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Ty x cxg p. ֘A @ \ r \ i Ƃ̋@ \ r \ ߂ ̔̔ X T. May 16, 11 254 CHAPTER 13 CALCULUS OF VECTORVALUED FUNCTIONS (LT CHAPTER 14) Use a computer algebra system to plot the projections onto the xy and xzplanes of the curve r(t) = t cost,tsin t,t in Exercise 17 In Exercises 19 and , let r(t) = sin t,cost,sin t cos2t as shown in Figure 12 y x z FIGURE 12 19. ̃X g x N ` TL ̐ ɉ A V GravititeCF Lifting Complex z p A b v ܂ I N ̌ Ղ f R e ̂ N ł B.

Free math problem solver answers your algebra, geometry, trigonometry, calculus, and statistics homework questions with stepbystep explanations, just like a math tutor. 116 = H > B R G B D g Z F b g g h _ h e h ` d b y m g b \ _ j k b l _ l “ K \ B \ Z g J b e k d b”, L h f 53, K \I 1 1, F _ o Z g b a Z p b y, _ e _ d l j b n. And f(1) = 5 Since f(x) is a polynomial, f is continuous on the real line We have f( 2) > 0 > f(0) So, by the Intermediate Value Theorem, there exists a number cbetween 2 and 0 such that f(c) = 0 Similarly, there exists a number dbetween 0 and 1 such that f(d) = 0 Note The choices x= 2;0;1 are not the only possibilities 13 Given y= 1 x2.

References ABeck,FirstOrder Methods in Optimization (17),chapter6 PLCombettesandJChPesquet,Proximal splitting methods in signal processing,inFixedPoint Algorithms for Inverse Problems in Science and Engineering (11) NParikhandSBoyd,Proximal algorithms (13) Theproximalmapping 624. F „x”= sup y2dom f „xTy f„y”” f isclosedandconvex fromFenchel’sinequality,xTy f„y” f„x”forally andx;therefore f „x” f„x” forallx equivalently,epi f epi f (forany f) if f isclosedandconvex,then f „x”= f„x” forallx equivalently,epi f = epi f (if f isclosedandconvex);proofonnextpage Conjugatefunctions 513. @ @ @ @ @ t Y s Y w O157 u X i b N ̂񂫁v @ @ @ @ @JR Y w @ k 5 d b ԍ F.

Let , → be a continuous function on the closed interval ,, and differentiable on the open interval (,), where. P R I O R I T Y D a t e 0 2 / 2 1 / 2 0 0 7. Tkrf(x)k2 will always be positive unless rf(x) = 0, this inequality implies that the objective function value strictly decreases with each iteration of gradient descent until it reaches the optimal value f(x) = f(x) Note that this convergence result only holds when we choose tto be small enough, ie t 1=L This explains.

\ JUDGE & COMMENTS \ i ΂ A ǂ j V 16 N P ɃI v s X g O X z e É ^ V F t Y C u L b ` t ́A x X g u C _ ЁE A g O C X O v ^ c ̃u C _ z e y уu b t F X g B. X, and e y only one point on the failure surface need to be computed Such point satisfies the following condition Pu =P and = uy y ux x ee ee 0 given load point load point ray b/2 α e y e ux e uy e x y x h/2 The procedure is summarized as follows 4 / 6. KÝ PDF°«x Õh)¦ xiv 04 A¢ àh Ý&4 /Zøüd Z 8!¢ Ýd × ä î 3é\îà#å\9Íh Ý`Î à9°d ñÕh Í 2 G«Àê Ý&Ía;4 ê Za Ý¥X C&!.

62 = H > B R G B D g Z F b g g h _ h e h ` d b y m g b \ _ j k b l _ l “ K \ B \ Z g J b e k d b”, L h f 53, K \I 1 1, F _ o Z g b a Z p b y, _ e _ d l j b n. GDP (Y) is the combination of consumption (C), investment (I), government spending (G), and net exports (exports (X) less imports (M)) The recovery that began in 09 in the United States has been tepid relative to previous recoveries (although better than in most other rich countries) though not because of weak consumption, which in fact has. Thus X = x,Y = b−y, and the curvature in question is k = b/a2 3 Let t → (x(t),y(t)) be a closed regular plane curve Let t → (˙x(t),y˙(t)) be the closed regular plane curve formed by the velocity vectors Prove that the integral 1 2π I xd˙ y˙ −yd˙ x˙ x˙2 ˙y2 is an integer Point out geometric interpretations of this integer in.

This means that x(t) can be written as a weighted integral of functions Cu (Lecture 3) ELE 301 Signals and Systems Fall 1112 9 / 55 Applying the system H to the input x(t), y(t) = H(x(t)) = H Z 1 1 x(˝) ˝(t)d˝ If the system obeys extended linearity we can interchange the order of the system operator and the integration y(t) = Z 1 1 x. The simplest case, apart from the trivial case of a constant function, is when y is a linear function of x, meaning that the graph of y is a line In this case, y = f(x) = mx b, for real numbers m and b, and the slope m is given by = =, where the symbol Δ is an abbreviation for "change in", and the combinations and refer to corresponding changes, ie. Y = C I G (XM) National income with government interference From the national income, some will be used for consumption, saving, and other’s used for paying taxes With adding of variable taxes on national income, then the formula is Y = C S T To see the balance of national income with the adding of government spending and tax.

116 = H > B R G B D g Z F b g g h _ h e h ` d b y m g b \ _ j k b l _ l “ K \ B \ Z g J b e k d b”, L h f 53, K \I 1 1, F _ o Z g b a Z p b y, _ e _ d l j b n. And since x(t) = y(t)F(t), we get x(t) = KeA(t) eA(t) Z t e A(s)b(s)ds is the general solution Example Solve x0= x=(1 t) 2t x(0) = 3 a(t) = 1=(1 t) Thus A(t) = ln(1 t) and F(t) = eA(t) = 1 tSo x(t) = K(1 t) (1 t) Z t 0 2s 1 s ds= K(1 t) 2t 2ln(1 t) The initial data yields, K= 3 so that x(t) = 3 5t 2ln(1 t. KÝ PDF°«x Õh)¦ xiv 04 A¢ àh Ý&4 /Zøüd Z 8!¢ Ýd × ä î 3é\îà#å\9Íh Ý`Î à9°d ñÕh Í 2 G«Àê Ý&Ía;4 ê Za Ý¥X C&!.

Simple and best practice solution for g=(xc)/x equation Check how easy it is, and learn it for the future Our solution is simple, and easy to understand, so don`t hesitate to use it as a solution of your homework. References ABeck,FirstOrder Methods in Optimization (17),chapter6 PLCombettesandJChPesquet,Proximal splitting methods in signal processing,inFixedPoint Algorithms for Inverse Problems in Science and Engineering (11) NParikhandSBoyd,Proximal algorithms (13) Theproximalmapping 624. Tbs $b%f%l%s$,$*fo$1$9$k!v (btv $b%.

Ό T Y 1960 N ܂ q v l Ƃ̔ I ̍ۂɉ̏ u z o v ̉ 𔭌@ A ^ I ɁA n N ̃T u } l W ۊǂ Ă J Z b g A n 񂪃J I P y މ @ B n N ̉̎ Ƃ Ă̐V Ȉ ʂ _ Ԍ A ̃v C x g S ȏ CD I Ό T Y ̃ h ̗w A n N ̈ D ̗w e } 1 ͐Ό T Y ̃I W i A J o ȁA I ł̉̏ B 1 n N ̃v C x g ƂȂ CD2 g B (C)RS. = −ˇ=6 (b) f(x;y)=xsin(xy);. The simplest case, apart from the trivial case of a constant function, is when y is a linear function of x, meaning that the graph of y is a line In this case, y = f(x) = mx b, for real numbers m and b, and the slope m is given by = =, where the symbol Δ is an abbreviation for "change in", and the combinations and refer to corresponding changes, ie.

Substitute Y(t) into y’’ p(t)y’ q(t)y = g(t) and determine the coefficients to satisfy the equation There is no solution of the form that we assumed Find a solution of Y(t) Determine the coefficients End N Y Second Order Linear Non Homogenous Differential Equations – Method of Undermined Coefficients –Block Diagram •. Signals and Systems P36 (c) Suppose that a second LTI system has the following output y(t) when the input is the unit step x(t) = u(t) y(t) = e'u(t) u(1 t) Determine and sketch the response of this system to the input x(t) shown in. Happy tree frinds ̎g ݂Ȃ aoi f p g Łbhappy tree frinds o b y ̒ʔ́b.

O ¬ ñ » @ ô ¬ × ï x u ¾ · ï @ ó Ô )odph lrqldwlrq ghwhfwru ô),' ó ³ è p Í Ð t y # ï Ý 7rwdo k\gurfduerq ô7& Æ ù \ ¬ ä \ ü é ó ³ § 0hwkdqh Æ ô d ó ³ § Æ Ð ú é · t d ú. Tkrf(x)k2 will always be positive unless rf(x) = 0, this inequality implies that the objective function value strictly decreases with each iteration of gradient descent until it reaches the optimal value f(x) = f(x) Note that this convergence result only holds when we choose tto be small enough, ie t 1=L This explains. And f(1) = 5 Since f(x) is a polynomial, f is continuous on the real line We have f( 2) > 0 > f(0) So, by the Intermediate Value Theorem, there exists a number cbetween 2 and 0 such that f(c) = 0 Similarly, there exists a number dbetween 0 and 1 such that f(d) = 0 Note The choices x= 2;0;1 are not the only possibilities 13 Given y= 1 x2.

For the function z(x,y)=yx^2xy with x(t)=log(t) and y(t)=t^2, we have Example For our introductory example, we can now find dP/dt Implicit Differentiation A special case of this chain rule allows us to find dy/dx for functions F(x,y)=0 that define y implicity as a function of x Suppose x is an independent variable and y=y(x). æ ä ù è P ´ Ù ¢ Ô Í ^ x · ø % ´ ­ ¯ ü ù ó æ ä ù · è { ¯ Ç ¢ t è ¨ þ Í ½ W t r Þ Ù Ù Ï ¼ ¦ · < ´ c ¢ Ô g ³ Í ¡ Ê t505 ° · ã v { · ´ Ì 9 ¬ ¯ ¸ ³ Ó Ç ¤ Ü u Ç ¨ t g r · " t Y " ¸ t V Ë ³ ã v { ´ ­ ³ Ó Ç ¢ u. Signals and Systems P36 (c) Suppose that a second LTI system has the following output y(t) when the input is the unit step x(t) = u(t) y(t) = e'u(t) u(1 t) Determine and sketch the response of this system to the input x(t) shown in.

From of 1, in this case (tan(x) sec(x))/ (tan(x) sec(x)), the integration could then proceed quite easily ∫ ∫ ∫ =∫ = = u du dx x x x x x dx x x x x xdx x sec tan sec tan sec tan sec tan sec sec sec 2 =ln uC=ln sec xtan xC Now back to the equation µ(t) y′ µ(t)p(t) y = µ(t)g(t) (*) On the right side there is. Example 6 X and Y are independent, each with an exponential(λ) distribution Find the density of Z = X Y and of W = Y −X2 Since X and Y are independent, we know that f(x,y) = fX(x)fY (y), giving us f(x,y) = ˆ λe−λxλe−λy if x,y ≥ 0 0 otherwise The first thing we do is draw a picture of the support set the first quadrant (a). 12 Let f(x;y) be a di erentiable function, and let u= x yand v= x−yFinda constant such that (fx) 2 (f y) 2 = ((f u) 2 (f v) 2) Solution By the chain rule (fx)2 (f y)2 =(f u f v)2 (f u−f v)2 =2((f u)2 (f v)2)Thus =2 13 Find the directional derivative D ~ufat the given point in the direction indicated by the angle (a) f(x;y)= p 5x−4y;(2;1);.

Simple and best practice solution for g=cx equation Check how easy it is, and learn it for the future Our solution is simple, and easy to understand, so don`t hesitate to use it as a solution of your homework. And since x(t) = y(t)F(t), we get x(t) = KeA(t) eA(t) Z t e A(s)b(s)ds is the general solution Example Solve x0= x=(1 t) 2t x(0) = 3 a(t) = 1=(1 t) Thus A(t) = ln(1 t) and F(t) = eA(t) = 1 tSo x(t) = K(1 t) (1 t) Z t 0 2s 1 s ds= K(1 t) 2t 2ln(1 t) The initial data yields, K= 3 so that x(t) = 3 5t 2ln(1 t. ,¿F9Ë€ C' B @;{ A )L Ë€5k _Ù€DE @L7 E @ ÿ A L ×€6m K €/@I· @ ß E '€4 @Y@ 1L ã€7l ^M© )é##@ ã E 3€E § E 2¼è Ü Ð Ä ¸ ª ž ‚ æ ^ 2 Ú ® p D * ö Ü Â ¨ Ž t Z @ & AOÿ@_ß„ ÿ C çZ/ÑFOé€ Le1 ow ƒ @ o@ ›w %€ D_ä@`Þ€ÏyOé@NßJ µBP·€ @J%a> „Ž0 µ@ «€ DùR> @0¡l FQ €m.

***** ̂ C x g ς !!!. I K j b N X C c J A G e b N ւ悤 I G e b N I K j b N X C c ́A w q S ĐH ׂ x Ƃ A u S v ɏ i ̑I s Ă ܂ B G e b N ͗L @JAS F H A I K j b N X C c J B L @ i ` r F ؂̃N I e B ͂ A q 낱 Ԗ{ ̔ ڎw i A X S Ă ܂ B. Tkrf(x)k2 will always be positive unless rf(x) = 0, this inequality implies that the objective function value strictly decreases with each iteration of gradient descent until it reaches the optimal value f(x) = f(x) Note that this convergence result only holds when we choose tto be small enough, ie t 1=L This explains.

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D µ è p Í ø x æ ¬ 8 ü ;. From of 1, in this case (tan(x) sec(x))/ (tan(x) sec(x)), the integration could then proceed quite easily ∫ ∫ ∫ =∫ = = u du dx x x x x x dx x x x x xdx x sec tan sec tan sec tan sec tan sec sec sec 2 =ln uC=ln sec xtan xC Now back to the equation µ(t) y′ µ(t)p(t) y = µ(t)g(t) (*) On the right side there is. Signals and Systems P36 (c) Suppose that a second LTI system has the following output y(t) when the input is the unit step x(t) = u(t) y(t) = e'u(t) u(1 t) Determine and sketch the response of this system to the input x(t) shown in.

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A For f(g(x) you take the f(x) equation, except, instead of the "x"s, you put in the "g(x)" equation Like this f(g(x)= 3(2x6) (then you multiply the 3 to the 2x6 and get) f(g(x)= 6x18 And that is your final answer For g(f(x) you do the exact reverse You start with the "g(x)" function, and then instead of the x, you stick in the f(x.