Problem 4 (a). 1 Evaluate dx, where i is some positive constant. -5teis +6e-ix Problem 4...
T (1 point) Evaluate f(x) dx, where J12) f(x) = { 2.2 -ASX < 0 | 3 sin(x), 0 < x < 1. [fle) de =
Problem 1: A signal f(t) is said to be periodic if for some positive constant To f(t) = f(t+%), for all t. Determine whether or not each of the following continuous-time signals is periodic. If the signal is period eriodic, determine the fundamental a) f(t) = cos2(20t + π35/180) + sin(40t-725/180) + cos( 10t + π/4) cos(20t-n/4) b) f(t) = 10 cos(t + π/4) + c) f(t) = Σ+0000(-1)"u(t-1-4m) 25 sin(V3t - T/2)
Evaluate ∫∫∫T 2xy dx dy dz where T is the solid in the first octant bounded above by the cylinder z = 4 − x^2 below by the x, y-plane, and on the sides by the planes x =0, y = 2x and y = 4. Answer: ∫ (4, 0) ∫ (y/2, 0) ∫ (4−x^2, 0) 2xy dz dx dy = ∫ (2, 0) ∫ (4, 2x) ∫ (4−x^2, 0) 2xy dz dy dx = 128/3
6. -1.25 points My Notes Evaluate (y 3 sin x) dx + (z2 +7 cos y) dy x3 dz COS JC where C is the curve r(t) - (sin t, cos t, sin 2t), 0 s t s 27. (Hint: Observe that C lies on the surface z - 2xy.) F dr-
6. -1.25 points My Notes Evaluate (y 3 sin x) dx + (z2 +7 cos y) dy x3 dz COS JC where C is the curve r(t) -...
Problem 5 Let f : [0,1] → R be continuous and assume f(zje (0, 1) for all x E (0,1). Let n E N with n 22. Show that there is eractly one solution in (0,1) for the equation 7L IC nx+f" (t) dt-n-f(t) dt.
For this problem, you will use the following equation in your calculations, where f(ti) = a, a > 0, f(t) = b, and both g and flare continuous on (t1, tz]. 5° y dx = *ocer ces de 1 g(t)f '(t) dt Find the area of the region. x = 2 sin2 y = 5 sin tane 03A - 2
4. (a) Assume a function h is differentiable at some point to. Is it true that h is continuous on some open-neighbourhood of xo? Provide either a proof or a counterexample. (b) Let f be twice differentiable on R and assume that f" is continuous. Show that for all x ER S(x) = S(0) + s°C)x + [ (x - 1))"(dt. (C) Deduce that for any twice continuously differentiable function f on R and any positive x > 0, x...
Problem 8. (1 point) If z2 = x2 + y2 with z > 0, dx/dt 4, and dyldt = 5, find dzldt when x = 12 and y = 35. dz Answer: dt =
Consider the figure below. y=f(x) Evaluate. 1 ['rs. O 2. /Ral dx = 0 Let A(x) = f(t) dt for f(x) as shown in blue in the figure below. y = f(x) + X 2 3 4 5 6 7 1 Calculate the following. A(2) = A(4) = A(2) = A (4) =
please simplify
Problem 2.3 Evaluate or simplify the following integrals or expression as much as possible (show your work). (a) L, 8(t)x(t – 1)dt (e) , 8(at)dt (i) cos(10zt) [8(t) + 8(t + 5)] sin (b) 8(t – T)x(t)dt (f) 8(2t – 5) sin nt dt (c) L 8(t)x(r – t)dt cos (x - 5)|6(x – 3)dx (sin ke (B) e*-2 8(w) (k) 6(r – t)x(t)dt (d) (h) Jt-11 t+9 8(1 – 3)đr Problem 2.3 Evaluate or simplify the following...