Show that the square integrable function f(x) = sin( πk log x/ log 2 )for k ≥ 1 are orthogonal over the interval 1 ≤ x ≤ 2 with respect to the weight function r(x) = 1/ x . Obtain the norms of the functions and construct the othornormal set.

Q: If f(x) is an integrable function defined on the closed interval [a, b] ], ƒ (x) dx represents the…

A:

Q: Let $c$ be a critical number of a function $f$ that is continuousinterval, except possibly at $c$,…

A:

Q: Prove that f(z)= 1/(1-z) in not uniformly continuous for |z|

A:

Q: If f has a critical point in the interior of a closed and bounded domain D then the minimum and…

A: Statement: If f has a critical point in the interior of a closed and bounded domain D then the…

Q: Let G(x) = tan 9t dt. In order to use the Fundamental Theorem of Calculus (Part 2), G(x) needs to be…

A: For the following integral,

Q: Let L2[0, 1] be all bounded Riemann integrable functions [0, 1] → R with inner product (f, g) Split…

A: Consider the provided question, Given that, Hn,kx=2n,    k2n≤x<k+122n-2n,   k+122n≤x<k+12n0,…

Q: Let f(x,y,z)=x² +x(2y+z)- 1 + 2y cos x. Find all the critical points of f and determine whether it…

A:

Q: if 0 < x < 1 Problem 1. Prove that f (x) is integrable using Riemann's 1 if x = 0 criterion…

A:

Q: If fi and fa are two bounded and integrable functions on [a, b] and there exists a number >0 such…

A:

Q: Suppose that f is a function on [a, b] such that f(x) = 0 for all x e [a, b]\T , where T is finite…

A:

Q: Suppose that g is a Riemann integrable function on [0.3, 31.5] and f(x) = g(x) except for the values…

A:

Q: 8. Use Simpsons' rule to numerically integrate for the n = 6, case f (x) = x on the unit interval,…

A:

Q: 10. Use Riemann sums and a limit to compute the exact area under the curve y = f(x) = 2x2 +1 on the…

A:

Q: . Apply Jensen's inequality (2) to f(x) = In x, with i 1 = ..= d = 1/n to prove that 1 Vx1x2...X 0,…

A: Given :-  Apply Jensen's inequality (2) to f  x  = ln x  , with λ1 = λ2 = ..........  = λn = 1n to…

Q: * The statement : Every bounded real valued function over [a, 6) is Riemann integrable over (a, b)…

A: Necessary condition for a function to be Riemann integrable is that function is bounded and interval…

Q: Show that the given set of functions is orthogonal on the indicated interval. Find the norm of each…

A: Here is the solution

Q: a) Show that the function defined as follows 1 <x<. when 2 1 f(x)= 2" where n = 0,1, 2,3,.. on 2"…

A: Given , fx=12n  , when 12n+1&lt;x&lt;12n   where n=0,1,2,3...  onf0=0  is interable in 0,1 has…

Q: Suppose that f: R-Cis a measurable function with 1+ |xf° for almost all x E R. Show that its Fourier…

A: Let, f⏜ be Fourier transform of fx. Then nth derivative of f⏜ is Fourier transform of…

Q: Is f(x) = x² uniformly continuous on (0,1)? Why or why not?

A:

Q: 9 Prove that if m(x) is differentiable on (-∞, 0) and its derivative m'(x) is bounded then m is…

A:

Q: Find the Wronskian for {x, -sin x, cos x}

A:

Q: 4 A function fx) = 1- x² + xr is defined in the closed interval [-1, 1]. The value of x in i onen…

A:

Q: Let (fn) be a sequence of functions that are integrable on [a, b] and that converge uniformly on [a,…

A:

Q: 2. Express the following limits as a definite integral of a suitable function over a finite…

A:

Q: 7. Prove that if the functions f and o are bounded and integrable in [a, b], then 6. 9. f(x) $(x) dx…

A:

Q: 8. Use Simpsons' rule to numerically integrate for the n = 6, case f(x) = x on the unit interval,…

A:

Q: Consider the sequence of functions n+ cos(x) 3n + sin(x) defined on the R. 1. Find the function f to…

A:

Q: 6. Suppose that f: R -R is uniformly continuous so that for some 6 > 0, and for all 1, y € R I/(=) –…

A: The objective here is to prove: if f(0)=0then for every x&gt;0 fx&lt;1+xδ Given: f:R→R is uniformly…

Q: Is the set {sin nx , cos mx} an orthogonal set with respect to the inner product on the set of all…

A:

Q: 5. Determine the interval on which f(x) = x+2 is integrable. %3D (-00, 2) [-2, co0) (-00,-2) U (-2,…

A:

Q: Show that the function f defined on [0, 1] by the conditions 1 f(x)=2rx when 7=1, 2, 3,...... És…

A:

Q: A set A C [0,1] is dense in [0, 1] iff every open interval that intersects [0, 1] contains a point…

A:

Q: (1) Give an example of a function f such that | f |is Riemann integrable but f is not.

A: Consider the function f on the interval 0,1 defined by:  fx=1, if x is rational-1, if x is…

Q: Let f(x) be a continuous function on [0, 1] and consider the limit 1 Xj +... f + In · drn. lim dx1…

A: As per the question we have to interpret the given integral as an expectation value and find the…

Q: (5) Prove that if f is continuously differentiable on [a, b], then |f(a) – f(b)| a – b max f'(x)|…

A:

Q: 10. Use Riemann sums and a limit to compute the exact area under the curve y = f(x) = 2x² + 1 on the…

A: Given÷

Q: Consider a two-dimensional circular metal plate of radius R and let u be its temperature…

A: The solution is given below

Q: Consider the sequence of functions {f} defined on the interval [0, 1], where fn(r)= nx 1+n²r²¹ Vx €…

A:

Q: Prove that f : [−4,2] → R defined by f(x) = 7/x+5 is uniformly continuous on the interval [−4, 2]…

A:

Q: If |fn(x) – fn(y)l 0 and all x, y in a compact interval, show that {fn} is uniformly…

A:

Q: Evaluate the definite integral: (2 – a²)dx using limit of the Riemann sum.

A:

Q: Suppose that f is integrable on [a,b], and that g(x) is defined for all x E[a,b] and ,exists and…

A: Given: To explain the given statement as follows, To prove that    exists and equals .

Q: 1. Express the following limits as definite integrals. (a) lim )V3(x;)² + In(x;) Ax, [2,7], where Ax…

A:

Q: Show that the square integrable function f(x) = sin( πk log x/ log 2 )for k ≥ 1 are orthogonal over…

A:

Q: Show that if f(x) is integrable on every interval of real numbers, and if a and b are real numbers…

A:

Q: 5. Consider the function if x € Q f(x)= if x # Q. Show that f is not Riemann integrable on [0, 1]. n…

A:

Q: Exercise 1. Express the limit as a definite integral on the given interval cosxi a) lim E-1 Xị Ax,…

A: We have to find the derivative.

Q: Consider the nonsmooth function: 1 f(x) = max x; +;|x||², 1<i<K where x E R", x; is the i-th…

A:

Q: To prove Theorem 1, (that If F is an antiderivative of f on an interval I, then the most general…

A: To prove the statement regarding the  antiderivatives of a function

Q: (5) Suppose that fis continuous on [a, b]- f(x) ≥ 0 Vx = [a,b] and [f(x)dx = 0. Prove that f(x) = 0…

A: