dx Determine x= f(t) for (t? +4t) 4x + 4,t> 0; f(1) = 3. dt For (1? + 4t) dx dt = 4x +4, x= f(t) =
Let S f(w)dt = 6, f(x)dx = -4, log(x)dt = 12, 9(x) dx = 9 Use these values to evaluate the given definite integral: -3 (f(x) f(x) + g(x)) dx
Useful Formula: Fourier Transform: F[f(t)] = F(w) sof(t)e-jw dt Inverse Fourier Transform: F-1[F(w)] = f (t) = 24., F(w)ejwidw Time Transformation property of Fourier Transform: f(at – to). FC)e=itoch Laplace Transform: L[f(t)] = F(s) = $© f(t)e-st dt Shifting property: L[f(t – to)u(t – to)] = e-toSF(s) e [tuce) = 1 and c [u(e) = ) Using the convolution property of Fourier Transform to find the following convolution: sinc(t) * sinc (4t) [Hint: sinc(t) or rect(w/2)] TC .
1. Let f(t) e-2/3. Show that f(t)dt = 1 and that if X is a random variable with density f, then for all a 〈 b
3) Let F(x) = {* In In(1+t) dt. t (a) Find the Maclaurin series for F: (b) Use the series in part (a) to evaluate F(-1) exactly and use the result to state its interval of convergence. (c) Approximate F(1) to three decimals. (Hint: Look for an alternating series. )
Let f(t) be the function with graph 2 ft - 72 -1 ol 1 2 3 -1 -2 Lat F(x) = 60° f(e) dt. Which one of the following statements about the function F(x) is true? F'(0) = 0 F(2) <F(3) F(-2) = 0 F(-1) > 0
7. Find the function f(t) that satisfies the equality f(t)dt = x-COS x + 1.
7. Find the function f(t) that satisfies the equality f(t)dt = x-COS x + 1.
Find the Fourier transform f(t) a. X(w-3) 8(+3) 3. 2cos3tx(t) with x(t)'s FT is X(a) 2X(w-3) + 2x(w + 3) ?(0-3) + ?(w + 3) c. d.
You have 3 attempts remaining. Let A(x) = S6 f(t) dt, with f(x) as in figure. A7- LL 1 1 1 A(x) has a local minimum on (0,6) at 2 = A(z) has a local maximum on (0,6) at x =
Sketch the product of these, viz., g(t)=u(t-3) u(-t-4) f g(t)dt = u(t-3) u(-t-4)dt Evaluate t=-oo t=-oo Evaluate ) f u(C)dt and (i) f u()dt t-500 t=-oo
Sketch the product of these, viz., g(t)=u(t-3) u(-t-4) f g(t)dt = u(t-3) u(-t-4)dt Evaluate t=-oo t=-oo Evaluate ) f u(C)dt and (i) f u()dt t-500 t=-oo