradient Vector Finding rates of cange in different directions Recall tat wen we first started considering derivatives of functions of more tan one variable,

More information

lecture 26: Richardson extrapolation

lecture 26: Richardson extrapolation 43 lecture 26: Ricardson extrapolation 35 Ricardson extrapolation, Romberg integration Trougout numerical analysis, one encounters procedures tat apply some simple approximation (eg, linear interpolation)

More information

University Mathematics 2

University Mathematics 2 University Matematics 2 1 Differentiability In tis section, we discuss te differentiability of functions. Definition 1.1 Differentiable function). Let f) be a function. We say tat f is differentiable at

More information

Click here to see an animation of the derivative

Click here to see an animation of the derivative Differentiation Massoud Malek Derivative Te concept of derivative is at te core of Calculus; It is a very powerful tool for understanding te beavior of matematical functions. It allows us to optimize functions,

More information

CS522 - Partial Di erential Equations

CS522 - Partial Di erential Equations CS5 - Partial Di erential Equations Tibor Jánosi April 5, 5 Numerical Di erentiation In principle, di erentiation is a simple operation. Indeed, given a function speci ed as a closed-form formula, its

More information