(a) Show that a function h: I→ R is differentiable at a I if and only if there exists afunction 1: I→ R which is continuous at x = a and satisfiesh(x) - h(a) = (x)(x − a)for all x € I.(b) Use this criterion for differentiability (in both directions) to prove the Chain Rule.

Answered: (a) Show that a function h: I→ R is…

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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2. Suppose that z is a function of r and y, and F(xz - y², xy - 2², yz - ²) = 0. Calculate and (Hint: Since F = 0, then dF = 0. Use a proper change of variables and chain rule to proceed)
L. Let f(z) be an entire function. (a) If f'(z)| ≤ z for all z € C, prove that there exists a, 3 € C such that f(z) = a + 3z² for all z EC and 3| ≤ 1. (b) If f(2)= f(z + 1) = f(z+i) for all z € C, prove that f(z) is constant. (c) If |f(z)| →∞ as z →→→∞, prove that f(z) is a polynomial function.
Let F be a function such that F(x) = x, for all real numbers x. Find the particular antiderivative F(x) that satisfies F(2) = 5. 9.
1. Let |x| = (x12 + x22)1/2. Define functions g(x1, x2) = x1x2e|x|2 , x = (x1, x2) ∈ R2 gradient ∇g and the points x where ∇g(x) = 0.
Suppose you are given the following theorem: Let f, g be two C² functions on R'. Fix x € R' and h > 0, and consider the closed interval [æ, æ + h]. Then for every y E [x, x + h] there exists & E (x, x +h) such that {f(y) – [f (x) + f' (x)(y – x)]}g"(£) = f"(E){g(y) – [g(x)+g'(x)(y – x)]}. Use this theorem to prove the 2"d order Mean Value Theorem. That is prove that if f is C² on R' then f (x + h) = f(x) +hf' (x) + f"(x + 0h) for some 0 E (0, 1). Hint: let g(y) = (y – x)².

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(a) Show that a function h: I→ R is differentiable at a I if and only if there exists a function 1: I→ R which is continuous at x = a and satisfies h(x) - h(a) = (x)(x − a) for all x € I. (b) Use this criterion for differentiability (in both directions) to prove the Chain Rule.