Solutions_3. 1 Exercise Exercise January 26, 2017
|
|
- ÍÊ Ζάρκος
- 6 χρόνια πριν
- Προβολές:
Transcript
1 s_3 Jnury 26, Exercise Apply composite Simpson s rule with m = 1, 2, 4 pnels to pproximte the integrls: () x 2 dx = 1 π/2 3, (b) cos(x) dx = 1, (c) e x dx = e 1, nd report the errors. () f(x) = x 2. For m = 1 we put h = (1 )/2 = 1/2, x =, x 1 = 1/2, x 2 = 1; y = f(x ) =, y 1 = f(x 1 ) = 1/4, y 2 = f(x 2 ) = 1. The pproximtion is then I h/3(y + 4y 1 + y 2 ) = ( + 4 1/4 + 1)/6 = 1/3. The error is thus. For m = 2 we hve h = (1 )/4 = 1/4, (x, x 1,..., x 4 ) = (, 1/4, 1/2, 3/4, 1), (y, y 1,..., y 4 ) = (, 1/16, 1/4, 9/16, 1). The pproximtion is then I h/3(y +4y 1 +2y 2 +4y 3 +y 4 ) = (+4 1/ / /16 + 1)/12 = (1/4 + 2/4 + 9/4 + 4/4)/12 = 16/(3 4 4) = 1/3. The error is thus. One could continue with m = 3, but the error hs to be zero. Indeed, Simpson s rule is bsed on qudrtic interpoltion polynomils, which mens tht x 2 will be represented exctly by the interpolting polynomil nd the qudrture will be exct. (b) f(x) = cos(x) For m = 1 we put h = (π/2 )/2 = π/4, x =, x 1 = π/4, x 2 = π/2; y = f(x ) = 1, y 1 = f(x 1 ) = 1/ 2, y 2 = f(x 2 ) =. The pproximtion is then I h/3(y + 4y 1 + y 2 ) = (1 + 4/ 2 + ) π/ The error is thus.227. For m = 2, h = π/8, (x,..., x 4 ) = (, π/8, π/4, 3π/8, π/2), (y,..., y 4 ) (1, , 1/ 2, , ). I h/3(y + 4y 1 + 2y 2 + 4y 3 + y 4 ) 1.135, giving the error.135. For m = 3, h = π/6, I , errror (c) f(x) = exp(x) For m = 1 we put h = (1 )/2 = 1/2, x =, x 1 = 1/2, x 2 = 1; y = f(x ) = 1, y 1 = f(x 1 ) = e 1/2, y 2 = f(x 2 ) = e. The pproximtion is then I h/3(y + 4y 1 + y 2 ) = (1 + 4 e + e)/ The error is thus e For m = 2, h = 1/4, (x,..., x 4 ) = (, 1/4, 1/2, 3/4, 1),... I h/3(y + 4y 1 + 2y 2 + 4y 3 + y 4 ) , giving the error For m = 3, h = 1/6, I , errror Exercise Find the degree of precision of the following pproximtion for 1 f(x) dx: 1
2 First of ll, let us evlute the integrl of monomil x n, n : [ ] { x x n n+1 x=1, for odd n, dx = = n + 1 2/(n + 1), for even n. 1 x= 1 () f(1) + f( 1): Let us pply the qudrture to the monomil x n, n : { 1 n + ( 1) n, for odd n, = 2, for even n, Therefore the qudrture grees with the integrl for n =, 1 (nd ll odd n), nd its degree of precision is 1. (One could lso observe tht this is trpezoid qudrture on [ 1, 1], which is known to be exct for polynomils up to degree 1.) (b) 2/3[f(-1) + f() + f(1)]. We check the qudrture on monomils of incresing degree: 2, for n =, 2/3[( 1) n + n + ( 1) n ] =, for odd n, 4/3, for even n >, which grees with the integrl for n =, 1 (nd odd n). Thus degree of precision is 1. (c) We check the qudrture on monomils of incresing degree: [( 1/ 3) n + (1/ { 3) n, for odd n, ] = 2/3 n/2, for even n, which grees with the integrl for n =, 1, 2, 3 (nd odd n). Thus degree of precision is 3. (This is n exmple of Gussin qudrture, which re exct for polynomils of degree up to 2n 1 when n points re used.) 3 Exercise Use the fct tht the error term of Boole s Rule (see Exercise ) is proportionl to f (6) (c) to find the exct error term. We pply the qudrture to the monomil x 6, for which f (6) (c) 6!, for ll c. Thus 4h [ x x 6 7 dx = 7 ] x=4h x= = 47 h 7 7 = 2h 45 ( h (2h) (3h) (4h) 6 ) + C6! nd we re interested in finding C. In [1]: from sympy import * init_printing() h,c=symbols('h C') integrl = (4*h)**7/7 qudrture = 2*h/45*(32*h**6 + 12*(2*h)**6 + 32*(3*h)**6 + 7*(4*h)**6) simplify(integrl-qudrture) 2
3 Out[1]: Thus C = 128h 7 /(21 6!) = 8h 7 / h Exercise Wee need to find c 1, c 2, c 3 such tht the rule hs degree of precision greter thn one. To hve degree of precision we need tht f(x) dx c 1 f() + c 2 f(.5) + c 3 f(1) 1 dx = 1 = c 1 f() + c 2 f(.5) + c 3 f(1) = c 1 + c 2 + c 3. Similrly, for degree of precision 1 we need x dx = 1/2 = c 1 f() + c 2 f(.5) + c 3 f(1) =.5c 2 + c 3. Finlly, for degree of precision 2 we need x 2 dx = 1/3 = c 1 f() + c 2 f(.5) + c 3 f(1) =.25c 2 + c 3. This gives us 3 liner equtions in 3 unknowns, whose solution is c 1 = c 3 = 1/6, c 2 = 2/3. This is the sme s Simpson s rule. 5 Exercise 3b), Exm L M [,b] f og T [,b] f være midpunkt og trpezoid kvdrturer med n = 1 pnel for funksjonen f på intervl [, b]. Feilestimtetene for disse kvdrturer er gitt v f(x) dx = M [,b] f + h3 24 f (c) + O(h 4 ), og f(x) dx = T [,b] f h3 12 f (c) + O(h 4 ), hvor c = ( + b)/2, og h = b. L oss definere en ny kvdrtur som Q [,b] f = αm [,b] f + βt [,b] f. Bestem verdiene α, β slik t : We hve f(x) dx = Q [,b] f + O(h 4 ). 3
4 nd therefore M [,b] f = T [,b] f = f(x) dx h3 24 f (c) + O(h 4 ), og f(x) dx + h3 12 f (c) + O(h 4 ), Q [,b] f = (α + β) As result we get system of equtions f(x) dx + (2β α) h O(h4 ). α + β = 1, 2β α =. Thus α = 2/3, β = 1/3, which in fct gives us Simpson s rule: 6 Exercise M [,b]f T [,b]f = 2h 3 f(c) + h 6 (f() + f(b)) = h [f() + 4f(c) + f(b)]. 6 Apply Adptive Qudrture by hnd, using the Trpezoid Rule with tolernce T OL =.5 to pproximte the integrls. Find the pproximtion error. () In this cse, f(x) = x 2, =, b = 1. We begin with n = 1 intervl, =, b = b. We will use S[, b] to denote the Trpezoid qudrture pplied on the intervl [, b]. c = ( + b)/2, S[, b] = ( )/2 = 1/2, S[, c] = ( 2 + (1/2) 2 )/4 = 1/16, S[c, b] = ((1/2) )/4 = 5/16. S[, b] S[, c] S[c, b] = 1/8 =.125 < 3.5 ((b )/(b )) =.15. Thus we stop with the pproximtion S[, c] + S[c, b] = 3/8 =.375 nd n error estimte (S[, b] S[, c] S[c, b])/3 = 1/ The ctul integrtion error is x2 3/8 = 1/3 3/8 = 1/24. Our estimte is exct becuse f is constnt in this cse, thus the pproximte eqution (5.38) in the book is ctully exct. (b) f(x) = cos(x), =, b = π/2. S[, π/2] = (π/2) (1 + )/2 = π/ , S[, π/4] = (π/4) (1 + 1/ 2)/2.7854, S[π/4, π, 2] = (π/4) (1/ 2 + )/ S[, b] S[, c] S[c, b] (π/2)/(π/2) =.15. Thus we need to split the intervl nd pply the dptive lgorithm recursively. S[, π/4] = (π/4) (1 + 1/ 2)/2.7854, S[, π/8] = (π/8) (1 + cos(π/8))/ , S[π/8, π/4] = (π/8) (cos(π/8) + cos(π/4))/2.3224, S[, π/4] S[, π/8] S[π/8, π/4] (π/4)/(π/2) =.75. Thus we need to split this intervl gin pply the dptive lgorithm recursively. S[, π/8] = (π/8) (1+cos(π/8))/ , S[, π/16] = (π/16) (1+cos(π/16))/ , S[π/16, π/8] = (π/16) (cos(π/16)+cos(π/8))/ , S[, π/8] S[, π/16] S[π/16, π/8].37 < 3.5 (π/8)/(π/2) =.375. Thus we re done on the intervl [, π/8] with the pproximtion S[, π/16] + S[π/16, π/8]
5 Let us look t the intervl [π/8, π/4] now. S[π/8, π/4] = (π/8) (cos(π/8) + cos(pi/4))/2.3224, S[π/8, 3π/16] = (π/16) (cos(π/8) + cos(3π/16))/ , S[3π/16, π/4] = (π/16) (cos(3π/16) + cos(π/4))/2.1515, S[π/8, π/4] S[π/8, 3π/16] S[3π/16, π/4].314 < 3.5 (π/8)/(π/2) =.375. Thus we re done on the intervl [π/8, π/4] with the pproximtion S[π/8, 3π/16] + S[3π/16, π/4] Let us look t the intervl [π/4, π/2] now. S[π/4, π/2] = (π/4) (cos(π/4) + cos(pi/2))/ , S[π/4, 3π/8] = (π/8) (cos(π/4) + cos(3π/8))/ , S[3π/8, π/2] = (π/8) (cos(3π/8) + cos(π/2))/2.7514, S[π/4, π/2] S[π/4, 3π/8] S[3π/8, π/2].1144 < 3.5 (π/4)/(π/2) =.75. Thus we re done on the intervl [π/4, π/2] with the pproximtion S[π/4, 3π/8] + S[3π/8, π/2] There re no intervls left, nd therefore the finl pproximtion is = wheres the exct integrl is π/2 cos(x) = 1. Thus the error is <.5. (c) In this cse, f(x) = exp(x), =, b = 1. S[, 1] = (1 + e)/ , S[,.5] = (1 + exp(.5))/ , S[.5, 1] = (exp(.5) + exp(1))/ S[, b] S[, c] S[c, b].152 < 3.5 ((b )/(b )) =.15. Thus we stop with the pproximtion S[, c] + S[c, b] nd n error estimte (S[, b] S[, c] S[c, b])/3.35. The ctul integrtion error is exp(x) (S[, c] + S[c, b]).36. Our estimte is quite good in this cse. 7 Exercise Develop n Adptive Qudrture method for rule (5.28). Let c = ( + b)/2, h = b nd pply the qudrture on [, b], [, c], nd [c, b] (we use the sme nottion s in the book): c f(x) dx + c f(x) dx = S[, b] + 14h5 45 f (4) (c 1 ), c 1 [, b], f(x) dx = S[, c] + S[c, b] + 14(h/2)5 45 f (4) (c 2 ) + 14(h/2)5 f (4) (c 3 ), 45 c 2 [, c], c3 [c, b]. Further ssuming tht f (4) (c 1 ) f (4) (c 2 ) f (4) (c 3 ) we obtin: c f(x) dx + c f(x) dx = S[, b] + 14h5 45 f (4) (c 1 ), f(x) dx S[, c] + S[c, b] h f (4) (c 1 ). We cn now subtrct the second eqution from the first to obtin: S[, b] S[, c] S[c, b] h f (4) (c 1 ). 5
6 Thus the number S[, b] S[, c] S[c, b] gives n pproximtion to 15 times the error of the qudrture S[, c] + S[c, b]. The rest is exctly the sme s for other qudrtures; one stops subdividing the intervl when S[, b] S[, c] S[c, b] < 15 T OL (b )/(b orig orig ) nd returns S[, c] + S[c, b] s the pproximtion of the integrl over the intervl [,b]. 8 Computer exercise Answers: () (b) (c) Computer exercise See the file uniform_refinement.py vilble on the wiki. Adpt the code to use Simpson s rule insted of Trpezoid. 6
Solutions 3. February 2, Apply composite Simpson s rule with m = 1, 2, 4 panels to approximate the integrals:
s Februry 2, 216 1 Exercise 5.2. Apply composite Simpson s rule with m = 1, 2, 4 pnels to pproximte the integrls: () x 2 dx = 1 π/2, (b) cos(x) dx = 1, (c) e x dx = e 1, nd report the errors. () f(x) =
Διαβάστε περισσότεραLecture 5: Numerical Integration
Lecture notes on Vritionl nd Approximte Metods in Applied Mtemtics - A Peirce UBC 1 Lecture 5: Numericl Integrtion Compiled 15 September 1 In tis lecture we introduce tecniques for numericl integrtion,
Διαβάστε περισσότεραOscillatory integrals
Oscilltory integrls Jordn Bell jordn.bell@gmil.com Deprtment of Mthemtics, University of Toronto August, 0 Oscilltory integrls Suppose tht Φ C R d ), ψ DR d ), nd tht Φ is rel-vlued. I : 0, ) C by Iλ)
Διαβάστε περισσότεραReview-2 and Practice problems. sin 2 (x) cos 2 (x)(sin(x)dx) (1 cos 2 (x)) cos 2 (x)(sin(x)dx) let u = cos(x), du = sin(x)dx. = (1 u 2 )u 2 ( du)
. Trigonometric Integrls. ( sin m (x cos n (x Cse-: m is odd let u cos(x Exmple: sin 3 (x cos (x Review- nd Prctice problems sin 3 (x cos (x Cse-: n is odd let u sin(x Exmple: cos 5 (x cos 5 (x sin (x
Διαβάστε περισσότεραΤΜΗΜΑ ΗΛΕΚΤΡΟΛΟΓΩΝ ΜΗΧΑΝΙΚΩΝ ΚΑΙ ΜΗΧΑΝΙΚΩΝ ΥΠΟΛΟΓΙΣΤΩΝ
ΗΜΥ ΔΙΑΚΡΙΤΗ ΑΝΑΛΥΣΗ ΚΑΙ ΔΟΜΕΣ ΤΜΗΜΑ ΗΛΕΚΤΡΟΛΟΓΩΝ ΜΗΧΑΝΙΚΩΝ ΚΑΙ ΜΗΧΑΝΙΚΩΝ ΥΠΟΛΟΓΙΣΤΩΝ ΗΜΥ Διακριτή Ανάλυση και Δομές Χειμερινό Εξάμηνο 6 Σειρά Ασκήσεων Ακέραιοι και Διαίρεση, Πρώτοι Αριθμοί, GCD/LC, Συστήματα
Διαβάστε περισσότεραCHAPTER 25 SOLVING EQUATIONS BY ITERATIVE METHODS
CHAPTER 5 SOLVING EQUATIONS BY ITERATIVE METHODS EXERCISE 104 Page 8 1. Find the positive root of the equation x + 3x 5 = 0, correct to 3 significant figures, using the method of bisection. Let f(x) =
Διαβάστε περισσότεραSection 7.6 Double and Half Angle Formulas
09 Section 7. Double and Half Angle Fmulas To derive the double-angles fmulas, we will use the sum of two angles fmulas that we developed in the last section. We will let α θ and β θ: cos(θ) cos(θ + θ)
Διαβάστε περισσότεραΣχολή Εφαρμοσμένων Μαθηματικών και Φυσικών Επιστημών. Εθνικό Μετσόβιο Πολυτεχνείο. Thales Workshop, 1-3 July 2015.
Σχολή Εφαρμοσμένων Μαθηματικών και Φυσικών Επιστημών Εθνικό Μετσόβιο Πολυτεχνείο Thles Worksho, 1-3 July 015 The isomorhism function from S3(L(,1)) to the free module Boštjn Gbrovšek Άδεια Χρήσης Το παρόν
Διαβάστε περισσότερα2 Composition. Invertible Mappings
Arkansas Tech University MATH 4033: Elementary Modern Algebra Dr. Marcel B. Finan Composition. Invertible Mappings In this section we discuss two procedures for creating new mappings from old ones, namely,
Διαβάστε περισσότεραEE512: Error Control Coding
EE512: Error Control Coding Solution for Assignment on Finite Fields February 16, 2007 1. (a) Addition and Multiplication tables for GF (5) and GF (7) are shown in Tables 1 and 2. + 0 1 2 3 4 0 0 1 2 3
Διαβάστε περισσότεραAreas and Lengths in Polar Coordinates
Kiryl Tsishchanka Areas and Lengths in Polar Coordinates In this section we develop the formula for the area of a region whose boundary is given by a polar equation. We need to use the formula for the
Διαβάστε περισσότεραTo find the relationships between the coefficients in the original equation and the roots, we have to use a different technique.
Further Conepts for Avne Mthemtis - FP1 Unit Ientities n Roots of Equtions Cui, Qurti n Quinti Equtions Cui Equtions The three roots of the ui eqution x + x + x + 0 re lle α, β n γ (lph, et n gmm). The
Διαβάστε περισσότερα3.4 SUM AND DIFFERENCE FORMULAS. NOTE: cos(α+β) cos α + cos β cos(α-β) cos α -cos β
3.4 SUM AND DIFFERENCE FORMULAS Page Theorem cos(αβ cos α cos β -sin α cos(α-β cos α cos β sin α NOTE: cos(αβ cos α cos β cos(α-β cos α -cos β Proof of cos(α-β cos α cos β sin α Let s use a unit circle
Διαβάστε περισσότεραSection 8.3 Trigonometric Equations
99 Section 8. Trigonometric Equations Objective 1: Solve Equations Involving One Trigonometric Function. In this section and the next, we will exple how to solving equations involving trigonometric functions.
Διαβάστε περισσότεραAMS 212B Perturbation Methods Lecture 14 Copyright by Hongyun Wang, UCSC. Example: Eigenvalue problem with a turning point inside the interval
AMS B Perturbtion Methods Lecture 4 Copyright by Hongyun Wng, UCSC Emple: Eigenvlue problem with turning point inside the intervl y + λ y y = =, y( ) = The ODE for y() hs the form y () + λ f() y() = with
Διαβάστε περισσότεραFinite Field Problems: Solutions
Finite Field Problems: Solutions 1. Let f = x 2 +1 Z 11 [x] and let F = Z 11 [x]/(f), a field. Let Solution: F =11 2 = 121, so F = 121 1 = 120. The possible orders are the divisors of 120. Solution: The
Διαβάστε περισσότεραAreas and Lengths in Polar Coordinates
Kiryl Tsishchanka Areas and Lengths in Polar Coordinates In this section we develop the formula for the area of a region whose boundary is given by a polar equation. We need to use the formula for the
Διαβάστε περισσότεραThe Simply Typed Lambda Calculus
Type Inference Instead of writing type annotations, can we use an algorithm to infer what the type annotations should be? That depends on the type system. For simple type systems the answer is yes, and
Διαβάστε περισσότεραConcrete Mathematics Exercises from 30 September 2016
Concrete Mathematics Exercises from 30 September 2016 Silvio Capobianco Exercise 1.7 Let H(n) = J(n + 1) J(n). Equation (1.8) tells us that H(2n) = 2, and H(2n+1) = J(2n+2) J(2n+1) = (2J(n+1) 1) (2J(n)+1)
Διαβάστε περισσότεραMath221: HW# 1 solutions
Math: HW# solutions Andy Royston October, 5 7.5.7, 3 rd Ed. We have a n = b n = a = fxdx = xdx =, x cos nxdx = x sin nx n sin nxdx n = cos nx n = n n, x sin nxdx = x cos nx n + cos nxdx n cos n = + sin
Διαβάστε περισσότεραExample Sheet 3 Solutions
Example Sheet 3 Solutions. i Regular Sturm-Liouville. ii Singular Sturm-Liouville mixed boundary conditions. iii Not Sturm-Liouville ODE is not in Sturm-Liouville form. iv Regular Sturm-Liouville note
Διαβάστε περισσότεραHomework 3 Solutions
Homework 3 Solutions Igor Yanovsky (Math 151A TA) Problem 1: Compute the absolute error and relative error in approximations of p by p. (Use calculator!) a) p π, p 22/7; b) p π, p 3.141. Solution: For
Διαβάστε περισσότεραSome definite integrals connected with Gauss s sums
Some definite integrls connected with Guss s sums Messenger of Mthemtics XLIV 95 75 85. If n is rel nd positive nd I(t where I(t is the imginry prt of t is less thn either n or we hve cos πtx coshπx e
Διαβάστε περισσότεραSCHOOL OF MATHEMATICAL SCIENCES G11LMA Linear Mathematics Examination Solutions
SCHOOL OF MATHEMATICAL SCIENCES GLMA Linear Mathematics 00- Examination Solutions. (a) i. ( + 5i)( i) = (6 + 5) + (5 )i = + i. Real part is, imaginary part is. (b) ii. + 5i i ( + 5i)( + i) = ( i)( + i)
Διαβάστε περισσότεραSrednicki Chapter 55
Srednicki Chapter 55 QFT Problems & Solutions A. George August 3, 03 Srednicki 55.. Use equations 55.3-55.0 and A i, A j ] = Π i, Π j ] = 0 (at equal times) to verify equations 55.-55.3. This is our third
Διαβάστε περισσότεραCHAPTER 48 APPLICATIONS OF MATRICES AND DETERMINANTS
CHAPTER 48 APPLICATIONS OF MATRICES AND DETERMINANTS EXERCISE 01 Page 545 1. Use matrices to solve: 3x + 4y x + 5y + 7 3x + 4y x + 5y 7 Hence, 3 4 x 0 5 y 7 The inverse of 3 4 5 is: 1 5 4 1 5 4 15 8 3
Διαβάστε περισσότεραPractice Exam 2. Conceptual Questions. 1. State a Basic identity and then verify it. (a) Identity: Solution: One identity is csc(θ) = 1
Conceptual Questions. State a Basic identity and then verify it. a) Identity: Solution: One identity is cscθ) = sinθ) Practice Exam b) Verification: Solution: Given the point of intersection x, y) of the
Διαβάστε περισσότερα2. THEORY OF EQUATIONS. PREVIOUS EAMCET Bits.
EAMCET-. THEORY OF EQUATIONS PREVIOUS EAMCET Bits. Each of the roots of the equation x 6x + 6x 5= are increased by k so that the new transformed equation does not contain term. Then k =... - 4. - Sol.
Διαβάστε περισσότεραFourier Series. MATH 211, Calculus II. J. Robert Buchanan. Spring Department of Mathematics
Fourier Series MATH 211, Calculus II J. Robert Buchanan Department of Mathematics Spring 2018 Introduction Not all functions can be represented by Taylor series. f (k) (c) A Taylor series f (x) = (x c)
Διαβάστε περισσότεραElectromagnetic Waves I
Electromgnetic Wves I Jnury, 03. Derivtion of wve eqution of string. Derivtion of EM wve Eqution in time domin 3. Derivtion of the EM wve Eqution in phsor domin 4. The complex propgtion constnt 5. The
Διαβάστε περισσότεραApproximation of distance between locations on earth given by latitude and longitude
Approximation of distance between locations on earth given by latitude and longitude Jan Behrens 2012-12-31 In this paper we shall provide a method to approximate distances between two points on earth
Διαβάστε περισσότεραMatrices and Determinants
Matrices and Determinants SUBJECTIVE PROBLEMS: Q 1. For what value of k do the following system of equations possess a non-trivial (i.e., not all zero) solution over the set of rationals Q? x + ky + 3z
Διαβάστε περισσότεραC.S. 430 Assignment 6, Sample Solutions
C.S. 430 Assignment 6, Sample Solutions Paul Liu November 15, 2007 Note that these are sample solutions only; in many cases there were many acceptable answers. 1 Reynolds Problem 10.1 1.1 Normal-order
Διαβάστε περισσότεραINTEGRAL INEQUALITY REGARDING r-convex AND
J Koren Mth Soc 47, No, pp 373 383 DOI 434/JKMS47373 INTEGRAL INEQUALITY REGARDING r-convex AND r-concave FUNCTIONS WdAllh T Sulimn Astrct New integrl inequlities concerning r-conve nd r-concve functions
Διαβάστε περισσότεραExercises 10. Find a fundamental matrix of the given system of equations. Also find the fundamental matrix Φ(t) satisfying Φ(0) = I. 1.
Exercises 0 More exercises are available in Elementary Differential Equations. If you have a problem to solve any of them, feel free to come to office hour. Problem Find a fundamental matrix of the given
Διαβάστε περισσότεραforms This gives Remark 1. How to remember the above formulas: Substituting these into the equation we obtain with
Week 03: C lassification of S econd- Order L inear Equations In last week s lectures we have illustrated how to obtain the general solutions of first order PDEs using the method of characteristics. We
Διαβάστε περισσότεραΑπόκριση σε Μοναδιαία Ωστική Δύναμη (Unit Impulse) Απόκριση σε Δυνάμεις Αυθαίρετα Μεταβαλλόμενες με το Χρόνο. Απόστολος Σ.
Απόκριση σε Δυνάμεις Αυθαίρετα Μεταβαλλόμενες με το Χρόνο The time integral of a force is referred to as impulse, is determined by and is obtained from: Newton s 2 nd Law of motion states that the action
Διαβάστε περισσότεραIf ABC is any oblique triangle with sides a, b, and c, the following equations are valid. 2bc. (a) a 2 b 2 c 2 2bc cos A or cos A b2 c 2 a 2.
etion 6. Lw of osines 59 etion 6. Lw of osines If is ny oblique tringle with sides, b, nd, the following equtions re vlid. () b b os or os b b (b) b os or os b () b b os or os b b You should be ble to
Διαβάστε περισσότεραST5224: Advanced Statistical Theory II
ST5224: Advanced Statistical Theory II 2014/2015: Semester II Tutorial 7 1. Let X be a sample from a population P and consider testing hypotheses H 0 : P = P 0 versus H 1 : P = P 1, where P j is a known
Διαβάστε περισσότεραES440/ES911: CFD. Chapter 5. Solution of Linear Equation Systems
ES440/ES911: CFD Chapter 5. Solution of Linear Equation Systems Dr Yongmann M. Chung http://www.eng.warwick.ac.uk/staff/ymc/es440.html Y.M.Chung@warwick.ac.uk School of Engineering & Centre for Scientific
Διαβάστε περισσότεραderivation of the Laplacian from rectangular to spherical coordinates
derivation of the Laplacian from rectangular to spherical coordinates swapnizzle 03-03- :5:43 We begin by recognizing the familiar conversion from rectangular to spherical coordinates (note that φ is used
Διαβάστε περισσότεραEcon 2110: Fall 2008 Suggested Solutions to Problem Set 8 questions or comments to Dan Fetter 1
Eon : Fall 8 Suggested Solutions to Problem Set 8 Email questions or omments to Dan Fetter Problem. Let X be a salar with density f(x, θ) (θx + θ) [ x ] with θ. (a) Find the most powerful level α test
Διαβάστε περισσότεραSecond Order RLC Filters
ECEN 60 Circuits/Electronics Spring 007-0-07 P. Mathys Second Order RLC Filters RLC Lowpass Filter A passive RLC lowpass filter (LPF) circuit is shown in the following schematic. R L C v O (t) Using phasor
Διαβάστε περισσότεραInverse trigonometric functions & General Solution of Trigonometric Equations. ------------------ ----------------------------- -----------------
Inverse trigonometric functions & General Solution of Trigonometric Equations. 1. Sin ( ) = a) b) c) d) Ans b. Solution : Method 1. Ans a: 17 > 1 a) is rejected. w.k.t Sin ( sin ) = d is rejected. If sin
Διαβάστε περισσότεραMathCity.org Merging man and maths
MathCity.org Merging man and maths Exercise 10. (s) Page Textbook of Algebra and Trigonometry for Class XI Available online @, Version:.0 Question # 1 Find the values of sin, and tan when: 1 π (i) (ii)
Διαβάστε περισσότεραb. Use the parametrization from (a) to compute the area of S a as S a ds. Be sure to substitute for ds!
MTH U341 urface Integrals, tokes theorem, the divergence theorem To be turned in Wed., Dec. 1. 1. Let be the sphere of radius a, x 2 + y 2 + z 2 a 2. a. Use spherical coordinates (with ρ a) to parametrize.
Διαβάστε περισσότεραSolutions to Exercise Sheet 5
Solutions to Eercise Sheet 5 jacques@ucsd.edu. Let X and Y be random variables with joint pdf f(, y) = 3y( + y) where and y. Determine each of the following probabilities. Solutions. a. P (X ). b. P (X
Διαβάστε περισσότεραCHAPTER 101 FOURIER SERIES FOR PERIODIC FUNCTIONS OF PERIOD
CHAPTER FOURIER SERIES FOR PERIODIC FUNCTIONS OF PERIOD EXERCISE 36 Page 66. Determine the Fourier series for the periodic function: f(x), when x +, when x which is periodic outside this rge of period.
Διαβάστε περισσότεραSecond Order Partial Differential Equations
Chapter 7 Second Order Partial Differential Equations 7.1 Introduction A second order linear PDE in two independent variables (x, y Ω can be written as A(x, y u x + B(x, y u xy + C(x, y u u u + D(x, y
Διαβάστε περισσότεραHOMEWORK 4 = G. In order to plot the stress versus the stretch we define a normalized stretch:
HOMEWORK 4 Problem a For the fast loading case, we want to derive the relationship between P zz and λ z. We know that the nominal stress is expressed as: P zz = ψ λ z where λ z = λ λ z. Therefore, applying
Διαβάστε περισσότερα6.3 Forecasting ARMA processes
122 CHAPTER 6. ARMA MODELS 6.3 Forecasting ARMA processes The purpose of forecasting is to predict future values of a TS based on the data collected to the present. In this section we will discuss a linear
Διαβάστε περισσότεραHomework 8 Model Solution Section
MATH 004 Homework Solution Homework 8 Model Solution Section 14.5 14.6. 14.5. Use the Chain Rule to find dz where z cosx + 4y), x 5t 4, y 1 t. dz dx + dy y sinx + 4y)0t + 4) sinx + 4y) 1t ) 0t + 4t ) sinx
Διαβάστε περισσότεραProblem Set 3: Solutions
CMPSCI 69GG Applied Information Theory Fall 006 Problem Set 3: Solutions. [Cover and Thomas 7.] a Define the following notation, C I p xx; Y max X; Y C I p xx; Ỹ max I X; Ỹ We would like to show that C
Διαβάστε περισσότεραVariational Wavefunction for the Helium Atom
Technische Universität Graz Institut für Festkörperphysik Student project Variational Wavefunction for the Helium Atom Molecular and Solid State Physics 53. submitted on: 3. November 9 by: Markus Krammer
Διαβάστε περισσότεραSection 9.2 Polar Equations and Graphs
180 Section 9. Polar Equations and Graphs In this section, we will be graphing polar equations on a polar grid. In the first few examples, we will write the polar equation in rectangular form to help identify
Διαβάστε περισσότεραPhys460.nb Solution for the t-dependent Schrodinger s equation How did we find the solution? (not required)
Phys460.nb 81 ψ n (t) is still the (same) eigenstate of H But for tdependent H. The answer is NO. 5.5.5. Solution for the tdependent Schrodinger s equation If we assume that at time t 0, the electron starts
Διαβάστε περισσότεραPARTIAL NOTES for 6.1 Trigonometric Identities
PARTIAL NOTES for 6.1 Trigonometric Identities tanθ = sinθ cosθ cotθ = cosθ sinθ BASIC IDENTITIES cscθ = 1 sinθ secθ = 1 cosθ cotθ = 1 tanθ PYTHAGOREAN IDENTITIES sin θ + cos θ =1 tan θ +1= sec θ 1 + cot
Διαβάστε περισσότερα9.09. # 1. Area inside the oval limaçon r = cos θ. To graph, start with θ = 0 so r = 6. Compute dr
9.9 #. Area inside the oval limaçon r = + cos. To graph, start with = so r =. Compute d = sin. Interesting points are where d vanishes, or at =,,, etc. For these values of we compute r:,,, and the values
Διαβάστε περισσότεραSOLUTIONS TO PROBLEMS IN LIE ALGEBRAS IN PARTICLE PHYSICS BY HOWARD GEORGI STEPHEN HANCOCK
SOLUTIONS TO PROBLEMS IN LIE ALGEBRAS IN PARTICLE PHYSICS BY HOWARD GEORGI STEPHEN HANCOCK STEPHEN HANCOCK Chpter 6 Solutions 6.A. Clerly NE α+β hs root vector α+β since H i NE α+β = NH i E α+β = N(α+β)
Διαβάστε περισσότεραStatistical Inference I Locally most powerful tests
Statistical Inference I Locally most powerful tests Shirsendu Mukherjee Department of Statistics, Asutosh College, Kolkata, India. shirsendu st@yahoo.co.in So far we have treated the testing of one-sided
Διαβάστε περισσότεραPg The perimeter is P = 3x The area of a triangle is. where b is the base, h is the height. In our case b = x, then the area is
Pg. 9. The perimeter is P = The area of a triangle is A = bh where b is the base, h is the height 0 h= btan 60 = b = b In our case b =, then the area is A = = 0. By Pythagorean theorem a + a = d a a =
Διαβάστε περισσότερα2. Μηχανικό Μαύρο Κουτί: κύλινδρος με μια μπάλα μέσα σε αυτόν.
Experiental Copetition: 14 July 011 Proble Page 1 of. Μηχανικό Μαύρο Κουτί: κύλινδρος με μια μπάλα μέσα σε αυτόν. Ένα μικρό σωματίδιο μάζας (μπάλα) βρίσκεται σε σταθερή απόσταση z από το πάνω μέρος ενός
Διαβάστε περισσότεραECE Spring Prof. David R. Jackson ECE Dept. Notes 2
ECE 634 Spring 6 Prof. David R. Jackson ECE Dept. Notes Fields in a Source-Free Region Example: Radiation from an aperture y PEC E t x Aperture Assume the following choice of vector potentials: A F = =
Διαβάστε περισσότεραΣυστήματα Διαχείρισης Βάσεων Δεδομένων
ΕΛΛΗΝΙΚΗ ΔΗΜΟΚΡΑΤΙΑ ΠΑΝΕΠΙΣΤΗΜΙΟ ΚΡΗΤΗΣ Συστήματα Διαχείρισης Βάσεων Δεδομένων Φροντιστήριο 9: Transactions - part 1 Δημήτρης Πλεξουσάκης Τμήμα Επιστήμης Υπολογιστών Tutorial on Undo, Redo and Undo/Redo
Διαβάστε περισσότεραJesse Maassen and Mark Lundstrom Purdue University November 25, 2013
Notes on Average Scattering imes and Hall Factors Jesse Maassen and Mar Lundstrom Purdue University November 5, 13 I. Introduction 1 II. Solution of the BE 1 III. Exercises: Woring out average scattering
Διαβάστε περισσότεραI Feel Pretty VOIX. MARIA et Trois Filles - N 12. BERNSTEIN Leonard Adaptation F. Pissaloux. ι œ. % α α α œ % α α α œ. œ œ œ. œ œ œ œ. œ œ. œ œ ƒ.
VOX Feel Pretty MARA et Trois Filles - N 12 BERNSTEN Leonrd Adpttion F. Pissloux Violons Contrebsse A 2 7 2 7 Allegro qd 69 1 2 4 5 6 7 8 9 B 10 11 12 1 14 15 16 17 18 19 20 21 22 2 24 C 25 26 27 28 29
Διαβάστε περισσότεραΚΥΠΡΙΑΚΗ ΕΤΑΙΡΕΙΑ ΠΛΗΡΟΦΟΡΙΚΗΣ CYPRUS COMPUTER SOCIETY ΠΑΓΚΥΠΡΙΟΣ ΜΑΘΗΤΙΚΟΣ ΔΙΑΓΩΝΙΣΜΟΣ ΠΛΗΡΟΦΟΡΙΚΗΣ 19/5/2007
Οδηγίες: Να απαντηθούν όλες οι ερωτήσεις. Αν κάπου κάνετε κάποιες υποθέσεις να αναφερθούν στη σχετική ερώτηση. Όλα τα αρχεία που αναφέρονται στα προβλήματα βρίσκονται στον ίδιο φάκελο με το εκτελέσιμο
Διαβάστε περισσότεραSection 7.7 Product-to-Sum and Sum-to-Product Formulas
Section 7.7 Product-to-Sum and Sum-to-Product Fmulas Objective 1: Express Products as Sums To derive the Product-to-Sum Fmulas will begin by writing down the difference and sum fmulas of the cosine function:
Διαβάστε περισσότεραSolution Series 9. i=1 x i and i=1 x i.
Lecturer: Prof. Dr. Mete SONER Coordinator: Yilin WANG Solution Series 9 Q1. Let α, β >, the p.d.f. of a beta distribution with parameters α and β is { Γ(α+β) Γ(α)Γ(β) f(x α, β) xα 1 (1 x) β 1 for < x
Διαβάστε περισσότεραTrigonometric Formula Sheet
Trigonometric Formula Sheet Definition of the Trig Functions Right Triangle Definition Assume that: 0 < θ < or 0 < θ < 90 Unit Circle Definition Assume θ can be any angle. y x, y hypotenuse opposite θ
Διαβάστε περισσότεραk A = [k, k]( )[a 1, a 2 ] = [ka 1,ka 2 ] 4For the division of two intervals of confidence in R +
Chapter 3. Fuzzy Arithmetic 3- Fuzzy arithmetic: ~Addition(+) and subtraction (-): Let A = [a and B = [b, b in R If x [a and y [b, b than x+y [a +b +b Symbolically,we write A(+)B = [a (+)[b, b = [a +b
Διαβάστε περισσότεραOther Test Constructions: Likelihood Ratio & Bayes Tests
Other Test Constructions: Likelihood Ratio & Bayes Tests Side-Note: So far we have seen a few approaches for creating tests such as Neyman-Pearson Lemma ( most powerful tests of H 0 : θ = θ 0 vs H 1 :
Διαβάστε περισσότερα1. For each of the following power series, find the interval of convergence and the radius of convergence:
Math 6 Practice Problems Solutios Power Series ad Taylor Series 1. For each of the followig power series, fid the iterval of covergece ad the radius of covergece: (a ( 1 x Notice that = ( 1 +1 ( x +1.
Διαβάστε περισσότεραNotes on the Open Economy
Notes on the Open Econom Ben J. Heijdra Universit of Groningen April 24 Introduction In this note we stud the two-countr model of Table.4 in more detail. restated here for convenience. The model is Table.4.
Διαβάστε περισσότεραLecture 2: Dirac notation and a review of linear algebra Read Sakurai chapter 1, Baym chatper 3
Lecture 2: Dirac notation and a review of linear algebra Read Sakurai chapter 1, Baym chatper 3 1 State vector space and the dual space Space of wavefunctions The space of wavefunctions is the set of all
Διαβάστε περισσότεραSelf and Mutual Inductances for Fundamental Harmonic in Synchronous Machine with Round Rotor (Cont.) Double Layer Lap Winding on Stator
Sel nd Mutul Inductnces or Fundmentl Hrmonc n Synchronous Mchne wth Round Rotor (Cont.) Double yer p Wndng on Sttor Round Rotor Feld Wndng (1) d xs s r n even r Dene S r s the number o rotor slots. Dene
Διαβάστε περισσότερα( y) Partial Differential Equations
Partial Dierential Equations Linear P.D.Es. contains no owers roducts o the deendent variables / an o its derivatives can occasionall be solved. Consider eamle ( ) a (sometimes written as a ) we can integrate
Διαβάστε περισσότεραNumerical Analysis FMN011
Numerical Analysis FMN011 Carmen Arévalo Lund University carmen@maths.lth.se Lecture 12 Periodic data A function g has period P if g(x + P ) = g(x) Model: Trigonometric polynomial of order M T M (x) =
Διαβάστε περισσότεραAnswer sheet: Third Midterm for Math 2339
Answer sheet: Third Midterm for Math 339 November 3, Problem. Calculate the iterated integrals (Simplify as much as possible) (a) e sin(x) dydx y e sin(x) dydx y sin(x) ln y ( cos(x)) ye y dx sin(x)(lne
Διαβάστε περισσότεραNowhere-zero flows Let be a digraph, Abelian group. A Γ-circulation in is a mapping : such that, where, and : tail in X, head in
Nowhere-zero flows Let be a digraph, Abelian group. A Γ-circulation in is a mapping : such that, where, and : tail in X, head in : tail in X, head in A nowhere-zero Γ-flow is a Γ-circulation such that
Διαβάστε περισσότεραCRASH COURSE IN PRECALCULUS
CRASH COURSE IN PRECALCULUS Shiah-Sen Wang The graphs are prepared by Chien-Lun Lai Based on : Precalculus: Mathematics for Calculus by J. Stuwart, L. Redin & S. Watson, 6th edition, 01, Brooks/Cole Chapter
Διαβάστε περισσότεραThe challenges of non-stable predicates
The challenges of non-stable predicates Consider a non-stable predicate Φ encoding, say, a safety property. We want to determine whether Φ holds for our program. The challenges of non-stable predicates
Διαβάστε περισσότεραExercises to Statistics of Material Fatigue No. 5
Prof. Dr. Christine Müller Dipl.-Math. Christoph Kustosz Eercises to Statistics of Material Fatigue No. 5 E. 9 (5 a Show, that a Fisher information matri for a two dimensional parameter θ (θ,θ 2 R 2, can
Διαβάστε περισσότεραPartial Differential Equations in Biology The boundary element method. March 26, 2013
The boundary element method March 26, 203 Introduction and notation The problem: u = f in D R d u = ϕ in Γ D u n = g on Γ N, where D = Γ D Γ N, Γ D Γ N = (possibly, Γ D = [Neumann problem] or Γ N = [Dirichlet
Διαβάστε περισσότεραTridiagonal matrices. Gérard MEURANT. October, 2008
Tridiagonal matrices Gérard MEURANT October, 2008 1 Similarity 2 Cholesy factorizations 3 Eigenvalues 4 Inverse Similarity Let α 1 ω 1 β 1 α 2 ω 2 T =......... β 2 α 1 ω 1 β 1 α and β i ω i, i = 1,...,
Διαβάστε περισσότεραChapter 7b, Torsion. τ = 0. τ T. T τ D'' A'' C'' B'' 180 -rotation around axis C'' B'' D'' A'' A'' D'' 180 -rotation upside-down C'' B''
Chpter 7b, orsion τ τ τ ' D' B' C' '' B'' B'' D'' C'' 18 -rottion round xis C'' B'' '' D'' C'' '' 18 -rottion upside-down D'' stright lines in the cross section (cross sectionl projection) remin stright
Διαβάστε περισσότεραOrdinal Arithmetic: Addition, Multiplication, Exponentiation and Limit
Ordinal Arithmetic: Addition, Multiplication, Exponentiation and Limit Ting Zhang Stanford May 11, 2001 Stanford, 5/11/2001 1 Outline Ordinal Classification Ordinal Addition Ordinal Multiplication Ordinal
Διαβάστε περισσότεραOscillation of Nonlinear Delay Partial Difference Equations. LIU Guanghui [a],*
Studies in Mthemtil Sienes Vol. 5, No.,, pp. [9 97] DOI:.3968/j.sms.938455.58 ISSN 93-8444 [Print] ISSN 93-845 [Online] www.snd.net www.snd.org Osilltion of Nonliner Dely Prtil Differene Equtions LIU Gunghui
Διαβάστε περισσότεραExercise 2: The form of the generalized likelihood ratio
Stats 2 Winter 28 Homework 9: Solutions Due Friday, March 6 Exercise 2: The form of the generalized likelihood ratio We want to test H : θ Θ against H : θ Θ, and compare the two following rules of rejection:
Διαβάστε περισσότερα4.6 Autoregressive Moving Average Model ARMA(1,1)
84 CHAPTER 4. STATIONARY TS MODELS 4.6 Autoregressive Moving Average Model ARMA(,) This section is an introduction to a wide class of models ARMA(p,q) which we will consider in more detail later in this
Διαβάστε περισσότεραDESIGN OF MACHINERY SOLUTION MANUAL h in h 4 0.
DESIGN OF MACHINERY SOLUTION MANUAL -7-1! PROBLEM -7 Statement: Design a double-dwell cam to move a follower from to 25 6, dwell for 12, fall 25 and dwell for the remader The total cycle must take 4 sec
Διαβάστε περισσότεραWritten Examination. Antennas and Propagation (AA ) April 26, 2017.
Written Examination Antennas and Propagation (AA. 6-7) April 6, 7. Problem ( points) Let us consider a wire antenna as in Fig. characterized by a z-oriented linear filamentary current I(z) = I cos(kz)ẑ
Διαβάστε περισσότεραΚΥΠΡΙΑΚΗ ΕΤΑΙΡΕΙΑ ΠΛΗΡΟΦΟΡΙΚΗΣ CYPRUS COMPUTER SOCIETY ΠΑΓΚΥΠΡΙΟΣ ΜΑΘΗΤΙΚΟΣ ΔΙΑΓΩΝΙΣΜΟΣ ΠΛΗΡΟΦΟΡΙΚΗΣ 6/5/2006
Οδηγίες: Να απαντηθούν όλες οι ερωτήσεις. Ολοι οι αριθμοί που αναφέρονται σε όλα τα ερωτήματα είναι μικρότεροι το 1000 εκτός αν ορίζεται διαφορετικά στη διατύπωση του προβλήματος. Διάρκεια: 3,5 ώρες Καλή
Διαβάστε περισσότεραEvery set of first-order formulas is equivalent to an independent set
Every set of first-order formulas is equivalent to an independent set May 6, 2008 Abstract A set of first-order formulas, whatever the cardinality of the set of symbols, is equivalent to an independent
Διαβάστε περισσότεραCHAPTER (2) Electric Charges, Electric Charge Densities and Electric Field Intensity
CHAPTE () Electric Chrges, Electric Chrge Densities nd Electric Field Intensity Chrge Configurtion ) Point Chrge: The concept of the point chrge is used when the dimensions of n electric chrge distriution
Διαβάστε περισσότερα(a,b) Let s review the general definitions of trig functions first. (See back cover of your book) sin θ = b/r cos θ = a/r tan θ = b/a, a 0
TRIGONOMETRIC IDENTITIES (a,b) Let s eview the geneal definitions of tig functions fist. (See back cove of you book) θ b/ θ a/ tan θ b/a, a 0 θ csc θ /b, b 0 sec θ /a, a 0 cot θ a/b, b 0 By doing some
Διαβάστε περισσότεραΕγκατάσταση λογισμικού και αναβάθμιση συσκευής Device software installation and software upgrade
Για να ελέγξετε το λογισμικό που έχει τώρα η συσκευή κάντε κλικ Menu > Options > Device > About Device Versions. Στο πιο κάτω παράδειγμα η συσκευή έχει έκδοση λογισμικού 6.0.0.546 με πλατφόρμα 6.6.0.207.
Διαβάστε περισσότεραΕγχειρίδια Μαθηµατικών και Χταποδάκι στα Κάρβουνα
[ 1 ] Πανεπιστήµιο Κύπρου Εγχειρίδια Μαθηµατικών και Χταποδάκι στα Κάρβουνα Νίκος Στυλιανόπουλος, Πανεπιστήµιο Κύπρου Λευκωσία, εκέµβριος 2009 [ 2 ] Πανεπιστήµιο Κύπρου Πόσο σηµαντική είναι η απόδειξη
Διαβάστε περισσότεραPart III - Pricing A Down-And-Out Call Option
Part III - Pricing A Down-And-Out Call Option Gary Schurman MBE, CFA March 202 In Part I we examined the reflection principle and a scaled random walk in discrete time and then extended the reflection
Διαβάστε περισσότερα