Find the incorrect step in the proof and explain why it is incorrect. Find the error in the following "Proof" and explain in a short paragraph why the reasoning is incorrect.Statement: If f is a function that is differentiable at the point w then the following are equivalent1. f'(w) = m2. there is a line y = mx +b containing the point (w, f(w)) and there is &> 0 such that either:f(x) < mx + b for all æ E (w – 8, w+ 8), or,• f(x) > mx +b for all x E (w – 8, w + 8).Proof: To prove that (2) implies (1) begin by observing that since it is known that the line y = mx +b contains thepoint (w, f(w)) it is possible to find b by evaluating y at w. This gives f(w) = mw + b and henceb = f(w) – mw and soy = mx + f(w) – mw.(5)The other hypothesis is that fis differentiable at w and sof(x) – f(w)lim= f'(w)(6)andf(x) – f(w)lim= f'(w)(7)Two cases must now be considered,Assume first that there is &> O such that f(æ) < mx + b for all æ € (w – , w + 8). It must be shown thatf'(w) = m. Note that by Equation (5)f(x) – f(w) < mx + f(w)f(w) = m(x – w).тwHence, if x > wthen 1/(x – w) >0 and sof(x) – f(w)m(x – w)and therefore by Equation (6)f'(w) = limf(x) – f(w)m(x – w)< lim= mI - wI - wOn the other hand, if a < w then 1/(x – w) <0 and sof(x) – f(w)m(x – w)wand hence by Equation (7)f'(w) = limf(x) – f(w)m(x – w)> limI - wIt follows that f'(w) = m in this case.Now assume that there is 8> O such that f() > mx + b for all z € (w – 8, w+ 8). It must again be shown thatf'(w) = m. Note that by Equation (5)f(x) – f(w) > mæ + f(w) -тw - f(w) — т (х — w).Hence, if æ > w then 1/(x – w) > 0 and sof(x) – f(w)т(т — w)and therefore by Equation (6)f'(w) = limf(x) – f(w)m(r – w)> lim= mOn the other hand, if a < w then 1/(x – w) < 0 and sof(x) – f(w)m(x – w)I - wand hence by Equation (7)f'(w) = limf(x) – f(w)m(x – w)< lim= mIt follows that f'(w) = m in this case too.To see that (1) implies (2) use Equation 3.1.2 of the text. This formula yields that the tangent line to f at w hasformula y – f(w) = f'(w)(x – w). Since f'(w) = m it follows that y - f(w) = m(x – w) and soy = mx + (f(w) – w). Since this is a tangent line it touches the graph of f at exactly one point on some intervalaround w and hence one of the two alternatives of (2) must hold with b = f(w) – w.

Answered: Find the incorrect step in the proof…

Find the incorrect step in the proof and explain why it is incorrect.

8th Edition

ISBN:9781285741550

Author:James Stewart

Publisher:James Stewart

Chapter1: Functions And Models

Section: Chapter Questions

Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...

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