How do you solve 3? Typed answers are preferred but not necessary, thank you! Will upvote answer! For f € R[0, M], M > 0, recall that the improper integral o f(x) dx converges if and only if· MI F(2) dælimM→+00,exists.Consider the function f(x) E C[0, +∞) defined bysin xf(r) = { 1,x > 0,x = 0.1. Show that “ f(x) dx converges if and only if (Sn) converges, whereSnsin xdx.Hint: | sin x /æ| < |1/x|.2. Write Sn = (-1)-'a; wherei=1insin xdx.a; = (-1)-–1(i-1)mShow that a; >0 and (an) is decreasing.3. Conclude that * f(x) dx converges.

Answered: Image /qna-images/answer/731b1376-5882-4a73-8d35-d7875075d032.jpg

Answered: 3. Conclude that * f(x) dx converges.

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Given a continuous function h : [0,2) —> R with h(x)>= 0 for all x : [0,2). Show that if there is a point c : (0,2) at which h(c) > 0, then (Integral of h from 0 to 2 is) > 0
Find the best linear approximation to f on th e interval [0, 1]. f (x) =√x
Let Show that hn → 0 uniformly on R but that the sequence of derivatives (hn) diverges for every x ∈ R.
Let L(x) be the liberalization of f(x)=e^x at a=0 Use L(x) to find the approximate value of e.
Redefine R(t) such that it is continuous at t = 0
If F(x) = ∫x-x et^2 dt, find f(x), where F is the antiderivative of f.
Answer the questions by Yes or No Let ID be an open, simple connected, bounded subset of R^d in all statements below. 1) Does every IVP of nonlinear ODE dx/dt = f(x), x(0) = x0 ∈ R^d with continuous f have a local, unique solution x? (2) Does every nonlinear ODE dx/dt = f(x), x(0) = x0 ∈ R^d with continuous f have a global solution x on [0, +∞)? (3) Does every nonlinear ODE dx/dt = f(x), x(0) = x0 ∈ R^d with polynomial f have a global solution x on [0, +∞)? (4) Is every rational function f also differentiable everywhere? (5) Is every continuous function f also differentiable? 6) Does the nonlinear ODE dx/dt = arctan(x) ∈R^1 have a global solution x on [0, +∞) for all finite initial values? (7) Does the IVP for the nonlinear ODE dx/dt = arctan(x) ∈R^1 have a unique solution x for all finite initial values x(0) = x0 ∈ R^1? (8) Is the function x ∈D = R^1→ f(x) = e^(−x) Lipschitz-continuous on D?
Consider the following function: f(x,y) = { (x2+y2)sin[1/(x2+y2)], for (x2+y2) not equal to 0; 0, for x = 0, y = 0}. (a) Investigate fx and fy (partial derivatives of f(x,y)) in a neighborhood of (0,0). Are fx and fy continuous at (0,0)? Are fx and fy bounded in a neighborhood of (0,0)? (b) Is the function f(x,y) differentiable at (0,0)?
Q2/ Evaluate the integral ∫ te^−2t dt , The limits of integration from 1 to 3

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