1. Let (a, bj be an interval in R and let be a continuous function on (a, b). (i) Check that the function z+ (b – a)r+ a is a bijection from [0,1] to [a, b) and its inverse is z+ (ii) For z € [0, 1], let g(r) = »((b-a)r+a) (which is equivalent to say that (z) = 9(=). Check that g is continuous on [0, 1]. (iii) By question 3), we know that there exists a sequence of polynomials, say (4n)n, which converges uniformly to g on [0, 1]. Construct a sequence of polynomials (p)n such that | – Pa|l.ja.bj = ||g -

1. Let (a, bj be an interval in R and let be a continuous function on (a, b). (i) Check that the function z+ (b – a)r+ a is a bijection from [0,1] to [a, b) and its inverse is z+ (ii) For z € [0, 1], let g(r) = »((b-a)r+a) (which is equivalent to say that (z) = 9(=). Check that g is continuous on [0, 1]. (iii) By question 3), we know that there exists a sequence of polynomials, say (4n)n, which converges uniformly to g on [0, 1]. Construct a sequence of polynomials (p)n such that | – Pa|l.ja.bj = ||g -

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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Differential Equations

Have you ever observed the movement of the satellites and rockets or how the planets go around the sun in their orbits? How will you determine their motion? Definitely they follow some scientific laws and these kinds of problems are framed as differential equations.

Question

4. Let [a, b] be an interval in R and let be a contimuous function on [a, b).
(i) Check that the function r+ (b- a)x + a is a bijection from [0, 1] to [a, b] and its inverse is I+E.
(ii) For r € [0, 1], let g(x) = »((b-a)x+a) (which is equivalent to say that (r) = g(). Check that
g is continuous on [0, 1].
(iii) By question 3), we know that there exists a sequence of polynomials, say (4n)n, which converges
uniformly to g on [0, 1]. Construct a sequence of polynomials (pn)n such that || – Pn|.ja.bj = ||g –
(iv) Deduce the Weiestrass approximation theorem for the continuous function on [a, b].

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