5. Let V-Pi(R), and, for p(x) E V, define f, f2 E V by 2 fi (p(x))p(t) dt and f2(p(xp(t) dt 0 0 Prove that (fi, f2) is a basis for V", and find a basis for V for which it is the dual basis
Let f be defined on an open interval I containing a point a (1) Prove that if f is differentiable on I and f"(a) exists, then lim h-+0 (a 2 h2 (2) Prove that if f is continuous at a and there exist constants α and β such that the limit L := lim h2 exists, then f(a)-α and f'(a)-β. Does f"(a) exist and equal to 2L? Let f be defined on an open interval I containing a point a...
11 1 11 3. Let f1, f2, ..., fn are differentiable functions from V HR with V a Hilbert space. Define F :V HRN by F(a) = [fi(x)] f2(2x) I:T, V x EV, [fn (2)] where R” is endowed with the standard inner product. Prove that T(V fi(2), y) | |(Vf2(2), y) | DF(x)(y) = V rEV. Liv fn(x), y)]
Real analysis 10 11 12 13 please (r 2 4.1 Limit of Function 129 se f: E → R, p is a limit point of E, and limf(x)-L. Prove that lim)ILI. h If, in addition, )o for all x E E, prove that lim b. Prove that lim (f(x))"-L" for each n E N. ethe limit theorems, examples, and previous exercises to find each of the following limits. State which theo- rems, examples, or exercises are used in each case....
2. (a) Let B = {f1, f2, f3} be a subset of P2 where fi(x) = x² – 3, f2(x) = x2 – 2x and f3(x) = x. Show that B is a basis of P2. (b) Determine whether or not the following sets are subspaces of F. (i) X = {f € F | f(x) = a(x + cos x), a € R}. (ii) Y = {f EF | f(x) = ax + sin x, a € R}. (c)...
1-> X- Let f :S → R and g:S → R be functions and c be a cluster point. Assume lim f (x), lim g(x) exists. Using the definition of the limit prove the following lim(af (x) + Bg(x)) =a lim f(x) + Blim g(x) for any a,ßeR xc XC X-> b. lim( f(x))} = (lim f(x)) f(x) lim f (x) c. If (Vxe S)g(x) # () and lim g(x)() then prove lim X-C XC 10 g(x) lim g(x) X-C
lim (x+1=0. Specify a relationship between e and & that guarantees the limit exists Use the precise definition of a limit to prove (Hint: Use the identityxxl.) State the steps for proving that lim f(x) - L xa to find a condition of the form Then, for any g>0, assume and use the relationship Let e be an arbitrary positive number. Use the inequality where depends only on the value of prove that between lim (x+1=0. Specify a relationship between...
3. The sequence (Fn) of Fibonacci numbers is defined by the recursive relation Fn+2 Fn+1+ F for all n E N and with Fi = F2= 1. to find a recursive relation for the sequence of ratios (a) Use the recursive relation for (F) Fn+ Fn an Hint: Divide by Fn+1 N (b) Show by induction that an 1 for all n (c) Given that the limit l = lim,0 an exists (so you do not need to prove that...
1. (3 points each) Answer each of the following statements as true or false a. If lim ) exists, then lim(lim() b. If lim f (x) exists, then fi (zo) exists. c. If f differentiable on la, b, then f is integrable on [a, b]. d. If f is continuous on [a, b] and differentiable on (a, b), then there exists a number X -To (a, b) such that f (b) f(a)- (b-a)f (x). e. If f is integrable on...
PLEASE ANSWER ALL! SHOWS STEPS 2. (a) Prove by using the definition of convergence only, without using limit theo- (b) Prove by using the definition of continuity, or by using the є_ó property, that 3. Let f be a twice differentiable function defined on the closed interval [0, 1]. Suppose rems, that if (S) is a sequence converging to s, then lim, 10 2 f (x) is a continuous function on R r,s,t e [0,1] are defined so that r...