Partial Di erential Equations
|
|
- Kristin Evans
- 7 years ago
- Views:
Transcription
1 Partial Di eretial Equatios Partial Di eretial Equatios Much of moder sciece, egieerig, ad mathematics is based o the study of partial di eretial equatios, where a partial di eretial equatio is a equatio ivolvig partial derivatives which implicitly de es a fuctio of or more variables. For example, if u (x; t) is the temperature of a metal bar at a distace x from the iitial ed of the bar, the uder suitable coditios u (x; t) is a solutio to the heat = u where k is a costat. As aother example, cosider that if u (x; t) is the displacemet of a strig a time t; the the vibratio of the strig is likely to satisfy the oe dimesioal wave equatio for a costat, which = u (1) Whe a partial di eretial equatio occurs i a applicatio, our goal is usually that of solvig the equatio, where a give fuctio is a solutio of a partial di eretial equatio if it is implicitly de ed by that equatio. That is, a solutio is a fuctio that satis es the equatio. EXAMPLE 1 Show that if a is a costat, the u (x; y) = si (at) cos (x) is a solutio = u () Solutio: Sice a is costat, the partials with respect to t = a cos (at) cos (x) = a si (at) si (x) (3) Moreover, u x = si (at) si (x) ad u xx = si (at) cos (x) ; so that u = a si (at) cos (x) (4) 1
2 Sice (3) ad (4) are the same, u (x; t) = si (at) cos (x) is a solutio to (). EXAMPLE Show that u (x; t) = e y si (x) is a solutio to Laplace s Equatio, = 0 Solutio: To begi with, u x = e y cos (x) ad u xx = e y si (x) : Moreover, u y = e y si (x) ad u yy = e y si (x) ; so that = ey si (x) + e y si (x) = 0 Check your Readig: Why are u; u y, ad u yy the same as u i example? Separatio of Variables Solutios to may (but ot all!) partial di eretial equatios ca be obtaied usig the techique kow as separatio of variables. It is based o the fact that if f (x) ad g (t) are fuctios of idepedet variables x; t respectively ad if f (x) = g (t) the there must be a costat for which f (x) = ad g (t) = : ( The proof is straightforward, f g (t) = 0 =) f 0 (x) = 0 =) f g f (x) = 0 =) g0 (t) = 0 =) g (x) costat) I separatio of variables, we rst assume that the solutio is of the separated form u (x; t) = X (x) T (t) We the substitute the separated form ito the equatio, ad if possible, move the x-terms to oe side ad the t-terms to the other. If ot possible, the this method will ot work; ad correspodigly, we say that the partial di eretial equatio is ot separable. Oce separated, the two sides of the equatio must be costat, thus requirig the solutios to ordiary di eretial equatios. A table of solutios to commo di eretial equatios is give below: Equatio Geeral Solutio y 00 +! y = 0 y (x) = A cos (!x) + B si (!x) y 0 = ky y (t) = P e kt y 00! y = 0 y (x) = A cosh (!x) + B sih (!x)
3 The product of X (x) ad T (t) is the separated solutio of the partial di eretial equatio. EXAMPLE 3 For k costat, d the separated solutio to the Heat = u Solutio: To do so, we substitute u (x; t) = X (x) T (t) ito the equatio @ (X (x) T (t)) = k (X (x) T Sice X (x) does ot deped o t; ad sice T (t) does ot deped o x; we obtai T (t) = kt X (x) which after evaluatig the derivatives simpli es to X (x) T 0 (t) = kt (t) X 00 (x) To separate the variables, we divide throughout by kx (x) T (t): This i tur simpli es to X (x) T 0 (t) kx (x) T (t) = kt (t) X00 (x) kx (x) T (t) T 0 (t) kt (t) = X00 (x) X (x) Thus, there is a costat such that T 0 kt = ad X 00 X = These i tur reduce to the di eretial equatios T 0 = kt ad X 00 = X The solutio to the rst is a expoetial fuctio of the form T (t) = P e kt If > 0; however, the temperature would grow to 1; which is ot physically possible. Thus, we assume that is egative, which is to say that =! for some umber!: As a result, we have X 00 =! X or X 00 +! X = 0 3
4 The equatio X 00 +! X = 0 is a harmoic oscillator, which has a solutio X (x) = A cos (!x) + B si (!x) Cosequetly, the separated solutio for the heat equatio is u (x; t) = X (x) T (t) = P e! kt (A cos (!x) + B si (!x)) It is importat to ote that i geeral a separated solutio to a partial di eretial equatio is ot the oly solutio or form of a solutio. Ideed, i the exercises, we will show that u (x; t) = 1 p kt e x =(4kt) is also a solutio to the heat equatio i example 3. As a simpler example, cosider that F (x; y) = y partial di eretial equatio F x + xf y = 0 This is because substitutig F x = x is a solutio to the x ad F y = 1 ito the equatio yields F x + xf y = x + x 1 = 0 Now let s obtai a di eret solutio by assumig a separated solutio of the form F (x; y) = X (x) Y (y) : EXAMPLE 4 Fid the separated solutio to F x + xf y = 0: Solutio: The separated form F (x; y) = X (x) Y (y) results i which i tur (X (x) Y (y)) + (X (x) Y (y)) = X 0 (x) Y (y) = xx (x) Y 0 (y) Dividig both sides by X (x) Y (y) leads to X 0 (x) xx (x) = Y 0 (y) Y (y) However, a fuctio of x ca be equal to a fuctio of y for all x ad y oly if both fuctios are costat. Thus, there is a costat such that X 0 (x) xx (x) = ad Y 0 (y) Y (y) = 4
5 It follows that Y 0 (y) = Y (y) ; which implies that Y (y) = C 1 e y : However, X 0 (x) = xx (x), so that separatio of variables yields dx dx = xx =) dx X = xdx Thus, R dx=x = R xdx; which yields l jxj = x + C jxj = e x +C X (x) = e C e x Thus, if we let C 3 = e C ; the Y (y) = C 3 exp x = ad the separated solutio is F (x; y) = Ce x e y = Ce (y x ) where C = C 1 C 3 is a arbitrary costat. Notice that there are similarities betwee the separated solutio F (x; y) = Ce (y x ) ad the other solutio we stated earlier, F (x; y) = y solutios are clearly ot the same. x : However, the two Check your Readig: Why is this method called separatio of variables? Boudary Coditios Partial di eretial equatios ofte occur with boudary coditios, which are costraits o the solutio at di eret poits i space. To illustrate how boudary coditios arise i applicatios, let us suppose that u (x; t) is the displacemet at x i [0; l] of a strig of legth l at time t: 5
6 Tesio o a short sectio of the strig over the iterval [x; x + x] is alog the tagets to the edpoits, y = u(x,t) x x+ x dy (x) = u x (x,t) dx x x+ x dy (x+ x) = u x (x+ x,t) dx Thus, the et tesio resposible for pullig the strig toward the x-axis is proportioal to the di erece i the slopes, Net T esio = k ( u x (x + x; t) u x (x; t) ) where k is the tesio costat (see for details). Cosequetly, if is the mass-desity of the strig (mass per uit legth), the mass times acceleratio equal to the force of tesio yields = k ( u x (x + x; t) u x (x; t) ) for arbitrarily small x: Solvig for u tt ad lettig x approach 0 = k lim x!0 u x (x + x; t) u x (x; t) x = k so that if we let a = k=; the the partial di eretial equatio describig the motio of the strig = u (5) which is the oe-dimesioal wave equatio. Moreover, sice the strig is xed at x = 0 ad x = l; we also have the boudary coditios u (0; t) = 0 ad u (l; t) = 0 (6) for all times t: If we avoid the trivial solutio (that of o vibratio, u = 0); the these boudary coditios ca be used to determie some of the arbitrary costats i the separated solutio. EXAMPLE 5 Fid the solutio of the oe dimesioal wave equatio (5) subject to the boudary coditios (6). 6
7 Solutio: To do so, we substitute u (x; t) = X (x) T (t) ito the equatio (X (x) T (t)) = (X (x) T (t)) =) X (x) T 00 (t) = a T (t) X 00 (x) To separate the variables, we the divide throughout by a X (x) T (t): This i tur simpli es to X (x) T 00 (t) a X (x) T (t) = a T (t) X 00 (x) a X (x) T (t) T 00 (t) a T (t) = X00 (x) X (x) As a result, there must be a costat such that T 00 a T = ad X 00 X = These i tur reduce to the di eretial equatios T 00 = a T ad X 00 = X If > 0; however, the oscillatios would become arbitrarily large i amplitude, which is ot physically possible. Thus, we assume that is egative, which is to say that =! for some umber!: As a result, we have T 00 = a! T ad X 00 =! X Both equatios are harmoic oscillators, so that the geeral solutios are T (t) = A 1 cos (a!t)+b 1 si (a!t) ad X (x) = A cos (!x)+b si (!x) where A 1 ; B 1 ; A ; ad B are arbitrary costats. Let s ow cocetrate o X (x) : The boudary coditios (6) imply that u (0; t) = X (0) T (t) = 0 ad u (l; t) = X (l) T (t) = 0 If we let T (t) = 0; the we will obtai the solutio u (x; t) = 0 for all t: This is called the trivial solutio sice it is the solutio correspodig to the strig ot movig at all. To avoid the trivial solutio, we thus assume that X (0) = 0 ad X (l) = 0 7
8 However, X (x) = A cos (!x) + B si (!x) ; so that X (0) = 0 implies that 0 = A cos (0) + B si (0) = A Thus, A = 0 ad X (x) = B si (!x) : The boudary coditio X (l) = 0 the implies that B si (!l) = 0 If we let B = 0; the we agai obtai the trivial solutio. To avoid the trivial solutio, we let si (!l) = 0; which i tur implies that!l = for ay iteger : Thus, there is a solutio for! = =l for each value of ; which meas that X (x) = B si l x is a solutio to the vibratig strig equatio for each : Cosequetly, for each iteger there is a separated solutio of the form h a a i u (x; t) = A 1 cos t + B 1 si t B si l l l x (7) Check your Readig: Where did the a=l come from i the al form of the separated solutio? Liearity ad Fourier Series We say that a partial di eretial equatio is liear if the liear combiatio of ay two solutios is also a solutio. For example, suppose that p (x; t) ad q (x; t) are both solutios to the heat equatio i.e., = = q (8) A liear combiatio of p ad q is of the form u (x; t) = Ap (x; t) + Bq (x; t) where A; B are both costats. Moreover, so that (8) implies (Ap (x; t) + Bq (x; t)) @t = = p + q = (Ap (x; t) + Bq (x; 8
9 That is, the liear combiatio u (x; t) = Ap (x; t) + Bq (x; t) is also a solutio to the heat equatio, ad cosequetly, we say that the heat equatio is a liear partial di eretial equatio. Suppose ow that a liear partial di eretial equatio has both boudary coditios ad iitial coditios, where iitial coditios are costraits o the solutio ad its derivatives at a xed poit i time. The a complete solutio to the partial di eretial equatio ca ofte be obtaied from the Fourier series decompositios of the iitial coditios. For example, let us suppose that the vibratig strig i example 5 is plucked at time t = 0, which is to say that it is released from rest at time t = 0 with a iitial shape give by the graph of the fuctio y = f (x): The the iitial coditios for the vibratig strig are u (x; 0) = f (x) ad (x; 0) = Let s apply the iitial coditios to the separated solutio (7). The iitial coditio u t (x; 0) = 0 implies that X (x) T 0 (0) = 0; so that to avoid the trivial solutio we suppose that T 0 (0) = 0. Thus, 0 = T 0 (0) = a!a 1 si (0) + B 1 a! cos (0) = B 1 As a result, we must have T (t) = A 1 cos (at=l) ; ad if we de e b = A 1 B ; the (7) reduces to a u (x; t) = b cos t si l l x (9) As will be show i the exercises, the 1 dimesioal wave equatio is liear. Thus, if u j (x; t) ad u k (x; t) are solutios (9) for itegers j ad k, the u j (x; t)+ u k (x; t) is also a solutio. I fact, the sum all possible solutios, which is the sum of all solutios for ay positive iteger value of ; is a solutio called the 9
10 geeral solutio. That is, the geeral solutio to the 1 dimesioal wave equatio with the give boudary ad iitial coditios is u (x; t) = 1X a b cos t si l l x =1 (10) Hece, the oly task left is that of determiig the values of the costats b : However, (10) implies that u (x; 0) = 1X =1 ad sice u (x; 0) = f (x) ; this reduces to f (x) = b cos (0) si l x 1X =1 b si l x As a result, if f (x) is cotiuous ad if f (0) = f (l) ; the the costats b are the Fourier Sie coe ciets of f (x) o [0; l] ; which are give by b = l Z l 0 f (x) si l x dx (11) For more o Fourier series ad their relatioship to partial di eretial equatios, see the Maple worksheet associated with this sectio. EXAMPLE 6 What is the solutio to the vibratig strig problem for a foot log strig which is iitially at rest ad which has a iitial shape that is the same as the graph of the fuctio u (x; 0) = 1 1 jx 1j 1 Solutio: We begi by dig the Fourier coe ciets b ; which accordig to (11) are for a l = foot log strig give by b = Z jx 1j si 1 x dx :Evaluatig usig the computer algebra system Maple the yields si si () b = 3 10
11 However, sice is a iteger, si () = 0 for all. Thus, b reduces to si b = 3 But si = 0 whe is eve, so that b0 = b = : : : = b = 0: Thus, we oly have odd coe ciets of the form b 1 = si ; b si 3 = 3 3 ; b si 5 = 3 5 ; : : : which simplify to b 1 = (1) 3 1 ; b 3 = ( 1) 3 3 ; b 5 = (1) 3 5 ; : : : Odd umbers are of the form + 1 for = 0; 1; : : : : Thus, we have ( 1) b +1 = 3 ( + 1) ad the solutio (10) is of the form 1X ( 1) a ( + 1) ( + 1) u (x; t) = 3 ( + 1) cos t si x l l =0 The Fourier series (10) is kow as the Harmoic Series i music theory. Ideed, if we write the Fourier series i expaded form a a 3a 3 u (x; t) = b 1 cos l t si l x +b cos t si l l x +b 3 cos t si l l x +: : : the the rst term is kow as the fudametal, which correspods to the strig shape of y = si (x=l) ; which is xed at x = 0 ad x = l, oscillatig at a amplitude of b 1 : The oscillatios themselves have a frequecy of f 1 = a l rad sec 1cycle rad = a l cycles sec A "cycle per secod" is kow as a Hertz ad recall that a = k=; so that f 1 = k l Hz Thus, icreases i tesio k cause the fudametal pitch to rise, while legtheig the strig lowers the pitch. A heavier strig (larger ) has a lower pitch tha a lighter strig. The secod term i the Harmoic Series of the strig oscillates at a amplitude b with twice the frequecy of the fudametal, f = a l 11 = f 1 :
12 It is kow as the rst harmoic or rst overtoe of the strig, ad it correspods to the oscillatio of a strig shape y = si (x=l) that is xed at x = 0, x = l=, ad x = l i.e., half as log as the fudametal. Similarly, the third term is the secod harmoic, which oscillates at a frequecy of f 3 = 3f 1 ad which correspods to oscillatios at amplitude b 3 of siusoidal shapes a third as log as the fudametal. For example, if the strig is at a legth, tesio, ad mass so as to oscillate with a frequecy of 440 hz ("A" above middle "C"), the we also hear a pitch of f = 880 hz (a octave above the fudametal), a pitch of f 3 = 3 (440) hz (a octave ad a fth above the fudametal) ad so o. Exercises Show that the give fuctio is a solutio to the give partial di eretial equatio. Assume that k;!; a; ad c are costats. 1. u (x; y) = x 3 is a solutio to u. u (x; y) = 3x y y 3 is a solutio = 0 = 0 3. u (x; t) = t + x is a solutio u 4. u (x; t) = x + t is a solutio 5. u (x; y) = e x si (y) is a solutio to u = 0 = 0 6. u (x; y) = ta 1 y x is a solutio to 7. u (x; t) = e!kt cos (!x) is a solutio = u 8. u (x; t) = si (!x) si (a!t) is a solutio 9. u (x; t) = f (x + ct) is a solutio 10. u (x; t) = f (x ct) is a solutio = u = u = u Fid the separated solutio to each of the followig partial di eretial equatios. 1
13 Assume that k; a; c; ad are costat. = = 0 = 15. F x + e x F y = F x + 3x F y = u x + u t = = 19. u = 0 0. u t = u xx @t V = 0. u t = u xx = 0 + = 0 5. Show that u (x; t) = 1 p t e x =(4t) is a solutio to the heat equatio u t = u xx : 6. Show that u (x; y; z) = x + y + z 1= is a solutio to the 3 dimesioal Laplace equatio = 0 7. Let i = 1 ad suppose that u (x; y) ad v (x; y) are such that (x + iy) = u (x; y) + i v (x; y) Fid u ad v ad show that both satisfy Laplace s equatio that is, that = 0 v = 0 I additio, show that u ad v satisfy the Cauchy-Riema Equatios u x = v y ; u y = v x 8. Let i = 1 ad suppose that u (x; y) ad v (x; y) are such that (x + iy) 4 = u (x; y) + i v (x; y) Fid u ad v ad show that both satisfy Laplace s equatio that is, that = 0 v = 0 I additio, show that u ad v satisfy the Cauchy-Riema Equatios u x = v y ; u y = v x 9. Suppose that a large populatio of micro-orgaisms (e.g., bacteria or plakto) is distributed alog the x-axis. If u (x; t) is the populatio per uit legth at locatio x ad at time t; the u (x; t) satis es a di usio equatio of the = + ru 13
14 where is the rate of dispersal ad r is the birthrate of the micro-orgaisms. If ad r are positive costats, the what is a separated solutio of this di usio equatio? (adapted from Mathematical Models i Biology, Leah Edelstei- Keshet, Radom House, 1988, p. 441). 30. Suppose that t deotes time ad x deotes the age of a cell ia give populatio of cells, ad let u (x; t) dx = umber of cells whose age at time t is betwee x ad x + dx The u (x; t) is the cell desity per uit age at time t; ad give appropriate assumptios, it satis + v = d 0 where v 0 ad d 0 are positive costats. What is the separated solutio to this equatio? (adapted from Mathematical Models i Biology, Leah Edelstei- Keshet, Radom House, 1988, p. 466). 31. Fid the separated solutio of the telegraph equatio with zero self iductace: + RSu Here u (x; t) is the electrostatic potetial at time t at a poit x uits from oe ed of a trasmissio lie, ad R, C; ad S are the resistace, capacitace, ad leakage coductace per uit legth, respectively. 3. If V (x; t) is the membrae voltage at time t i secods ad at a distace x from the distal (i.e., iitial) ed of a uiform, cylidrical, ubrached sectio of a dedrite, the V (x; t) satis es d 4R V = + 1 V (1) R m where d is the diameter of the cylidrical dedritic sectio, R i is the resistivity of the itracellular uid, C m is the membrae capacitace, ad R m is the membrae resistivity. Fid a separated solutio to (1) give that C m ; R m ; ad R i are positive costats. 33. I Quatum mechaics, a particle movig i a straight lie is said to be i a state (x; t) if Z b a j (x; t)j dx represets the probability of the particle beig i the iterval [a; b] o the lie at time t: If a subatomic particle is travelig i a straight lie close to the speed of light, the it s state satis es the oe dimesioal @t = 14
15 where > 0 is costat. Fid the separated solutio of the oe dimesioal Klei-Gordo equatio. 34. If a subatomic particle is travelig i a straight lie much slower tha the speed of light ad o forces are actig o that particle, the its state (as explaied i problem 33) satis es the oe dimesioal Schrödiger equatio of a sigle = i@ (13) where i = 1: Fid the separated solutio of (13) (Hit: you will eed to use Euler s idetity e it = cos (t) + i si (t) 35. Show that if u (x; t) ad v (x; t) are both solutios to the oe dimesioal wave = u the so also is the fuctio w (x; t) = Au (x; t) + Bv (x; t) where A ad B are costats. What does this say about the wave equatio? 36. Show that if u (x; y) ad v (x; y) are both solutios to Laplace s equatio = 0 the so also is the fuctio w (x; y) = Au (x; y) + Bv (x; y) where A ad B are costats. What does this tell us about Laplace s equatio? 37. Suppose that the iitial coditios for the guitar strig i example 6 are x u (x; 0) = si ad (x; 0) = What are the coe ciets b i the solutio (10) for these iitial coditios? 38. Solve the vibratig strig problem for the boudary (0; t) = 0 ad (l; t) = ad for the iitial coditios u (x; 0) = f (x) ad u t (x; 0) = 0: 39. Heat Equatio I: Fid the geeral solutio to the heat equatio subject to the boudary = u u (0; t) = 0 u (; t) = Heat Equatio II: If the iitial coditio is u (x; 0) = x x ; the what are the Fourier coe ciets i the geeral solutio foud i exercise 39? 15
16 41. Laplace s Equatio I: Fid the geeral solutio to the Laplace equatio subject to the boudary coditios = 0 u (0; y) = 0 u (; y) = 0 4. Laplace s Equatio II: If the iitial coditios are u (x; 0) = si (x=) ad u y (x; 0) = 0; the what are the Fourier coe ciets i the geeral solutio foud i exercise 41? 43. Write to Lear: I a short essay, explai i your ow words why a equatio of the form f (x; y) = g (t) implies that both f (x; y) ad g (t) are costat. (x; y; ad t are both idepedet variables). 44. *What is a separated solutio of the -dimesioal wave = u a@ + u 45. Fid a separated solutio of the followig oliear wave = 16
In nite Sequences. Dr. Philippe B. Laval Kennesaw State University. October 9, 2008
I ite Sequeces Dr. Philippe B. Laval Keesaw State Uiversity October 9, 2008 Abstract This had out is a itroductio to i ite sequeces. mai de itios ad presets some elemetary results. It gives the I ite Sequeces
More informationTrigonometric Form of a Complex Number. The Complex Plane. axis. ( 2, 1) or 2 i FIGURE 6.44. The absolute value of the complex number z a bi is
0_0605.qxd /5/05 0:45 AM Page 470 470 Chapter 6 Additioal Topics i Trigoometry 6.5 Trigoometric Form of a Complex Number What you should lear Plot complex umbers i the complex plae ad fid absolute values
More informationSection 11.3: The Integral Test
Sectio.3: The Itegral Test Most of the series we have looked at have either diverged or have coverged ad we have bee able to fid what they coverge to. I geeral however, the problem is much more difficult
More informationHeat (or Diffusion) equation in 1D*
Heat (or Diffusio) equatio i D* Derivatio of the D heat equatio Separatio of variables (refresher) Worked eamples *Kreysig, 8 th Ed, Sectios.4b Physical assumptios We cosider temperature i a log thi wire
More informationSoving Recurrence Relations
Sovig Recurrece Relatios Part 1. Homogeeous liear 2d degree relatios with costat coefficiets. Cosider the recurrece relatio ( ) T () + at ( 1) + bt ( 2) = 0 This is called a homogeeous liear 2d degree
More informationOur aim is to show that under reasonable assumptions a given 2π-periodic function f can be represented as convergent series
8 Fourier Series Our aim is to show that uder reasoable assumptios a give -periodic fuctio f ca be represeted as coverget series f(x) = a + (a cos x + b si x). (8.) By defiitio, the covergece of the series
More informationSECTION 1.5 : SUMMATION NOTATION + WORK WITH SEQUENCES
SECTION 1.5 : SUMMATION NOTATION + WORK WITH SEQUENCES Read Sectio 1.5 (pages 5 9) Overview I Sectio 1.5 we lear to work with summatio otatio ad formulas. We will also itroduce a brief overview of sequeces,
More informationApproximating Area under a curve with rectangles. To find the area under a curve we approximate the area using rectangles and then use limits to find
1.8 Approximatig Area uder a curve with rectagles 1.6 To fid the area uder a curve we approximate the area usig rectagles ad the use limits to fid 1.4 the area. Example 1 Suppose we wat to estimate 1.
More informationConvexity, Inequalities, and Norms
Covexity, Iequalities, ad Norms Covex Fuctios You are probably familiar with the otio of cocavity of fuctios. Give a twicedifferetiable fuctio ϕ: R R, We say that ϕ is covex (or cocave up) if ϕ (x) 0 for
More informationI. Chi-squared Distributions
1 M 358K Supplemet to Chapter 23: CHI-SQUARED DISTRIBUTIONS, T-DISTRIBUTIONS, AND DEGREES OF FREEDOM To uderstad t-distributios, we first eed to look at aother family of distributios, the chi-squared distributios.
More informationSAMPLE QUESTIONS FOR FINAL EXAM. (1) (2) (3) (4) Find the following using the definition of the Riemann integral: (2x + 1)dx
SAMPLE QUESTIONS FOR FINAL EXAM REAL ANALYSIS I FALL 006 3 4 Fid the followig usig the defiitio of the Riema itegral: a 0 x + dx 3 Cosider the partitio P x 0 3, x 3 +, x 3 +,......, x 3 3 + 3 of the iterval
More informationUniversity of California, Los Angeles Department of Statistics. Distributions related to the normal distribution
Uiversity of Califoria, Los Ageles Departmet of Statistics Statistics 100B Istructor: Nicolas Christou Three importat distributios: Distributios related to the ormal distributio Chi-square (χ ) distributio.
More informationOverview of some probability distributions.
Lecture Overview of some probability distributios. I this lecture we will review several commo distributios that will be used ofte throughtout the class. Each distributio is usually described by its probability
More information1. MATHEMATICAL INDUCTION
1. MATHEMATICAL INDUCTION EXAMPLE 1: Prove that for ay iteger 1. Proof: 1 + 2 + 3 +... + ( + 1 2 (1.1 STEP 1: For 1 (1.1 is true, sice 1 1(1 + 1. 2 STEP 2: Suppose (1.1 is true for some k 1, that is 1
More informationFind the inverse Laplace transform of the function F (p) = Evaluating the residues at the four simple poles, we find. residue at z = 1 is 4te t
Homework Solutios. Chater, Sectio 7, Problem 56. Fid the iverse Lalace trasform of the fuctio F () (7.6). À Chater, Sectio 7, Problem 6. Fid the iverse Lalace trasform of the fuctio F () usig (7.6). Solutio:
More informationChapter 6: Variance, the law of large numbers and the Monte-Carlo method
Chapter 6: Variace, the law of large umbers ad the Mote-Carlo method Expected value, variace, ad Chebyshev iequality. If X is a radom variable recall that the expected value of X, E[X] is the average value
More informationAP Calculus AB 2006 Scoring Guidelines Form B
AP Calculus AB 6 Scorig Guidelies Form B The College Board: Coectig Studets to College Success The College Board is a ot-for-profit membership associatio whose missio is to coect studets to college success
More information1 Correlation and Regression Analysis
1 Correlatio ad Regressio Aalysis I this sectio we will be ivestigatig the relatioship betwee two cotiuous variable, such as height ad weight, the cocetratio of a ijected drug ad heart rate, or the cosumptio
More informationChapter 5: Inner Product Spaces
Chapter 5: Ier Product Spaces Chapter 5: Ier Product Spaces SECION A Itroductio to Ier Product Spaces By the ed of this sectio you will be able to uderstad what is meat by a ier product space give examples
More informationSequences and Series
CHAPTER 9 Sequeces ad Series 9.. Covergece: Defiitio ad Examples Sequeces The purpose of this chapter is to itroduce a particular way of geeratig algorithms for fidig the values of fuctios defied by their
More information0.7 0.6 0.2 0 0 96 96.5 97 97.5 98 98.5 99 99.5 100 100.5 96.5 97 97.5 98 98.5 99 99.5 100 100.5
Sectio 13 Kolmogorov-Smirov test. Suppose that we have a i.i.d. sample X 1,..., X with some ukow distributio P ad we would like to test the hypothesis that P is equal to a particular distributio P 0, i.e.
More informationChapter 7 Methods of Finding Estimators
Chapter 7 for BST 695: Special Topics i Statistical Theory. Kui Zhag, 011 Chapter 7 Methods of Fidig Estimators Sectio 7.1 Itroductio Defiitio 7.1.1 A poit estimator is ay fuctio W( X) W( X1, X,, X ) of
More informationCS103A Handout 23 Winter 2002 February 22, 2002 Solving Recurrence Relations
CS3A Hadout 3 Witer 00 February, 00 Solvig Recurrece Relatios Itroductio A wide variety of recurrece problems occur i models. Some of these recurrece relatios ca be solved usig iteratio or some other ad
More informationS. Tanny MAT 344 Spring 1999. be the minimum number of moves required.
S. Tay MAT 344 Sprig 999 Recurrece Relatios Tower of Haoi Let T be the miimum umber of moves required. T 0 = 0, T = 7 Iitial Coditios * T = T + $ T is a sequece (f. o itegers). Solve for T? * is a recurrece,
More informationProperties of MLE: consistency, asymptotic normality. Fisher information.
Lecture 3 Properties of MLE: cosistecy, asymptotic ormality. Fisher iformatio. I this sectio we will try to uderstad why MLEs are good. Let us recall two facts from probability that we be used ofte throughout
More informationA Theoretical and Experimental Analysis of the Acoustic Guitar. Eric Battenberg ME 173 5-18-09
A Theoretical ad Experimetal Aalysis of the Acoustic Guitar Eric Batteberg ME 173 5-18-09 1 Itroductio ad Methods The acoustic guitar is a striged musical istrumet frequetly used i popular music. Because
More informationwhere: T = number of years of cash flow in investment's life n = the year in which the cash flow X n i = IRR = the internal rate of return
EVALUATING ALTERNATIVE CAPITAL INVESTMENT PROGRAMS By Ke D. Duft, Extesio Ecoomist I the March 98 issue of this publicatio we reviewed the procedure by which a capital ivestmet project was assessed. The
More informationThe Binomial Multi- Section Transformer
4/15/21 The Bioial Multisectio Matchig Trasforer.doc 1/17 The Bioial Multi- Sectio Trasforer Recall that a ulti-sectio atchig etwork ca be described usig the theory of sall reflectios as: where: Γ ( ω
More informationTHE ARITHMETIC OF INTEGERS. - multiplication, exponentiation, division, addition, and subtraction
THE ARITHMETIC OF INTEGERS - multiplicatio, expoetiatio, divisio, additio, ad subtractio What to do ad what ot to do. THE INTEGERS Recall that a iteger is oe of the whole umbers, which may be either positive,
More informationExample 2 Find the square root of 0. The only square root of 0 is 0 (since 0 is not positive or negative, so those choices don t exist here).
BEGINNING ALGEBRA Roots ad Radicals (revised summer, 00 Olso) Packet to Supplemet the Curret Textbook - Part Review of Square Roots & Irratioals (This portio ca be ay time before Part ad should mostly
More informationSystems Design Project: Indoor Location of Wireless Devices
Systems Desig Project: Idoor Locatio of Wireless Devices Prepared By: Bria Murphy Seior Systems Sciece ad Egieerig Washigto Uiversity i St. Louis Phoe: (805) 698-5295 Email: bcm1@cec.wustl.edu Supervised
More informationTHE REGRESSION MODEL IN MATRIX FORM. For simple linear regression, meaning one predictor, the model is. for i = 1, 2, 3,, n
We will cosider the liear regressio model i matrix form. For simple liear regressio, meaig oe predictor, the model is i = + x i + ε i for i =,,,, This model icludes the assumptio that the ε i s are a sample
More informationCHAPTER 7: Central Limit Theorem: CLT for Averages (Means)
CHAPTER 7: Cetral Limit Theorem: CLT for Averages (Meas) X = the umber obtaied whe rollig oe six sided die oce. If we roll a six sided die oce, the mea of the probability distributio is X P(X = x) Simulatio:
More information.04. This means $1000 is multiplied by 1.02 five times, once for each of the remaining sixmonth
Questio 1: What is a ordiary auity? Let s look at a ordiary auity that is certai ad simple. By this, we mea a auity over a fixed term whose paymet period matches the iterest coversio period. Additioally,
More informationUC Berkeley Department of Electrical Engineering and Computer Science. EE 126: Probablity and Random Processes. Solutions 9 Spring 2006
Exam format UC Bereley Departmet of Electrical Egieerig ad Computer Sciece EE 6: Probablity ad Radom Processes Solutios 9 Sprig 006 The secod midterm will be held o Wedesday May 7; CHECK the fial exam
More information1 Computing the Standard Deviation of Sample Means
Computig the Stadard Deviatio of Sample Meas Quality cotrol charts are based o sample meas ot o idividual values withi a sample. A sample is a group of items, which are cosidered all together for our aalysis.
More informationNormal Distribution.
Normal Distributio www.icrf.l Normal distributio I probability theory, the ormal or Gaussia distributio, is a cotiuous probability distributio that is ofte used as a first approimatio to describe realvalued
More informationLecture 5: Span, linear independence, bases, and dimension
Lecture 5: Spa, liear idepedece, bases, ad dimesio Travis Schedler Thurs, Sep 23, 2010 (versio: 9/21 9:55 PM) 1 Motivatio Motivatio To uderstad what it meas that R has dimesio oe, R 2 dimesio 2, etc.;
More informationAP Calculus BC 2003 Scoring Guidelines Form B
AP Calculus BC Scorig Guidelies Form B The materials icluded i these files are iteded for use by AP teachers for course ad exam preparatio; permissio for ay other use must be sought from the Advaced Placemet
More informationModified Line Search Method for Global Optimization
Modified Lie Search Method for Global Optimizatio Cria Grosa ad Ajith Abraham Ceter of Excellece for Quatifiable Quality of Service Norwegia Uiversity of Sciece ad Techology Trodheim, Norway {cria, ajith}@q2s.tu.o
More informationCase Study. Normal and t Distributions. Density Plot. Normal Distributions
Case Study Normal ad t Distributios Bret Halo ad Bret Larget Departmet of Statistics Uiversity of Wiscosi Madiso October 11 13, 2011 Case Study Body temperature varies withi idividuals over time (it ca
More informationA probabilistic proof of a binomial identity
A probabilistic proof of a biomial idetity Joatho Peterso Abstract We give a elemetary probabilistic proof of a biomial idetity. The proof is obtaied by computig the probability of a certai evet i two
More informationHow To Solve An Old Japanese Geometry Problem
116 Taget circles i the ratio 2 : 1 Hiroshi Okumura ad Masayuki Wataabe I this article we cosider the followig old Japaese geometry problem (see Figure 1), whose statemet i [1, p. 39] is missig the coditio
More informationMaximum Likelihood Estimators.
Lecture 2 Maximum Likelihood Estimators. Matlab example. As a motivatio, let us look at oe Matlab example. Let us geerate a radom sample of size 00 from beta distributio Beta(5, 2). We will lear the defiitio
More information4.3. The Integral and Comparison Tests
4.3. THE INTEGRAL AND COMPARISON TESTS 9 4.3. The Itegral ad Compariso Tests 4.3.. The Itegral Test. Suppose f is a cotiuous, positive, decreasig fuctio o [, ), ad let a = f(). The the covergece or divergece
More informationCHAPTER 3 DIGITAL CODING OF SIGNALS
CHAPTER 3 DIGITAL CODING OF SIGNALS Computers are ofte used to automate the recordig of measuremets. The trasducers ad sigal coditioig circuits produce a voltage sigal that is proportioal to a quatity
More information5: Introduction to Estimation
5: Itroductio to Estimatio Cotets Acroyms ad symbols... 1 Statistical iferece... Estimatig µ with cofidece... 3 Samplig distributio of the mea... 3 Cofidece Iterval for μ whe σ is kow before had... 4 Sample
More informationWHEN IS THE (CO)SINE OF A RATIONAL ANGLE EQUAL TO A RATIONAL NUMBER?
WHEN IS THE (CO)SINE OF A RATIONAL ANGLE EQUAL TO A RATIONAL NUMBER? JÖRG JAHNEL 1. My Motivatio Some Sort of a Itroductio Last term I tought Topological Groups at the Göttige Georg August Uiversity. This
More informationLecture 4: Cauchy sequences, Bolzano-Weierstrass, and the Squeeze theorem
Lecture 4: Cauchy sequeces, Bolzao-Weierstrass, ad the Squeeze theorem The purpose of this lecture is more modest tha the previous oes. It is to state certai coditios uder which we are guarateed that limits
More informationNon-life insurance mathematics. Nils F. Haavardsson, University of Oslo and DNB Skadeforsikring
No-life isurace mathematics Nils F. Haavardsso, Uiversity of Oslo ad DNB Skadeforsikrig Mai issues so far Why does isurace work? How is risk premium defied ad why is it importat? How ca claim frequecy
More informationAsymptotic Growth of Functions
CMPS Itroductio to Aalysis of Algorithms Fall 3 Asymptotic Growth of Fuctios We itroduce several types of asymptotic otatio which are used to compare the performace ad efficiecy of algorithms As we ll
More informationLecture 4: Cheeger s Inequality
Spectral Graph Theory ad Applicatios WS 0/0 Lecture 4: Cheeger s Iequality Lecturer: Thomas Sauerwald & He Su Statemet of Cheeger s Iequality I this lecture we assume for simplicity that G is a d-regular
More informationConfidence Intervals. CI for a population mean (σ is known and n > 30 or the variable is normally distributed in the.
Cofidece Itervals A cofidece iterval is a iterval whose purpose is to estimate a parameter (a umber that could, i theory, be calculated from the populatio, if measuremets were available for the whole populatio).
More informationCS103X: Discrete Structures Homework 4 Solutions
CS103X: Discrete Structures Homewor 4 Solutios Due February 22, 2008 Exercise 1 10 poits. Silico Valley questios: a How may possible six-figure salaries i whole dollar amouts are there that cotai at least
More informationCHAPTER 3 THE TIME VALUE OF MONEY
CHAPTER 3 THE TIME VALUE OF MONEY OVERVIEW A dollar i the had today is worth more tha a dollar to be received i the future because, if you had it ow, you could ivest that dollar ad ear iterest. Of all
More informationRunning Time ( 3.1) Analysis of Algorithms. Experimental Studies ( 3.1.1) Limitations of Experiments. Pseudocode ( 3.1.2) Theoretical Analysis
Ruig Time ( 3.) Aalysis of Algorithms Iput Algorithm Output A algorithm is a step-by-step procedure for solvig a problem i a fiite amout of time. Most algorithms trasform iput objects ito output objects.
More informationCooley-Tukey. Tukey FFT Algorithms. FFT Algorithms. Cooley
Cooley Cooley-Tuey Tuey FFT Algorithms FFT Algorithms Cosider a legth- sequece x[ with a -poit DFT X[ where Represet the idices ad as +, +, Cooley Cooley-Tuey Tuey FFT Algorithms FFT Algorithms Usig these
More informationMath 113 HW #11 Solutions
Math 3 HW # Solutios 5. 4. (a) Estimate the area uder the graph of f(x) = x from x = to x = 4 usig four approximatig rectagles ad right edpoits. Sketch the graph ad the rectagles. Is your estimate a uderestimate
More informationLecture 13. Lecturer: Jonathan Kelner Scribe: Jonathan Pines (2009)
18.409 A Algorithmist s Toolkit October 27, 2009 Lecture 13 Lecturer: Joatha Keler Scribe: Joatha Pies (2009) 1 Outlie Last time, we proved the Bru-Mikowski iequality for boxes. Today we ll go over the
More informationDetermining the sample size
Determiig the sample size Oe of the most commo questios ay statisticia gets asked is How large a sample size do I eed? Researchers are ofte surprised to fid out that the aswer depeds o a umber of factors
More informationLesson 17 Pearson s Correlation Coefficient
Outlie Measures of Relatioships Pearso s Correlatio Coefficiet (r) -types of data -scatter plots -measure of directio -measure of stregth Computatio -covariatio of X ad Y -uique variatio i X ad Y -measurig
More informationMEI Structured Mathematics. Module Summary Sheets. Statistics 2 (Version B: reference to new book)
MEI Mathematics i Educatio ad Idustry MEI Structured Mathematics Module Summary Sheets Statistics (Versio B: referece to ew book) Topic : The Poisso Distributio Topic : The Normal Distributio Topic 3:
More information2-3 The Remainder and Factor Theorems
- The Remaider ad Factor Theorems Factor each polyomial completely usig the give factor ad log divisio 1 x + x x 60; x + So, x + x x 60 = (x + )(x x 15) Factorig the quadratic expressio yields x + x x
More informationInfinite Sequences and Series
CHAPTER 4 Ifiite Sequeces ad Series 4.1. Sequeces A sequece is a ifiite ordered list of umbers, for example the sequece of odd positive itegers: 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29...
More information5.4 Amortization. Question 1: How do you find the present value of an annuity? Question 2: How is a loan amortized?
5.4 Amortizatio Questio 1: How do you fid the preset value of a auity? Questio 2: How is a loa amortized? Questio 3: How do you make a amortizatio table? Oe of the most commo fiacial istrumets a perso
More information3. Greatest Common Divisor - Least Common Multiple
3 Greatest Commo Divisor - Least Commo Multiple Defiitio 31: The greatest commo divisor of two atural umbers a ad b is the largest atural umber c which divides both a ad b We deote the greatest commo gcd
More informationResearch Article Sign Data Derivative Recovery
Iteratioal Scholarly Research Network ISRN Applied Mathematics Volume 0, Article ID 63070, 7 pages doi:0.540/0/63070 Research Article Sig Data Derivative Recovery L. M. Housto, G. A. Glass, ad A. D. Dymikov
More informationVladimir N. Burkov, Dmitri A. Novikov MODELS AND METHODS OF MULTIPROJECTS MANAGEMENT
Keywords: project maagemet, resource allocatio, etwork plaig Vladimir N Burkov, Dmitri A Novikov MODELS AND METHODS OF MULTIPROJECTS MANAGEMENT The paper deals with the problems of resource allocatio betwee
More informationBasic Elements of Arithmetic Sequences and Series
MA40S PRE-CALCULUS UNIT G GEOMETRIC SEQUENCES CLASS NOTES (COMPLETED NO NEED TO COPY NOTES FROM OVERHEAD) Basic Elemets of Arithmetic Sequeces ad Series Objective: To establish basic elemets of arithmetic
More informationEkkehart Schlicht: Economic Surplus and Derived Demand
Ekkehart Schlicht: Ecoomic Surplus ad Derived Demad Muich Discussio Paper No. 2006-17 Departmet of Ecoomics Uiversity of Muich Volkswirtschaftliche Fakultät Ludwig-Maximilias-Uiversität Müche Olie at http://epub.ub.ui-mueche.de/940/
More informationSEQUENCES AND SERIES
Chapter 9 SEQUENCES AND SERIES Natural umbers are the product of huma spirit. DEDEKIND 9.1 Itroductio I mathematics, the word, sequece is used i much the same way as it is i ordiary Eglish. Whe we say
More informationHypothesis testing. Null and alternative hypotheses
Hypothesis testig Aother importat use of samplig distributios is to test hypotheses about populatio parameters, e.g. mea, proportio, regressio coefficiets, etc. For example, it is possible to stipulate
More informationFOUNDATIONS OF MATHEMATICS AND PRE-CALCULUS GRADE 10
FOUNDATIONS OF MATHEMATICS AND PRE-CALCULUS GRADE 10 [C] Commuicatio Measuremet A1. Solve problems that ivolve liear measuremet, usig: SI ad imperial uits of measure estimatio strategies measuremet strategies.
More informationINFINITE SERIES KEITH CONRAD
INFINITE SERIES KEITH CONRAD. Itroductio The two basic cocepts of calculus, differetiatio ad itegratio, are defied i terms of limits (Newto quotiets ad Riema sums). I additio to these is a third fudametal
More informationHere are a couple of warnings to my students who may be here to get a copy of what happened on a day that you missed.
This documet was writte ad copyrighted by Paul Dawkis. Use of this documet ad its olie versio is govered by the Terms ad Coditios of Use located at http://tutorial.math.lamar.edu/terms.asp. The olie versio
More informationDepartment of Computer Science, University of Otago
Departmet of Computer Sciece, Uiversity of Otago Techical Report OUCS-2006-09 Permutatios Cotaiig May Patters Authors: M.H. Albert Departmet of Computer Sciece, Uiversity of Otago Micah Colema, Rya Fly
More informationEscola Federal de Engenharia de Itajubá
Escola Federal de Egeharia de Itajubá Departameto de Egeharia Mecâica Pós-Graduação em Egeharia Mecâica MPF04 ANÁLISE DE SINAIS E AQUISÇÃO DE DADOS SINAIS E SISTEMAS Trabalho 02 (MATLAB) Prof. Dr. José
More informationListing terms of a finite sequence List all of the terms of each finite sequence. a) a n n 2 for 1 n 5 1 b) a n for 1 n 4 n 2
74 (4 ) Chapter 4 Sequeces ad Series 4. SEQUENCES I this sectio Defiitio Fidig a Formula for the th Term The word sequece is a familiar word. We may speak of a sequece of evets or say that somethig is
More information1. C. The formula for the confidence interval for a population mean is: x t, which was
s 1. C. The formula for the cofidece iterval for a populatio mea is: x t, which was based o the sample Mea. So, x is guarateed to be i the iterval you form.. D. Use the rule : p-value
More informationAnnuities Under Random Rates of Interest II By Abraham Zaks. Technion I.I.T. Haifa ISRAEL and Haifa University Haifa ISRAEL.
Auities Uder Radom Rates of Iterest II By Abraham Zas Techio I.I.T. Haifa ISRAEL ad Haifa Uiversity Haifa ISRAEL Departmet of Mathematics, Techio - Israel Istitute of Techology, 3000, Haifa, Israel I memory
More informationExploratory Data Analysis
1 Exploratory Data Aalysis Exploratory data aalysis is ofte the rst step i a statistical aalysis, for it helps uderstadig the mai features of the particular sample that a aalyst is usig. Itelliget descriptios
More informationCME 302: NUMERICAL LINEAR ALGEBRA FALL 2005/06 LECTURE 8
CME 30: NUMERICAL LINEAR ALGEBRA FALL 005/06 LECTURE 8 GENE H GOLUB 1 Positive Defiite Matrices A matrix A is positive defiite if x Ax > 0 for all ozero x A positive defiite matrix has real ad positive
More informationMath C067 Sampling Distributions
Math C067 Samplig Distributios Sample Mea ad Sample Proportio Richard Beigel Some time betwee April 16, 2007 ad April 16, 2007 Examples of Samplig A pollster may try to estimate the proportio of voters
More information3. If x and y are real numbers, what is the simplified radical form
lgebra II Practice Test Objective:.a. Which is equivalet to 98 94 4 49?. Which epressio is aother way to write 5 4? 5 5 4 4 4 5 4 5. If ad y are real umbers, what is the simplified radical form of 5 y
More informationTheorems About Power Series
Physics 6A Witer 20 Theorems About Power Series Cosider a power series, f(x) = a x, () where the a are real coefficiets ad x is a real variable. There exists a real o-egative umber R, called the radius
More informationTaking DCOP to the Real World: Efficient Complete Solutions for Distributed Multi-Event Scheduling
Taig DCOP to the Real World: Efficiet Complete Solutios for Distributed Multi-Evet Schedulig Rajiv T. Maheswara, Milid Tambe, Emma Bowrig, Joatha P. Pearce, ad Pradeep araatham Uiversity of Souther Califoria
More informationInstitute of Actuaries of India Subject CT1 Financial Mathematics
Istitute of Actuaries of Idia Subject CT1 Fiacial Mathematics For 2014 Examiatios Subject CT1 Fiacial Mathematics Core Techical Aim The aim of the Fiacial Mathematics subject is to provide a groudig i
More informationA Note on Sums of Greatest (Least) Prime Factors
It. J. Cotemp. Math. Scieces, Vol. 8, 203, o. 9, 423-432 HIKARI Ltd, www.m-hikari.com A Note o Sums of Greatest (Least Prime Factors Rafael Jakimczuk Divisio Matemática, Uiversidad Nacioal de Luá Bueos
More informationNATIONAL SENIOR CERTIFICATE GRADE 11
NATIONAL SENIOR CERTIFICATE GRADE MATHEMATICS P NOVEMBER 007 MARKS: 50 TIME: 3 hours This questio paper cosists of 9 pages, diagram sheet ad a -page formula sheet. Please tur over Mathematics/P DoE/November
More informationDescriptive Statistics
Descriptive Statistics We leared to describe data sets graphically. We ca also describe a data set umerically. Measures of Locatio Defiitio The sample mea is the arithmetic average of values. We deote
More informationConfidence Intervals for One Mean
Chapter 420 Cofidece Itervals for Oe Mea Itroductio This routie calculates the sample size ecessary to achieve a specified distace from the mea to the cofidece limit(s) at a stated cofidece level for a
More informationComplex Numbers. where x represents a root of Equation 1. Note that the ± sign tells us that quadratic equations will have
Comple Numbers I spite of Calvi s discomfiture, imagiar umbers (a subset of the set of comple umbers) eist ad are ivaluable i mathematics, egieerig, ad sciece. I fact, i certai fields, such as electrical
More information3 Basic Definitions of Probability Theory
3 Basic Defiitios of Probability Theory 3defprob.tex: Feb 10, 2003 Classical probability Frequecy probability axiomatic probability Historical developemet: Classical Frequecy Axiomatic The Axiomatic defiitio
More informationChapter 14 Nonparametric Statistics
Chapter 14 Noparametric Statistics A.K.A. distributio-free statistics! Does ot deped o the populatio fittig ay particular type of distributio (e.g, ormal). Sice these methods make fewer assumptios, they
More informationYour organization has a Class B IP address of 166.144.0.0 Before you implement subnetting, the Network ID and Host ID are divided as follows:
Subettig Subettig is used to subdivide a sigle class of etwork i to multiple smaller etworks. Example: Your orgaizatio has a Class B IP address of 166.144.0.0 Before you implemet subettig, the Network
More informationChapter 5 Unit 1. IET 350 Engineering Economics. Learning Objectives Chapter 5. Learning Objectives Unit 1. Annual Amount and Gradient Functions
Chapter 5 Uit Aual Amout ad Gradiet Fuctios IET 350 Egieerig Ecoomics Learig Objectives Chapter 5 Upo completio of this chapter you should uderstad: Calculatig future values from aual amouts. Calculatig
More informationHow To Solve The Homewor Problem Beautifully
Egieerig 33 eautiful Homewor et 3 of 7 Kuszmar roblem.5.5 large departmet store sells sport shirts i three sizes small, medium, ad large, three patters plaid, prit, ad stripe, ad two sleeve legths log
More informationGCSE STATISTICS. 4) How to calculate the range: The difference between the biggest number and the smallest number.
GCSE STATISTICS You should kow: 1) How to draw a frequecy diagram: e.g. NUMBER TALLY FREQUENCY 1 3 5 ) How to draw a bar chart, a pictogram, ad a pie chart. 3) How to use averages: a) Mea - add up all
More informationNATIONAL SENIOR CERTIFICATE GRADE 12
NATIONAL SENIOR CERTIFICATE GRADE MATHEMATICS P EXEMPLAR 04 MARKS: 50 TIME: 3 hours This questio paper cosists of 8 pages ad iformatio sheet. Please tur over Mathematics/P DBE/04 NSC Grade Eemplar INSTRUCTIONS
More information1 The Gaussian channel
ECE 77 Lecture 0 The Gaussia chael Objective: I this lecture we will lear about commuicatio over a chael of practical iterest, i which the trasmitted sigal is subjected to additive white Gaussia oise.
More information