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(c) x(t) = sin wt (sin 2wt 1 X(W) = 21 XI) * X2(W), where Xi(w)XM=1, IwI < 7, = 0, otherwise IwI < 27, = l X2(W) 10, otherwise Signals and Systems S910 Hence, X(w) is given by the convolution shown in Figure S910 XI) X 2 (W) TM a r k e t e d H i l l C o u r t e v e n t s , a n d d e s i g n e d H i l l C o u r t C o u n c i l b o a r d s , s h i r t s , & f l y e r s Title Resume 18C Mdx Ndy Parametrize the curve x= t, y= t, with tfrom 0 to 1 Put everything in terms of t dx= dt dy= dt M= x2y= t3 N= x 2y= t Now we put this into the integral I= Z C Mdx Ndy= Z 1 0 t3 dt tdt= Z 1 0 t3 tdt= 1 4 This is a di erent value from Example GT3, which leads to the important principle Important principle for line integrals

This site contains scenes that may cause you to shut down the computer and go play outside with strangers s o l o m a x c o m m u n i c a t i o n s(c) Consider the equation y' = f(at by c) where a, b, and c are constants Show that the substitution x = at by c changes the equation to the separable equation x' = a bf(x) Use this method to find the general solution of the equation y' = (y t?Determine whether T is onetoone or onto Define T R2 → R3 by T(a 1,a 2) = (a 1 a 2,0,2a 1 −a 2) Solution We first prove that T is a linear transformation Let x = (x 1,x 2),y = (y 1,y 2) ∈ R2 and let c ∈ R T(cxy) = T(c(x 1,x 2)(y 1,y 2)) = T(cx 1 y 1,cx 2 y 2) = ((cx 1 y 1)(cx 2 y 2),0,(2(cx 1 y 1)−(cx 2 y 2)) = c(x

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X opt is the set of optimal points x is locally optimal if there is an R > 0 such that x is optimal forSignals and Systems Part 11/ Solutions S313 We see that the system is timeinvariant from T 2T 1x(t T) = T 2y (t T)l = y 2(t T), Tx(t T) = y 2(t T) (b) False Two nonlinear systems in cascade can be linear, as shown in Figure S310Title Microsoft Word Revd Anthony Everett Author nmiller Created Date AM

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From this geometry, we can see that line integral ∫ C f (x, y) d s ∫ C f (x, y) d s does not depend on the parameterization r (t) r (t) of C As long as the curve is traversed exactly once by the parameterization, the area of the sheet formed by the function and the curve is the sameT O X I C A March 30 at 725 AM Si Soy 1212 1 Share Like Comment Share T O X I C A March 29 at 810 AM 5454Note Remember when we have the consumption function in the form C = a b(Y – T) that autonomous consumption is a and the marginal propensity to consume is b To solve for the consumption and savings functions with respect to real output rather than disposable

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Optimal and locally optimal points x is feasible if x ∈ domf 0 and it satisfies the constraints a feasible x is optimal if f 0(x) = p⋆;(x(t),y(t)) as the parameter t ranges over an interval I C = (x(t),y(t)) t ∈ I Examples 1 • The graph of a function y = f(x), x ∈ I, is a curve C that is parametrized by x(t) = t, y(t) = f(t), t ∈ I • The graph of a polar equation r = ρ(θ), θ ∈ I, is a curve C that is parametrized by the functions x(t) =T(x) = Ax Then solving the system amounts to nding all of the vectors x 2Rn such that T(x) = 0 Solving the di erential equation y00y = 0 is equivalent to nding functions y such that T(y) = 0, where T is de ned as T(y) = y00y

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