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QUESTION 4 (4.1) Consider the variational problem with Lagrangan function 2r Sin t and endpont conditions z(0) 0, z...
Problem 4.9 (e) /(z) = and γ is parametrized by r(t), 0 z + t 1, and satisfies Imr(t)> 0, r(0) -4 + i, and γ(1) 6 + 2i (f) f(s) sin(z) and γ is some piecewise smooth path from 1 to π. 4.2 and the fact that the length of γ does not change under 4.9. Prove Proposi reparametrization. (Hint: Assume γ, σ, and τ are smooth. Start with the definition off, f, apply the chain rule to σ...
Problem 4. For r E [0, 1, fnd F)-(t)dt, where fr) 3 2r. Verify that F is continuous on [0,1] and F"(z) =f(z) at all points where f is continuous. Problern 5. Suppose that g, h : [c, d] → [a,b] are differentiable. ForエE [c,d] define h(a) Find H'(r) Problem 4. For r E [0, 1, fnd F)-(t)dt, where fr) 3 2r. Verify that F is continuous on [0,1] and F"(z) =f(z) at all points where f is continuous. Problern...
Problem 1. Consider the nonhomogencous heat equation for u(a,t) subject to the nonhomogeneous boundary conditions u(0,t1, t)- 0, and the initial condition 1--+ sin(z) u(z,0) = e solution u(z, t) by completing each of the following steps Find the equilibrium temperature distribution we r) Find th (b) Denote v, t)t) - ()Derive the IBVP for the function vz,t). (c) Find v(x, t) (d) Find u(x, t) Problem 1. Consider the nonhomogencous heat equation for u(a,t) subject to the nonhomogeneous boundary...
- Problem 6. Consider the function y2 m) sin ((10/s) t+π/4). lndicate whether each of the following waveforms is equivalent to y? Briefly justify your answers 1. (2 m) cos(10/s)t+/4) 2. (2 m) cos( (10/s)t+3 T/4) 3. (2m) cos( (10/s)t -/4) 4. (2 m) sin((10/ s) t + π/4 + 4 π) 5.-(2 m) sin ((10/s) t + π /4-3r) 6. (-2m) cos((10/s) t + 12n/4) 7. (v ฐ m) [cos((10/s) t) + sin((10/s) t)] 8. (2 m) cos ((-10/s)...
Problem 6. Consider the function y=(2m) sin ((10/s) t+π/4). Indicate whether each of the following waveforms is equivalent to y? Briefly justify your answers. 1. (2 m) cos((10/s)1+π/4) 2. (2 m) cos((10/s) t +3r/4) 3. (2 m) cos((10/s) t-r/4) 4. (2 m) sin(10/s)t +/4+4T) 5.-(2 m) sin ((10/s)t+/4-3m) 6. (-2m) oos((10/s) t+12r/4) 7. (VZn) [cos((10/s) t) + sin((10/s)t)] 8. (2m) cos ((-10/s)t+r/4) m) 1-cos((10/s) t + π)-sin((10/s) t-π
Problem 1. Consider the function f(x)- 3.12 show that f is Riemann integrable on [0.2] and use the definition to find .后f(x)dr Problem 2. Consider the function -2, zEQ 2, O f(r) = Show that f is not Riemann integrable on 0,1 but s Reemann integrable on this interval. Problem 3. (a) Let f be a real-valued function on a, b] such thatf()0 for all c, where c E [a, b Prove that f is Riemann integrable on a, b...
Please show all your works. Thanks. 4.(25 pts) Consider a periodic function X(t) = Sin(3t). Cos . Express x(t) in Exponential Fourier Series form and calculate Fourier Coefficients Co, C1, C-1,C2, C-2 ... etc (as many Fourier Coefficients as needed). What is the fundamental frequency (wo) of the x(t)? (hint: Use Euler's formula to express Sin(.) and Cos(.) in exponential forms)
please explain and do in matlab Problem 3. Consider the function f(x) e cos(2r). (1) Sketch its graph over the interval [0, m) by the following commands: (2) Using h = 0.01 π/6 in [0, π]. The commands are: to compute the difference quotient for z And the difference quotient is: ( 6 (3) Using h-0.01 to approximate the second derivative by computing the difdifquo for in [0, π). The commands are: And the difdifquo is: Problem 3. Consider the...
Problem 1 (20 points) Consider the differential equation for the function y given by 4 cos(4y) 40e 2e) cos(8t)+5 eu 2t) sin(8t)/ - 12e - 0. 8 sin(4y) y a. (4/20) Just by reordering terms on the left hand side above, write the equation as Ny + M 0 for appropriate functions N, M. Then compute: aN(t, y) ayM(t, y) b. (8/20) Find an integrating factor If you keep an integrating constant, call it c (t) N and M M,...
Using Mathematica Consider the vector-valued function r(t)=et cos t i+(sin t)/(t+4) j +t k. a) Plot the curve with t going over the interval [-2, 2]. b) Plot the curve again over the same interval, but this time add the velocity vector in blue at (1, 0, 0) to the graph. c) Plot the curve again over the same interval, along with the blue velocity vector at (1, 0, 0), but this time add the acceleration vector in red at...