1. For each of the following power series, find the interval of convergence and the radius of convergence:

Μέγεθος: px
Εμφάνιση ξεκινά από τη σελίδα:

Download "1. For each of the following power series, find the interval of convergence and the radius of convergence:"

Transcript

1 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. The lim a = lim ( x +1 + x = lim x + = x lim = x lim = x, so this series coverges absolutely for 1 < x < 1. Notice whe x = 1, we have ( 1 1 = ( 1 which diverges by the th term test. Similarly, whe x = 1, we have ( 1 ( 1 = ( 1 = 1 which diverges by the th term test. Hece, the iterval of covergece is: ( 1,1 ad the radius covergece is: R = 1. (b (x 3 Notice that = +1 ( (x 3+1. The lim a = lim +1 x 3 +1 ( x 3 (c = lim x 3 + = x 3 lim whe x 3 < 1, or for 5 < x < 7. Notice whe x = 5, we have ( 1 = series coverges absolutely. Similarly, whe x = 7, we have (1 = Hece, the iterval of covergece is: 3 (x [ 5, 7 ] + = x 3 lim = x 3, so this series coverges absolutely ( 1 Thus, sice (1 = 1 is a coverget p-series, the origial 1, which is a coverget p-series. ad the radius covergece is: R = 1. (3 Notice that = 3 +1 (x The lim a = lim ( 3 x x + 1 = 1 ( 3 x + 1 lim 3 3, which, after a few applicatios of L Hôpital s Rule, is absolutely whe x + 1 < 3 or for 4 < x <. 3 Notice whe x = 4, we have 3 ( 3 = ( 1 3, which diverges by the th term test. Similarly, whe x =, we have = 3 which diverges by the th term test. Hece, the iterval of covergece is: ( 4, ad the radius covergece is: R = 3. x + 1, so this series coverges 3

2 (d ( 1 (x Notice that = ( 1 (! (x +1. The lim = 0 = x lim Hece the iterval of covergece is (, ad R =. a (e ( 1 1 (x Notice that = ( ( +1 (x +1. The lim = 1 x lim = lim +1 x +1 (! a = lim x +1 ( +1 x x = 1 x, so this series coverges absolutely whe x < or for 8 < x < 1. Notice whe x = 8, we have ( 1 1 ( = ( ( 1 =, which diverges sice it is the harmoic series. Similarly, whe x =, we have ( 1 1 = ( 1 1 which coverges by the Alteratig Series Test. Hece, the iterval of covergece is: ( 8,] ad the radius covergece is: R =.. Use a kow series to fid a power series i x that has the give fuctio as its sum: (a xsi(x 3 (b Recall the Maclauri series for siu = Therefore, si(x 3 = Hece xsi(x 3 = l(1 + x x ( 1 u+1 (! ( 1 (x3 +1 (! = ( 1 (x6+4 (!. Recall the Maclauri series for l(1 + x = Therefore, l(1 + x x = ( 1 x ( 1 (x6+3 (!. ( 1 x+1 (c x arcta x x 3 Recall the Maclauri series for arcta(x = ( 1 x+1 = x x3 3 + x5 5 x7 7 + Therefore, x arcta(x = x (x x3 3 + x5 5 x x+1 = ( 1 Hece x arcta x +1 x x 3 = ( 1

3 3. Use a power series to approximate each of the followig to withi 3 decimal places: (a arcta 1 Notice that the Maclauri series arcta(x = is a alteratig series satisfyig the hypotheses of ( 1 x+1 the alteratig series test whe x = 1 (.5+1. The to fid our approximatio, we eed to fid such that < a 0 = 1, a 1 = , a 3 = = , a 4 = , ad a Hece arcta (b l(1.01 Notice that the Maclauri series l(1 + x = ( 1 x+1 is a alteratig series satisfyig the hypotheses of the alteratig series test whe x = The to fid our approximatio, we eed to fid such that ( < a 0 = 0.01, a 1 = Hece l( (c si ( π Notice that the Maclauri series si x = ( 1 x+1 is a alteratig series satisfyig the hypotheses of the (! alteratig series test whe x = π. The to fid our approximatio, we eed to fid such that ( π +1 (! < a 0 = π , a , a ( π Hece si For each of the followig fuctios, fid the Taylor Series about the idicated ceter ad also determie the iterval of covergece for the series. (a f(x = e x 1, c = 1 Notice that f (x = e x 1 ad f (x = e x 1. I fact, f ( (x = e x 1 for every. The f ( (1 = e 0 = 1 for every, ad hece a = 1 for every. Thus e x 1 (x 1 =. Thus this series coverges o (, ad R =. a = lim x 1 +1 (! 1 = x 1 lim x 1 = 0 (b f(x = cos x, c = π f (x = si x, f (x = cos x, f (x = si x, f 4 (x = cos x, ad the same patter cotiues from there. ( π Therefore, f = cos π ( = 0 f π = si π ( = 1, f π = cos π ( = 0, f π = si π ( = 1, π f4 = cos π = 0, ad the patter cotiues from there. Therefore, a 0 = 0, a 1 = 1, a = 0, a 3 = 1 3! = 1 6 Hece the series is: cos x = ( (! (x π +1 a = lim x π +1 (! ( + 3! x π = x π lim 1 ( + 3( + = 0 Thus this series coverges o (, ad R =.

4 (c f(x = 1 x, c = 1 f (x = x, f (x = x 3, f (x = 6x 4, so f (x = ( 1 x (+1 The f( 1 = 1, f ( 1 = 1, f ( 1 =, f ( 1 = 6, ad f ( 1 =. Therefore, a 0 = 1, a 1 = 1, a = 1, a 3 = 1, ad, i fact, a = 1 for all. Hece 1 x = ( 1(x 1 a = lim ( 1 x 1 +1 ( 1 x 1 coverges absolutely for 0 x Whe x = 0, we have ( 1( 1, which diverges by the th term test. Similarly, whe x = we have ( 1(1, which also diverges by the th term test. Thus this series coverges o (0, ad R = 1. = x 1, so this series 5. For each of the followig fuctios, fid the Taylor Polyomial for the fuctio at the idicated ceter c. Also fid the Remaider term. (a f(x = x, c = 1, = 3. First, f (x = 1 x 1, f (x = 1 4 x 3, f (x = 3 8 x 5, ad f (4 (x = x 7. The f(1 = 1, f (1 = 1, f (1 = 1 4, f (1 = 3 8. Hece a 0 = 1, a 1 = 1, a = 1 8, ad a 3 = 1 16 Thus P 3 (x = (x (x (x 13 ad R 3 (x = f(4 (z 4! (x = 5z 18 (x 14 (b f(x = lx, c = 1, = 4. First, f (x = x 1, f (x = x, f (x = x 3, f (4 (x = 6x 4, ad f (5 (x = 4x 5. The f(1 = 0, f (1 = 1, f (1 = 1, f (1 =, ad f (4 (1 = 6. Hece a 0 = 0, a 1 = 1, a = 1, a 3 = 1 3, ad a 4 = 1 4 Thus P 4 (x = 0 + (x 1 1 (x (x (x 14 ad R 4 (x = f(5 (z (x 1 5 = 4z 5 (x 15 = z 5 5 (x 15 (c f(x = 1 + x, c = 0, = 4. First, f (x = x(1 + x 1, f (x = (1 + x 1 x (1 + x 3, f (x = 3x(1 + x 3 + 3x 3 (1 + x 5, f (4 (x = 3(1 + x x (1 + x 5 15x 4 (1 + x 7, ad f (5 (x = 45x(1 + x 5 150x 3 (1 + x 7 + 5x 5 (1 + x 9 The f(0 = 1, f (0 = 0, f (0 = 1, f (0 = 0, ad f (4 (0 = 3. Hece a 0 = 1, a 1 = 0, a = 1, a 3 = 0, ad a 4 = 1 8 Thus P 4 (x = x 1 8 x4 ad R 4 (x = f(5 (z x 5 = 45z(1+z 5 150z 3 (1+z 7 +5z 5 (1+z 9 x 5 6. Estimate each of the followig usig a Taylor Polyomial of degree 4. Also fid the error for your approximatio. Fially, fid the umber of terms eeded to guaratee a accuracy or at least 5 decimal places. (a e 0.1 Recall that e x = x. The P 4 (x = 1 + x + x + x3 6 + x4 4, ad R 4 = ez x5

5 Whe x = 0.1, P 4 (x = I geeral, R (x = f(+1(z e z (! x+1 = (! (0.1+1, where 0 z 0.1. Sice e x is icreasig, we eed to fid so that e 0.1 (! (0.1+1 < Whe we use P 4 (x, our error is at most e0.1 ( (i fact, oe would oly eed P 3 (x to get withi 5 decimal places. (b l 0.9 Recall that l(1 + x = ( 1 x+1. We will take x = 0.1 so that l(1 + x = l(.9 The P 4 (x = x x + x3 3 x4 4. Also, f(5 (x = 4(1 + x 5. 4(1 + z 5 Therefore, R 4 = x 5 (1 + z (+1. I geeral, R (x = ( 1 x +1. Whe x = 0.1, P 4 (x = Sice R (x = f(+1(z (! x+1 (1 + z (+1 = ( 1 x +1, where 0.1 z 0. (1.1 (+1 Sice l(1+x is egative ad icreasig whe.1 < x < 0, we eed to fid so that ( 1 x +1 < (1.1 (5 Whe we use P 4 (x, our error is at most ( (1.1 (6 If we use P 5 (x, our error is at most ( , so this is a sufficiet umber of terms to 6 approximate to at least 5 decimal places. (c 1. We will use f(x = x cetered at c = 1 ad we will take x = 1.. The f (x = 1 x 1, f (x = 1 4 x 3, f (x = 3 8 x 5, f (4 (x = x 7, ad f (5 (x = 5 3 x 9. The f(1 = 1, f (1 = 1, f (1 = 1 4, f (1 = 3 15, ad f(4(1= Hece a 0 = 1, a 1 = 1, a = 1 8, a 3 = 1 16, ad a 4 = 5 18 Thus P 4 (x = (x (x (x (x 14 ad R 4 (x = f(5 (z (x = 7z 56 (x 15 Thus 1. P 4 (1. = ( ( ( ( (1. 9 The error of this approximatio is at most: ( Hece this estimate is already sufficiet to approximate to 5 decimal places (oe ca easly verify that P 3 (x is oly accurate to 4 decimal places.

L.K.Gupta (Mathematic Classes) www.pioeermathematics.com MOBILE: 985577, 4677 + {JEE Mai 04} Sept 0 Name: Batch (Day) Phoe No. IT IS NOT ENOUGH TO HAVE A GOOD MIND, THE MAIN THING IS TO USE IT WELL Marks:

Διαβάστε περισσότερα

CHAPTER 103 EVEN AND ODD FUNCTIONS AND HALF-RANGE FOURIER SERIES

CHAPTER 103 EVEN AND ODD FUNCTIONS AND HALF-RANGE FOURIER SERIES CHAPTER 3 EVEN AND ODD FUNCTIONS AND HALF-RANGE FOURIER SERIES EXERCISE 364 Page 76. Determie the Fourier series for the fuctio defied by: f(x), x, x, x which is periodic outside of this rage of period.

Διαβάστε περισσότερα

Last Lecture. Biostatistics Statistical Inference Lecture 19 Likelihood Ratio Test. Example of Hypothesis Testing.

Last Lecture. Biostatistics Statistical Inference Lecture 19 Likelihood Ratio Test. Example of Hypothesis Testing. Last Lecture Biostatistics 602 - Statistical Iferece Lecture 19 Likelihood Ratio Test Hyu Mi Kag March 26th, 2013 Describe the followig cocepts i your ow words Hypothesis Null Hypothesis Alterative Hypothesis

Διαβάστε περισσότερα

3.4 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. 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 περισσότερα

Homework for 1/27 Due 2/5

Homework for 1/27 Due 2/5 Name: ID: Homework for /7 Due /5. [ 8-3] I Example D of Sectio 8.4, the pdf of the populatio distributio is + αx x f(x α) =, α, otherwise ad the method of momets estimate was foud to be ˆα = 3X (where

Διαβάστε περισσότερα

n r f ( n-r ) () x g () r () x (1.1) = Σ g() x = Σ n f < -n+ r> g () r -n + r dx r dx n + ( -n,m) dx -n n+1 1 -n -1 + ( -n,n+1)

n r f ( n-r ) () x g () r () x (1.1) = Σ g() x = Σ n f < -n+ r> g () r -n + r dx r dx n + ( -n,m) dx -n n+1 1 -n -1 + ( -n,n+1) 8 Higher Derivative of the Product of Two Fuctios 8. Leibiz Rule about the Higher Order Differetiatio Theorem 8.. (Leibiz) Whe fuctios f ad g f g are times differetiable, the followig epressio holds. r

Διαβάστε περισσότερα

IIT JEE (2013) (Trigonomtery 1) Solutions

IIT JEE (2013) (Trigonomtery 1) Solutions L.K. Gupta (Mathematic Classes) www.pioeermathematics.com MOBILE: 985577, 677 (+) PAPER B IIT JEE (0) (Trigoomtery ) Solutios TOWARDS IIT JEE IS NOT A JOURNEY, IT S A BATTLE, ONLY THE TOUGHEST WILL SURVIVE

Διαβάστε περισσότερα

Presentation of complex number in Cartesian and polar coordinate system

Presentation of complex number in Cartesian and polar coordinate system 1 a + bi, aεr, bεr i = 1 z = a + bi a = Re(z), b = Im(z) give z = a + bi & w = c + di, a + bi = c + di a = c & b = d The complex cojugate of z = a + bi is z = a bi The sum of complex cojugates is real:

Διαβάστε περισσότερα

CHAPTER 25 SOLVING EQUATIONS BY ITERATIVE METHODS

CHAPTER 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 περισσότερα

Στα επόμενα θεωρούμε ότι όλα συμβαίνουν σε ένα χώρο πιθανότητας ( Ω,,P) Modes of convergence: Οι τρόποι σύγκλισης μιας ακολουθίας τ.μ.

Στα επόμενα θεωρούμε ότι όλα συμβαίνουν σε ένα χώρο πιθανότητας ( Ω,,P) Modes of convergence: Οι τρόποι σύγκλισης μιας ακολουθίας τ.μ. Στα πόμνα θωρούμ ότι όλα συμβαίνουν σ ένα χώρο πιθανότητας ( Ω,,). Modes of covergece: Οι τρόποι σύγκλισης μιας ακολουθίας τ.μ. { } ίναι οι ξής: σ μια τ.μ.. Ισχυρή σύγκλιση strog covergece { } lim = =.

Διαβάστε περισσότερα

Homework 3 Solutions

Homework 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 περισσότερα

Fourier Series. constant. The ;east value of T>0 is called the period of f(x). f(x) is well defined and single valued periodic function

Fourier Series. constant. The ;east value of T>0 is called the period of f(x). f(x) is well defined and single valued periodic function Fourier Series Periodic uctio A uctio is sid to hve period T i, T where T is ve costt. The ;est vlue o T> is clled the period o. Eg:- Cosider we kow tht, si si si si si... Etc > si hs the periods,,6,..

Διαβάστε περισσότερα

Homework 4.1 Solutions Math 5110/6830

Homework 4.1 Solutions Math 5110/6830 Homework 4. Solutios Math 5/683. a) For p + = αp γ α)p γ α)p + γ b) Let Equilibria poits satisfy: p = p = OR = γ α)p ) γ α)p + γ = α γ α)p ) γ α)p + γ α = p ) p + = p ) = The, we have equilibria poits

Διαβάστε περισσότερα

SUPERPOSITION, MEASUREMENT, NORMALIZATION, EXPECTATION VALUES. Reading: QM course packet Ch 5 up to 5.6

SUPERPOSITION, MEASUREMENT, NORMALIZATION, EXPECTATION VALUES. Reading: QM course packet Ch 5 up to 5.6 SUPERPOSITION, MEASUREMENT, NORMALIZATION, EXPECTATION VALUES Readig: QM course packet Ch 5 up to 5. 1 ϕ (x) = E = π m( a) =1,,3,4,5 for xa (x) = πx si L L * = πx L si L.5 ϕ' -.5 z 1 (x) = L si

Διαβάστε περισσότερα

Srednicki Chapter 55

Srednicki 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 περισσότερα

Bessel function for complex variable

Bessel function for complex variable Besse fuctio for compex variabe Kauhito Miuyama May 4, 7 Besse fuctio The Besse fuctio Z ν () is the fuctio wich satisfies + ) ( + ν Z ν () =. () Three kids of the soutios of this equatio are give by {

Διαβάστε περισσότερα

Solution Series 9. i=1 x i and i=1 x i.

Solution 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 περισσότερα

EE512: Error Control Coding

EE512: 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 περισσότερα

C.S. 430 Assignment 6, Sample Solutions

C.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 περισσότερα

Biorthogonal Wavelets and Filter Banks via PFFS. Multiresolution Analysis (MRA) subspaces V j, and wavelet subspaces W j. f X n f, τ n φ τ n φ.

Biorthogonal Wavelets and Filter Banks via PFFS. Multiresolution Analysis (MRA) subspaces V j, and wavelet subspaces W j. f X n f, τ n φ τ n φ. Chapter 3. Biorthogoal Wavelets ad Filter Baks via PFFS 3.0 PFFS applied to shift-ivariat subspaces Defiitio: X is a shift-ivariat subspace if h X h( ) τ h X. Ex: Multiresolutio Aalysis (MRA) subspaces

Διαβάστε περισσότερα

MATH 38061/MATH48061/MATH68061: MULTIVARIATE STATISTICS Solutions to Problems on Matrix Algebra

MATH 38061/MATH48061/MATH68061: MULTIVARIATE STATISTICS Solutions to Problems on Matrix Algebra MATH 38061/MATH48061/MATH68061: MULTIVARIATE STATISTICS Solutios to Poblems o Matix Algeba 1 Let A be a squae diagoal matix takig the fom a 11 0 0 0 a 22 0 A 0 0 a pp The ad So, log det A t log A t log

Διαβάστε περισσότερα

On Generating Relations of Some Triple. Hypergeometric Functions

On Generating Relations of Some Triple. Hypergeometric Functions It. Joural of Math. Aalysis, Vol. 5,, o., 5 - O Geeratig Relatios of Some Triple Hypergeometric Fuctios Fadhle B. F. Mohse ad Gamal A. Qashash Departmet of Mathematics, Faculty of Educatio Zigibar Ade

Διαβάστε περισσότερα

Solve the difference equation

Solve the difference equation Solve the differece equatio Solutio: y + 3 3y + + y 0 give tat y 0 4, y 0 ad y 8. Let Z{y()} F() Taig Z-trasform o both sides i (), we get y + 3 3y + + y 0 () Z y + 3 3y + + y Z 0 Z y + 3 3Z y + + Z y

Διαβάστε περισσότερα

Solutions: Homework 3

Solutions: Homework 3 Solutios: Homework 3 Suppose that the radom variables Y,, Y satisfy Y i = βx i + ε i : i,, where x,, x R are fixed values ad ε,, ε Normal0, σ ) with σ R + kow Fid ˆβ = MLEβ) IND Solutio: Observe that Y

Διαβάστε περισσότερα

Section 8.3 Trigonometric Equations

Section 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 περισσότερα

Finite Field Problems: Solutions

Finite 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 περισσότερα

FREE VIBRATION OF A SINGLE-DEGREE-OF-FREEDOM SYSTEM Revision B

FREE VIBRATION OF A SINGLE-DEGREE-OF-FREEDOM SYSTEM Revision B FREE VIBRATION OF A SINGLE-DEGREE-OF-FREEDOM SYSTEM Revisio B By Tom Irvie Email: tomirvie@aol.com February, 005 Derivatio of the Equatio of Motio Cosier a sigle-egree-of-freeom system. m x k c where m

Διαβάστε περισσότερα

derivation of the Laplacian from rectangular to spherical coordinates

derivation 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 περισσότερα

CHAPTER 101 FOURIER SERIES FOR PERIODIC FUNCTIONS OF PERIOD

CHAPTER 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 περισσότερα

Areas and Lengths in Polar Coordinates

Areas 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 περισσότερα

F19MC2 Solutions 9 Complex Analysis

F19MC2 Solutions 9 Complex Analysis F9MC Solutions 9 Complex Analysis. (i) Let f(z) = eaz +z. Then f is ifferentiable except at z = ±i an so by Cauchy s Resiue Theorem e az z = πi[res(f,i)+res(f, i)]. +z C(,) Since + has zeros of orer at

Διαβάστε περισσότερα

Math221: HW# 1 solutions

Math221: 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 περισσότερα

2 Composition. Invertible Mappings

2 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 περισσότερα

Section 7.6 Double and Half Angle Formulas

Section 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 περισσότερα

Example Sheet 3 Solutions

Example 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 περισσότερα

Areas and Lengths in Polar Coordinates

Areas 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 περισσότερα

Exercises 10. Find a fundamental matrix of the given system of equations. Also find the fundamental matrix Φ(t) satisfying Φ(0) = I. 1.

Exercises 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 περισσότερα

Statistical Inference I Locally most powerful tests

Statistical 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 περισσότερα

4.6 Autoregressive Moving Average Model ARMA(1,1)

4.6 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 περισσότερα

Fourier Series. MATH 211, Calculus II. J. Robert Buchanan. Spring Department of Mathematics

Fourier 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 περισσότερα

Matrices and Determinants

Matrices 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 περισσότερα

HOMEWORK 4 = G. In order to plot the stress versus the stretch we define a normalized stretch:

HOMEWORK 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 περισσότερα

MATH423 String Theory Solutions 4. = 0 τ = f(s). (1) dτ ds = dxµ dτ f (s) (2) dτ 2 [f (s)] 2 + dxµ. dτ f (s) (3)

MATH423 String Theory Solutions 4. = 0 τ = f(s). (1) dτ ds = dxµ dτ f (s) (2) dτ 2 [f (s)] 2 + dxµ. dτ f (s) (3) 1. MATH43 String Theory Solutions 4 x = 0 τ = fs). 1) = = f s) ) x = x [f s)] + f s) 3) equation of motion is x = 0 if an only if f s) = 0 i.e. fs) = As + B with A, B constants. i.e. allowe reparametrisations

Διαβάστε περισσότερα

Solutions to Exercise Sheet 5

Solutions 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 περισσότερα

ST5224: Advanced Statistical Theory II

ST5224: 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 περισσότερα

Practice Exam 2. Conceptual Questions. 1. State a Basic identity and then verify it. (a) Identity: Solution: One identity is csc(θ) = 1

Practice 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 περισσότερα

Econ 2110: Fall 2008 Suggested Solutions to Problem Set 8 questions or comments to Dan Fetter 1

Econ 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 περισσότερα

Introduction of Numerical Analysis #03 TAGAMI, Daisuke (IMI, Kyushu University)

Introduction of Numerical Analysis #03 TAGAMI, Daisuke (IMI, Kyushu University) Itroductio of Numerical Aalysis #03 TAGAMI, Daisuke (IMI, Kyushu Uiversity) web page of the lecture: http://www2.imi.kyushu-u.ac.jp/~tagami/lec/ Strategy of Numerical Simulatios Pheomea Error modelize

Διαβάστε περισσότερα

p n r.01.05.10.15.20.25.30.35.40.45.50.55.60.65.70.75.80.85.90.95

p n r.01.05.10.15.20.25.30.35.40.45.50.55.60.65.70.75.80.85.90.95 r r Table 4 Biomial Probability Distributio C, r p q This table shows the probability of r successes i idepedet trials, each with probability of success p. p r.01.05.10.15.0.5.30.35.40.45.50.55.60.65.70.75.80.85.90.95

Διαβάστε περισσότερα

Section 9.2 Polar Equations and Graphs

Section 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 περισσότερα

Homework 8 Model Solution Section

Homework 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 περισσότερα

If we restrict the domain of y = sin x to [ π, π ], the restrict function. y = sin x, π 2 x π 2

If we restrict the domain of y = sin x to [ π, π ], the restrict function. y = sin x, π 2 x π 2 Chapter 3. Analytic Trigonometry 3.1 The inverse sine, cosine, and tangent functions 1. Review: Inverse function (1) f 1 (f(x)) = x for every x in the domain of f and f(f 1 (x)) = x for every x in the

Διαβάστε περισσότερα

PARTIAL NOTES for 6.1 Trigonometric Identities

PARTIAL 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 περισσότερα

If we restrict the domain of y = sin x to [ π 2, π 2

If we restrict the domain of y = sin x to [ π 2, π 2 Chapter 3. Analytic Trigonometry 3.1 The inverse sine, cosine, and tangent functions 1. Review: Inverse function (1) f 1 (f(x)) = x for every x in the domain of f and f(f 1 (x)) = x for every x in the

Διαβάστε περισσότερα

Other Test Constructions: Likelihood Ratio & Bayes Tests

Other 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 περισσότερα

Fourier Series. Fourier Series

Fourier Series. Fourier Series ECE 37 Z. Aliyazicioglu Elecrical & Compuer Egieerig Dep. Cal Poly Pomoa Periodic sigal is a fucio ha repeas iself every secods. x() x( ± ) : period of a fucio, : ieger,,3, x() 3 x() x() Periodic sigal

Διαβάστε περισσότερα

Section 8.2 Graphs of Polar Equations

Section 8.2 Graphs of Polar Equations Section 8. Graphs of Polar Equations Graphing Polar Equations The graph of a polar equation r = f(θ), or more generally F(r,θ) = 0, consists of all points P that have at least one polar representation

Διαβάστε περισσότερα

DESIGN OF MACHINERY SOLUTION MANUAL h in h 4 0.

DESIGN 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 περισσότερα

Solution to Review Problems for Midterm III

Solution to Review Problems for Midterm III Solution to Review Problems for Mierm III Mierm III: Friday, November 19 in class Topics:.8-.11, 4.1,4. 1. Find the derivative of the following functions and simplify your answers. (a) x(ln(4x)) +ln(5

Διαβάστε περισσότερα

A study on generalized absolute summability factors for a triangular matrix

A study on generalized absolute summability factors for a triangular matrix Proceedigs of the Estoia Acadey of Scieces, 20, 60, 2, 5 20 doi: 0.376/proc.20.2.06 Available olie at www.eap.ee/proceedigs A study o geeralized absolute suability factors for a triagular atrix Ere Savaş

Διαβάστε περισσότερα

The Heisenberg Uncertainty Principle

The Heisenberg Uncertainty Principle Chemistry 460 Sprig 015 Dr. Jea M. Stadard March, 015 The Heiseberg Ucertaity Priciple A policema pulls Werer Heiseberg over o the Autobah for speedig. Policema: Sir, do you kow how fast you were goig?

Διαβάστε περισσότερα

Απόκριση σε Μοναδιαία Ωστική Δύναμη (Unit Impulse) Απόκριση σε Δυνάμεις Αυθαίρετα Μεταβαλλόμενες με το Χρόνο. Απόστολος Σ.

Απόκριση 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 περισσότερα

w o = R 1 p. (1) R = p =. = 1

w o = R 1 p. (1) R = p =. = 1 Πανεπιστήµιο Κρήτης - Τµήµα Επιστήµης Υπολογιστών ΗΥ-570: Στατιστική Επεξεργασία Σήµατος 205 ιδάσκων : Α. Μουχτάρης Τριτη Σειρά Ασκήσεων Λύσεις Ασκηση 3. 5.2 (a) From the Wiener-Hopf equation we have:

Διαβάστε περισσότερα

2. THEORY OF EQUATIONS. PREVIOUS EAMCET Bits.

2. 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 περισσότερα

Pg 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 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 περισσότερα

) 2. δ δ. β β. β β β β. r k k. tll. m n Λ + +

) 2. δ δ. β β. β β β β. r k k. tll. m n Λ + + Techical Appedix o Hamig eposis ad Helpig Bowes: The ispaae Impac of Ba Cosolidaio (o o be published bu o be made available upo eques. eails of Poofs of Poposiios 1 ad To deive Poposiio 1 s exac ad sufficie

Διαβάστε περισσότερα

Overview. Transition Semantics. Configurations and the transition relation. Executions and computation

Overview. Transition Semantics. Configurations and the transition relation. Executions and computation Overview Transition Semantics Configurations and the transition relation Executions and computation Inference rules for small-step structural operational semantics for the simple imperative language Transition

Διαβάστε περισσότερα

α β

α β 6. Eerg, Mometum coefficiets for differet velocit distributios Rehbock obtaied ) For Liear Velocit Distributio α + ε Vmax { } Vmax ε β +, i which ε v V o Give: α + ε > ε ( α ) Liear velocit distributio

Διαβάστε περισσότερα

Lecture 17: Minimum Variance Unbiased (MVUB) Estimators

Lecture 17: Minimum Variance Unbiased (MVUB) Estimators ECE 830 Fall 2011 Statistical Sigal Processig istructor: R. Nowak, scribe: Iseok Heo Lecture 17: Miimum Variace Ubiased (MVUB Estimators Ultimately, we would like to be able to argue that a give estimator

Διαβάστε περισσότερα

LAD Estimation for Time Series Models With Finite and Infinite Variance

LAD Estimation for Time Series Models With Finite and Infinite Variance LAD Estimatio for Time Series Moels With Fiite a Ifiite Variace Richar A. Davis Colorao State Uiversity William Dusmuir Uiversity of New South Wales 1 LAD Estimatio for ARMA Moels fiite variace ifiite

Διαβάστε περισσότερα

Fractional Colorings and Zykov Products of graphs

Fractional Colorings and Zykov Products of graphs Fractional Colorings and Zykov Products of graphs Who? Nichole Schimanski When? July 27, 2011 Graphs A graph, G, consists of a vertex set, V (G), and an edge set, E(G). V (G) is any finite set E(G) is

Διαβάστε περισσότερα

b. Use the parametrization from (a) to compute the area of S a as S a ds. Be sure to substitute for ds!

b. 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 περισσότερα

Potential Dividers. 46 minutes. 46 marks. Page 1 of 11

Potential Dividers. 46 minutes. 46 marks. Page 1 of 11 Potential Dividers 46 minutes 46 marks Page 1 of 11 Q1. In the circuit shown in the figure below, the battery, of negligible internal resistance, has an emf of 30 V. The pd across the lamp is 6.0 V and

Διαβάστε περισσότερα

Γιάννης Σαριδάκης Σχολή Μ.Π.Δ., Πολυτεχνείο Κρήτης

Γιάννης Σαριδάκης Σχολή Μ.Π.Δ., Πολυτεχνείο Κρήτης 2 η Διάλεξη Ακολουθίες 29 Νοεµβρίου 206 Γιάννης Σαριδάκης Σχολή Μ.Π.Δ., Πολυτεχνείο Κρήτης ΑΠΕΙΡΟΣΤΙΚΟΣ ΛΟΓΙΣΜΟΣ, ΤΟΜΟΣ Ι - Fiey R.L. / Weir M.D. / Giordao F.R. Πανεπιστημιακές Εκδόσεις Κρήτης 2 Όρια Ακολουθιών

Διαβάστε περισσότερα

B.A. (PROGRAMME) 1 YEAR

B.A. (PROGRAMME) 1 YEAR Graduate Course B.A. (PROGRAMME) YEAR ALGEBRA AND CALCULUS (PART-A : ALGEBRA) CONTENTS Lesso Lesso Lesso Lesso Lesso Lesso : Complex Numbers : De Moivre s Theorem : Applicatios of De Moivre s Theorem 4

Διαβάστε περισσότερα

Ψηφιακή Επεξεργασία Εικόνας

Ψηφιακή Επεξεργασία Εικόνας ΠΑΝΕΠΙΣΤΗΜΙΟ ΙΩΑΝΝΙΝΩΝ ΑΝΟΙΚΤΑ ΑΚΑΔΗΜΑΪΚΑ ΜΑΘΗΜΑΤΑ Ψηφιακή Επεξεργασία Εικόνας Φιλτράρισμα στο πεδίο των συχνοτήτων Διδάσκων : Αναπληρωτής Καθηγητής Νίκου Χριστόφορος Άδειες Χρήσης Το παρόν εκπαιδευτικό

Διαβάστε περισσότερα

ΤΜΗΜΑ ΗΛΕΚΤΡΟΛΟΓΩΝ ΜΗΧΑΝΙΚΩΝ ΚΑΙ ΜΗΧΑΝΙΚΩΝ ΥΠΟΛΟΓΙΣΤΩΝ

ΤΜΗΜΑ ΗΛΕΚΤΡΟΛΟΓΩΝ ΜΗΧΑΝΙΚΩΝ ΚΑΙ ΜΗΧΑΝΙΚΩΝ ΥΠΟΛΟΓΙΣΤΩΝ ΤΜΗΜΑ ΗΛΕΚΤΡΟΛΟΓΩΝ ΜΗΧΑΝΙΚΩΝ ΚΑΙ ΜΗΧΑΝΙΚΩΝ ΥΠΟΛΟΓΙΣΤΩΝ ΗΜΥ 311: Διακριτή Ανάλυση και Δομές Χειμερινό Εξάμηνο 016 Σειρά Ασκήσεων : Συναρτήσεις, Σχέσεις, Σειρές και Αθροίσματα, Αλγόριθμοι και Πολυπλοκότητα

Διαβάστε περισσότερα

ΤΜΗΜΑ ΗΛΕΚΤΡΟΛΟΓΩΝ ΜΗΧΑΝΙΚΩΝ ΚΑΙ ΜΗΧΑΝΙΚΩΝ ΥΠΟΛΟΓΙΣΤΩΝ

ΤΜΗΜΑ ΗΛΕΚΤΡΟΛΟΓΩΝ ΜΗΧΑΝΙΚΩΝ ΚΑΙ ΜΗΧΑΝΙΚΩΝ ΥΠΟΛΟΓΙΣΤΩΝ ΗΜΥ 3: ΔΙΑΚΡΙΤΗ ΑΝΑΛΥΣΗ ΚΑΙ ΔΟΜΕΣ ΤΜΗΜΑ ΗΛΕΚΤΡΟΛΟΓΩΝ ΜΗΧΑΝΙΚΩΝ ΚΑΙ ΜΗΧΑΝΙΚΩΝ ΥΠΟΛΟΓΙΣΤΩΝ ΗΜΥ 3: Διακριτή Ανάλυση και Δομές Χειμερινό Εξάμηνο 06 Σειρά Ασκήσεων : Συναρτήσεις, Σχέσεις, Σειρές και Αθροίσματα,

Διαβάστε περισσότερα

Q1a. HeavisideTheta x. Plot f, x, Pi, Pi. Simplify, n Integers

Q1a. HeavisideTheta x. Plot f, x, Pi, Pi. Simplify, n Integers 2 M2 Fourier Series answers in Mathematica Note the function HeavisideTheta is for x>0 and 0 for x

Διαβάστε περισσότερα

Μια εισαγωγή στα Μαθηματικά για Οικονομολόγους

Μια εισαγωγή στα Μαθηματικά για Οικονομολόγους Μια εισαγωγή στα Μαθηματικά για Οικονομολόγους Μαθηματικά Ικανές και αναγκαίες συνθήκες Έστω δυο προτάσεις Α και Β «Α είναι αναγκαία συνθήκη για την Β» «Α είναι ικανή συνθήκη για την Β» Α is ecessary for

Διαβάστε περισσότερα

Approximation of distance between locations on earth given by latitude and longitude

Approximation 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 περισσότερα

Inverse trigonometric functions & General Solution of Trigonometric Equations. ------------------ ----------------------------- -----------------

Inverse 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 περισσότερα

SCITECH Volume 13, Issue 2 RESEARCH ORGANISATION Published online: March 29, 2018

SCITECH Volume 13, Issue 2 RESEARCH ORGANISATION Published online: March 29, 2018 Journal of rogressive Research in Mathematics(JRM) ISSN: 2395-028 SCITECH Volume 3, Issue 2 RESEARCH ORGANISATION ublished online: March 29, 208 Journal of rogressive Research in Mathematics www.scitecresearch.com/journals

Διαβάστε περισσότερα

Degenerate Perturbation Theory

Degenerate Perturbation Theory R.G. Griffi BioNMR School page 1 Degeerate Perturbatio Theory 1.1 Geeral Whe cosiderig the CROSS EFFECT it is ecessary to deal with degeerate eergy levels ad therefore degeerate perturbatio theory. The

Διαβάστε περισσότερα

The Simply Typed Lambda Calculus

The 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 περισσότερα

ΚΥΠΡΙΑΚΗ ΕΤΑΙΡΕΙΑ ΠΛΗΡΟΦΟΡΙΚΗΣ CYPRUS COMPUTER SOCIETY ΠΑΓΚΥΠΡΙΟΣ ΜΑΘΗΤΙΚΟΣ ΔΙΑΓΩΝΙΣΜΟΣ ΠΛΗΡΟΦΟΡΙΚΗΣ 19/5/2007

ΚΥΠΡΙΑΚΗ 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 περισσότερα

On a four-dimensional hyperbolic manifold with finite volume

On a four-dimensional hyperbolic manifold with finite volume BULETINUL ACADEMIEI DE ŞTIINŢE A REPUBLICII MOLDOVA. MATEMATICA Numbers 2(72) 3(73), 2013, Pages 80 89 ISSN 1024 7696 On a four-dimensional hyperbolic manifold with finite volume I.S.Gutsul Abstract. In

Διαβάστε περισσότερα

ΠΑΝΕΠΙΣΤΗΜΙΟ ΚΥΠΡΟΥ ΤΜΗΜΑ ΠΛΗΡΟΦΟΡΙΚΗΣ. ΕΠΛ342: Βάσεις Δεδομένων. Χειμερινό Εξάμηνο Φροντιστήριο 10 ΛΥΣΕΙΣ. Επερωτήσεις SQL

ΠΑΝΕΠΙΣΤΗΜΙΟ ΚΥΠΡΟΥ ΤΜΗΜΑ ΠΛΗΡΟΦΟΡΙΚΗΣ. ΕΠΛ342: Βάσεις Δεδομένων. Χειμερινό Εξάμηνο Φροντιστήριο 10 ΛΥΣΕΙΣ. Επερωτήσεις SQL ΠΑΝΕΠΙΣΤΗΜΙΟ ΚΥΠΡΟΥ ΤΜΗΜΑ ΠΛΗΡΟΦΟΡΙΚΗΣ ΕΠΛ342: Βάσεις Δεδομένων Χειμερινό Εξάμηνο 2013 Φροντιστήριο 10 ΛΥΣΕΙΣ Επερωτήσεις SQL Άσκηση 1 Για το ακόλουθο σχήμα Suppliers(sid, sname, address) Parts(pid, pname,

Διαβάστε περισσότερα

CHAPTER 48 APPLICATIONS OF MATRICES AND DETERMINANTS

CHAPTER 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 περισσότερα

COMMON RANDOM FIXED POINT THEOREMS IN SYMMETRIC SPACES

COMMON RANDOM FIXED POINT THEOREMS IN SYMMETRIC SPACES Iteratioal Joural of Avacemets i Research & Techology, Volume, Issue, Jauary-03 ISSN 78-7763 COMMON RANDOM FIXED POINT THEOREMS IN SYMMETRIC SPACES Dr Neetu Vishwakarma a Dr M S Chauha Sagar Istitute of

Διαβάστε περισσότερα

Similarly, we may define hyperbolic functions cosh α and sinh α from the unit hyperbola

Similarly, we may define hyperbolic functions cosh α and sinh α from the unit hyperbola Universit of Hperbolic Functions The trigonometric functions cos α an cos α are efine using the unit circle + b measuring the istance α in the counter-clockwise irection along the circumference of the

Διαβάστε περισσότερα

CRASH COURSE IN PRECALCULUS

CRASH 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 περισσότερα

SCHOOL OF MATHEMATICAL SCIENCES G11LMA Linear Mathematics Examination Solutions

SCHOOL 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 περισσότερα

Outline. M/M/1 Queue (infinite buffer) M/M/1/N (finite buffer) Networks of M/M/1 Queues M/G/1 Priority Queue

Outline. M/M/1 Queue (infinite buffer) M/M/1/N (finite buffer) Networks of M/M/1 Queues M/G/1 Priority Queue Queueig Aalysis Outlie M/M/ Queue (ifiite buffer M/M//N (fiite buffer M/M// (Erlag s B forula M/M/ (Erlag s C forula Networks of M/M/ Queues M/G/ Priority Queue M/M/ M: Markovia/Meoryless Arrival process

Διαβάστε περισσότερα

Strain gauge and rosettes

Strain gauge and rosettes Strain gauge and rosettes Introduction A strain gauge is a device which is used to measure strain (deformation) on an object subjected to forces. Strain can be measured using various types of devices classified

Διαβάστε περισσότερα

the total number of electrons passing through the lamp.

the total number of electrons passing through the lamp. 1. A 12 V 36 W lamp is lit to normal brightness using a 12 V car battery of negligible internal resistance. The lamp is switched on for one hour (3600 s). For the time of 1 hour, calculate (i) the energy

Διαβάστε περισσότερα

( )( ) ( ) ( )( ) ( )( ) β = Chapter 5 Exercise Problems EX α So 49 β 199 EX EX EX5.4 EX5.5. (a)

( )( ) ( ) ( )( ) ( )( ) β = Chapter 5 Exercise Problems EX α So 49 β 199 EX EX EX5.4 EX5.5. (a) hapter 5 xercise Problems X5. α β α 0.980 For α 0.980, β 49 0.980 0.995 For α 0.995, β 99 0.995 So 49 β 99 X5. O 00 O or n 3 O 40.5 β 0 X5.3 6.5 μ A 00 β ( 0)( 6.5 μa) 8 ma 5 ( 8)( 4 ) or.88 P on + 0.0065

Διαβάστε περισσότερα

Ordinal Arithmetic: Addition, Multiplication, Exponentiation and Limit

Ordinal 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 περισσότερα

9.09. # 1. Area inside the oval limaçon r = cos θ. To graph, start with θ = 0 so r = 6. Compute dr

9.09. # 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 περισσότερα