4. (a) Let A € R² be open and let f: AR be C². Let (a, b) A and suppose the rectangle R= [a, a +h] x [b, b+k] C A. Show that there exist p, q = R such that: f(a, b) f(a, b) f(a, b + k) − f(a+h, b) + f(a+h, b+ k) = ddyf (p)hk f(a,b+k) − f(a + h,b) + f(a+h,b+ k) = Əyəxf (q)hk (b) Let g: R → R be C². Use part (a) to show that: g" (a) Hint: Consider f(x, y) = g(x+y). lim h→0 g(a+h)- - 2g(a) + g(a - h) h²
4. (a) Let A € R² be open and let f: AR be C². Let (a, b) A and suppose the rectangle R= [a, a +h] x [b, b+k] C A. Show that there exist p, q = R such that: f(a, b) f(a, b) f(a, b + k) − f(a+h, b) + f(a+h, b+ k) = ddyf (p)hk f(a,b+k) − f(a + h,b) + f(a+h,b+ k) = Əyəxf (q)hk (b) Let g: R → R be C². Use part (a) to show that: g" (a) Hint: Consider f(x, y) = g(x+y). lim h→0 g(a+h)- - 2g(a) + g(a - h) h²
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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![4. (a) Let A € R² be open and let f: AR be C². Let (a, b) A and suppose the rectangle
R = [a, a + h] × [b, b+k] CA. Show that there exist p, q € R such that:
f(a, b)
f(a, b)
f(a, b + k) − f(a+h, b) + f(a+h, b+ k) = ddyf (p)hk
f(a,b+k) − f(a + h,b) + f(a+h,b+ k) = Əyəxf (q)hk
(b) Let g: R → R be C². Use part (a) to show that:
g" (a):
Hint: Consider f(x, y) = g(x + y).
lim
h→0
g(a+h)- - 2g(a) + g(a - h)
h²](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F24d2d084-e1b8-4965-aed3-95c7dd8fa933%2F22362b57-2cf5-4ba8-87e9-b77c11ef8d0d%2Feeb904b_processed.png&w=3840&q=75)
Transcribed Image Text:4. (a) Let A € R² be open and let f: AR be C². Let (a, b) A and suppose the rectangle
R = [a, a + h] × [b, b+k] CA. Show that there exist p, q € R such that:
f(a, b)
f(a, b)
f(a, b + k) − f(a+h, b) + f(a+h, b+ k) = ddyf (p)hk
f(a,b+k) − f(a + h,b) + f(a+h,b+ k) = Əyəxf (q)hk
(b) Let g: R → R be C². Use part (a) to show that:
g" (a):
Hint: Consider f(x, y) = g(x + y).
lim
h→0
g(a+h)- - 2g(a) + g(a - h)
h²
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