Summary - it is basic problem so I have shown step by step solution .
Using the properties of the δ-function a) Evaluate6(az - b) f(x) dx for a 0; consider both a > 0 ...
A1. A few useful properties of the Dirac delta function The Dirac δ(z) functions is defined by δ(z) = 0 if |メ0, δ(z) = oo ifx-0 but the integral of the function over any interval containing the zero of the argument is unity, Equivalent, if f(x) is continuous at the origin One should treat the Dirac δ function as a limit of a sequence of functions peaked at x-0 As the limit is approached, the height of the peak increases...
2. Prove the following useful properties of Dirac δ-functions (a) δ(ax) = (z) (b) zfic) =0 (c) f(x)5(-a) f(a)5(r d) δ(z-a (aメ0) a) ( dz9(x-a) ) where θ(x) is the step function defined as 1 if r 0 0 if r <0 θ(z) = 2. Prove the following useful properties of Dirac δ-functions (a) δ(ax) = (z) (b) zfic) =0 (c) f(x)5(-a) f(a)5(r d) δ(z-a (aメ0) a) ( dz9(x-a) ) where θ(x) is the step function defined as 1 if...
4. (a) Let A [0, oo) and let f.g:AR be functions which are continuous at 0 and are such that f(0) 9(0)-1. Show that there exists some δ > 0 such that ifTE 0,d) then (b) Consider the function 0 l if z e R is rational, if zER is irrational f(z) Show that limfr) does not exists for any ceR. 4. (a) Let A [0, oo) and let f.g:AR be functions which are continuous at 0 and are such...
(1 point) Using Properties of Definite Integrals. Given S f(x) fo dx = 0 and f(x) dx = 6 evaluate (a) f(x) dx = (b) f(x) dx - Liro f(x) dx = (c) L.ro 3f(x) dx = (d) $350 38(x) dx Note: You can earn partial credit on this problem. Preview Mv Answers Submit Answers 19
the function y=f(x)={ 0-4), 14x+16, x20 x<0 Consider 1. (a) Sketch the graph off. (3 pts.) (b) Verify that the function is continuous everywhere using the properties of the definition and possibly calculating the limit at a particular point. (2 pts.) (c) Show f'(x) is not continuous at x-0. (5 pts.) the function y=f(x)={ 0-4), 14x+16, x20 x
(8). The one dimensional neutron diffusion equation with a (plane) source at x-0 is d'f(x) n (2) +002 f (x)-00(x) dx where f(x) is the flux of neutrons (f(x)→0 as x→±o), Q δ (x) is the (plane) source at x-0 (5(x) is the Dirac delta function), and o is a constant. This problem involves finding the solution to this equation using Fourier Transforms. You may use the formulas derived in class for the Fourier Transform of derivatives, but otherwise compute...
Each function f(x) changes value when x changes from x0 to x0 + dx. Find the change Δ.fX0 + dx)-foo), the value of the estimate df- The change Δ-1.902 (Round to the nearest thousandth.) r (%) d, and the approximate error la-dfl. The value of the estimate df f(x)-6x-4, X,--1.1 , dx=0.1 (Round to the nearest thousandth.) dx Tangent 0 to Each function f(x) changes value when x changes from x0 to x0 + dx. Find the change Δ.fX0 +...
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).
4. f(x) is a continuous function on [0, 1] and So f (x)dx = a, where a is constant. Evaluate the following double integral f(x)f(y)dydx. (Hint: Change the order of the integration and use the property of the double integral, so that you can apply Fubini's theorem.)
show by steps, definitions and theorems " f(x) dx = 0 for all integers Let f(x) be a continuous function on (a,0). If n> 0, then show that f(x) = 0 on [a, b].