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40. Prove the mean value theorem for functions of two variables.

40. Prove the mean value theorem for functions of two variables.

Advanced Engineering Mathematics
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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6.40) my professor says I have to explain the steps in the solved problems in the picture. Not just copy eveything down from the text.

Mean value theorem 6.40. Prove the mean value theorem for functions of two variables. Let f(t) = f(x, + ht, Yo + kt). By the mean value theorem for functions of one variable, F(1) = F(0) = F'(0) 0 < 0 < 1 (1) CHAPTER 6 Partial Derivatives 151 If x = x, + ht, y = y + kt, then F(t) = f(x, y), so that by Problem 6.17, F'(1) =f. (dv/dt) +f,(dy/dt) = hf, + kf, and F(0) = hf,(xo + Oh, yo + Ok) + kf,(xo + Oh, yo + Ok) where 0 <0<1. Thus, (1) becomes f(x, + h, yo + k) – f (Xp» Yo) = hf,(x, + Oh, yo + Ok) + kf,(x, + 0h, yo + Ok) (2) where 0 <0<1 as required. Note that Equation (2), which is analogous to Equation (1) of Problem 6.14, where h= Ax, has the advan- tage of being more symmetric (and also more useful), since only a single number 0 is involved.
Transcribed Image Text:Mean value theorem 6.40. Prove the mean value theorem for functions of two variables. Let f(t) = f(x, + ht, Yo + kt). By the mean value theorem for functions of one variable, F(1) = F(0) = F'(0) 0 < 0 < 1 (1) CHAPTER 6 Partial Derivatives 151 If x = x, + ht, y = y + kt, then F(t) = f(x, y), so that by Problem 6.17, F'(1) =f. (dv/dt) +f,(dy/dt) = hf, + kf, and F(0) = hf,(xo + Oh, yo + Ok) + kf,(xo + Oh, yo + Ok) where 0 <0<1. Thus, (1) becomes f(x, + h, yo + k) – f (Xp» Yo) = hf,(x, + Oh, yo + Ok) + kf,(x, + 0h, yo + Ok) (2) where 0 <0<1 as required. Note that Equation (2), which is analogous to Equation (1) of Problem 6.14, where h= Ax, has the advan- tage of being more symmetric (and also more useful), since only a single number 0 is involved.
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