Problem # 8: Consider the functions fl (x)-x and f2(x)-6-10cx on the interval [0, 1] a)...
PART 1A PART 1B Let f(t) = S4 0 < t < -4 < t < 27 and assume that when f(t) is extended to the negative t-axis in a periodic manner, the resulting function is even. Consider the following differential equation. dax 3 2 + 6x = f(t) dt Find a particular solution of the above differential equation of the form Xp(t) = Ë Ancos nat, a s(t,n) n=1 р n=1 and enter the function g(t, n) into the...
Consider the space V of continuous functions on (0, 1] with the 2-norm 12 J f2 We saw in class that V is an incomplete normed linear space. (a) For a continuous function p on [0, 1], define a linear map Mp: V-V by Mpf-pf. Show that Mp is bounded and calculate its norm. (b) Is A = (Mplp E C(0,1)) a Banach algebra? Note that B(V) is necessarily incomplete, so it is not enough to prove that A is...
2. (10 Points) Fine an arrangement of the following functions fl, f2u10 so that fl-0(f2), f2 -O(f3),.., f(9)-0(f10). Also indicate which functions grow at the same asymptotic rate. lg[n!), In(n), n, 2(2n), 2(n*), nlg(n) lg(n), n2, 1, lg(n)
Problem #10: Consider the following function. 8(x,y) = 8x? - 7y2 + 16V7x (a) Find the critical point of g. If the critical point is (a, b) then enter a b (without the quotes) into the answer box. (b) Using your critical point in (a), find the value of D(a,b) from the Second Partials test that is used to classify the critical point. (c) Use the Second Partials test to classify the critical point from (a). Problem #10(a): Enter your...
Problem #10: Consider the following function. 8(x,y) = {2x2 – 3y2 +6V6 y (a) Find the critical point of g. If the critical point is (a, b) then enter 'ab' (without the quotes) into the answer box. (b) Using your critical point in (a), find the value of D(a,b) from the Second Partials test that is used to classify the critical point. (c) Use the Second Partials test to classify the critical point from (a). Problem #10(a): Enter your answer...
Problem #8 : A lamina with constant density ρ(r.))-5 occupies the region under the curve y-sin(m/8) from x-0 to x-8. Find the moments of inertia 4 and Enter the values of 4 and ly (in that order) into the answer box below, separated with a comma. Enter your answer symbolically, as in these examples Problem #8: Just save Submit Problem #8 for Grading Problem #8 | Attempt #1 | Attempt #2 Attempt #3 Attempt #4 Attempt #5 Your Answer: Your...
Consider the function on the interval (0, 2). f(x) = sin(x) cos(x) + 8 (a) Find the open interval(s) on which the function is increasing or decreasing. (Enter your answers using interval notation.) increasing decreasing (b) Apply the First Derivative Test to identify all relative extrema. relative maxima (x, y) = (smaller x-value) (larger x-value) (X,Y)= (x, y) = (1 (x, y) = relative minima (smaller x-value) (larger x-value)
2. Consider the vector space C([0, 1]) consisting of all continuous functions f: [0,1]-R with the weighted inner product, (f.g)-f(x) g(x) x dr. (a) Let Po(z) = 1, Pi(z) = x-2, and P2(x) = x2-6r + 흡 Show that {Po, pi,r) are orthogonal with respect to this inner product b) Use Gram-Schmidt on f(x)3 to find a polynomial pa(r) which is orthogonal to each of po P1 P2 You may use your favorite web site or software to calculate the...
Problem #8: A rod of length 9 coincides with the interval [0,9] on the x-axis. Consider the heat equation in the special case when k=1 if both ends are held at temperature zero for all t> 0. The initial temperature is f(x) throughout where f(x) = a sin(876x) + b sin(4x) The solution to the heat equation under the above conditions is of the form u (x, t) = a g1(x, t) + b g2(x, t) (a) Enter the function...
the below is the previous question solution: 1. Recall the following boundary-value problem on the interval [0, 1] from Homework 2: f" =-Xf, f'(1) =-f(1). f(0) = 0, Show that if (Anh) and to this boundary-value problem, λι, λ2 〉 0, λιメÂn then fi and f2 are orthogonal with respect to the standard inner product (.9)J( gr)dr. (You may use the solution posted on the course website, or work directly from the equation and boundary conditions above.) (λ2'J2) are two...