Hint: tional on CR(0, 1], then there e edis t E [0, 1] such that α(f)-f(t)....
18. Let T be the matrix transformation T -1 2 0 -1 2 2 -1 h 2 -3 k 4 a. What are the domain and codomain of T? b. Find the REF of [T]. Hint: You'll need the REF in some of the following questions. -1 -1 -1 -3 (REF of [7]= 0 2 2 4 is given here so that you can correctly answer the following 0 0 h – 2 k-6 questions.) c. Define the range of...
(1 point) The Taylor series for f(x) = e' at a = -2 is Cr(x + 2)" n=0 Find the first few coefficients. Co = C1 = C2 = C3 = C4 = x 5 (1 point) Find the first four terms of the Taylor series for the function - about the point a = 1. (Your answers should include the variable x when appropriate and be listed in increasing degree, starting with the constant term) 5 II + +...
n× n×r Rrxp. E Rnxp E ~ X,xp(0. În ⓧΣ). Find the Hotelling's T 2 test statistif for Ho : 3j-0 for any particular j є (1. r-1). Note that -(A,A, , 3r-1)T and 3, E RP. n× n×r Rrxp. E Rnxp E ~ X,xp(0. În ⓧΣ). Find the Hotelling's T 2 test statistif for Ho : 3j-0 for any particular j є (1. r-1). Note that -(A,A, , 3r-1)T and 3, E RP.
3. (25 Points) Find f(t). f(0) + f(t - 1)f(t)dt = t. Hint: The second term on the left side is a convolution and it might be helpful to use the Laplace Transform. 1 4. (10 Points) Solve the initial value problem by Laplace transform techniques. x" + 5x' + 4x = 0;x(0) = 1,x'(0) = 0. I 5. (15 Points) Find a series solution for the following differential equation. Calculate the radius of convergence. 2(x - 1)y' = 3y...
Exercise 41.2 Consider the signal f(t) window w(t)e amat, α E R, and the Gaussian (a) Verify that is well-defined (even though f f L'(R)). (b) Compute Ws(A, b) using the following result: e-ra(1+iz)2 dt = a-2 For a > 0 and E R, (c) Show that l W, (A, b)12 attains its maximum when λ a. Exercise 41.2 Consider the signal f(t) window w(t)e amat, α E R, and the Gaussian (a) Verify that is well-defined (even though f...
Piecewise function f(t) = 1 when 0 < t < 1, and f(t) =-1 when-1 < t < 0. Also f(t) = 0 for any other t (t < 1 or t 2 1). Answer the following questions: 1. Sketch the graph of f (t) 2. Calculate Fourier Transform F(j) 3. If g(t) = f(t) + 1, what is G(jw), ie. Fourier transform of g(t)? 4, extra 3-point credit: h(t) = f(t) + sin(kt), find the Fourier Transform of h(t).
он e) Na Cr 0. H2SO4, H,0 (excess) NaBH EtOH он Hint: esterification H+ 1 LDA 2) PhCH Br 14 /36
5. Let y E C2([0, T]; R), T > 0 satisfy y"(t) = 피t, y(0) = y'(0) = 0 e R. Use Picard-Lindelöf 1+t' to prove that a unique solution to the IVP exists for short time, as follows: (a) Let b E R2, A E M2 (R) . Show that any function g : R2 -R2.9(x) = Ax+b is Lipschitz. 1 mark (b) Transform the DE for y into a(t) Az(t) +b(t) for a suitable z, A, b. 2...
9·Let m, n E Z+ with (m, n) 1. Let f : Zmn-t Zrn x Zn by, for all a є z /([a]mn) = ([a]rn , [a]n). (a) Prove that f is well-defined. (b) Let m- 4 and n - 7. Find a Z such that f ([al28) (34,(517). (c) Prove that f is a bijection.2 (HINT: To prove that f is onto, given (bm, [cm) E Zm x Zn, consider z - cmr + bns, where 1 mr +ns.)
Find the Laplace transform of f(0) = 1, for 0 <t<1 5, for 1<t<2. e-l for t > 2