
Problem 3: Consider a continuous function x(t), defined for t 0. The Laplace Transform (LT) for x(t) is defined as: X(s) - Ix(t)e-st dt. Derive the following properties: a) LT(6(t))-1, the ?(t) is the Dirac-delta function b) LT(u(t))-1/s, where u(t) is the unit-step function c) LT(sin(wt))-u/(s2 + ?2) d) LT(x(t-t)u(t-t)) = e-stx(s), ? > 0. e LT(tx)-4x(s).
Linear algebra
Show that the transformation T defined by T(X), x)) = (2x - 3X2, X, +4,6x) is not linear. If T is a linear transformation, then T(0) = and T(cu + dv) = CT(u) + dT(v) for all vectors u, v in the domain of T and all scalars c, d.
JO # 2. Let the linear operator K : C([0, 1]) + C([0, 1]) be defined for each continuous real valued function f on [0, 1] by (Kf)(x) = 1 | tf(t) dt. a. Find a range of values for the parameter 1 for which the operator norm of K is strictly less than 1 with respect to the norm on C([0, 1]) given by Il gl. = sup{lg(t)| :te [0, 1]}. b. Describe an iterative process for solving (generating...
1. Let f:R → R be the function defined as: 32 0 if x is rational if x is irrational Prove that lim -70 f(x) = 0. Prove that limc f(x) does not exist for every real number c + 0. 2. Let f:R + R be a continuous function such that f(0) = 0 and f(2) = 0. Prove that there exists a real number c such that f(c+1) = f(c). 3 Let f. (a,b) R be a function...
Q 2(a) -3, --<tso, [13 Marks] Q 2(b) Show that the Fourier series amplitude coefficients (Am) n-1,3,5, .., of x(t), as defined in question Q2(a), decrease in value in proportion to 1/n.
Q 2(a) -3, --
(5) Let qe Q. Suppose that a <b, 0<c<d, and that f : [a, b] → [c, d]. If f is integrable on [a,b], then prove that * (t)dt) = f'(x) for all 3 € (a, b).
2. Show that a. P.(-x) =(-1)*P(x) b. Px.(0) - (-1) 2(!) c. P(0) - 0 3. Prove that j «P.com j«P«6?P.:(}é 4. Evaluate j <P:6)Ps(x)dt 5. Use Rodrigue's formula to calculate P (4)
The graph of f is shown to the right. The function F(x) is
defined by
for .
a) Find F(0) and F(3).
b) Find F'(1).
c) For what value of x does F(x) have its maximum value? What is
this maximum value?
d) Sketch a possible graph of F. Do not attempt to find a
formula for F. (You could, but it is more work than necessary.)
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Show state diagram & Complete the timing trace q* z q x =0 x =1 x=0 x=1 A B C 0 1 B C A 0 0 C A B 1 0 x 0 0 1 1 1 0 0 0 0 0 1 0 q A z
Let T : C([0, 1]) → R be a (not necessarily bounded) linear
functional.
Show that T is positive if and only if
=
(here 1 denotes the constant function [0, 1] → R, x → 1).
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