Prove Theorem 1.9 from Theorems 1.7 and 1.8. Theorem 1.9 If f(x,y) has continuous first and second partial deriva- tives in a neighborhood D of the point (a,b) in the (x,y) plane, then f(x, y) = f(a, b) + ƒ,(a, b)(x − a) + ƒ,(a, b)( y − b) + R₂(x, y) - (1.37) Theorem 1.7: Taylor's formula with (integral) remainder If f(x) has n+ 1 continuous derivatives on [a,b] and c is some point in [a,b]. then for all x E [a, b] f(x) = f(c) + f'(c)(x − c) + ƒ˜(c)(x − c)² + ... 2! ƒ^(c)(x − c)² + R₂+1(x) n! (1.32) Theorem 1.8: Chain rule If the function f(x,y. =) has continuous first partial derivatives with respect to each of its variables, and x = x(t), y = y(t)., = = =(t) are continuously differentiable func- tions of 1, then g(t) = f(x(t). y(t)..(t)) is also continuously dif- ferentiable, and 8′(1) = 3 x'(1) + y²(1) ++ 2/12 (1) ax
Prove Theorem 1.9 from Theorems 1.7 and 1.8. Theorem 1.9 If f(x,y) has continuous first and second partial deriva- tives in a neighborhood D of the point (a,b) in the (x,y) plane, then f(x, y) = f(a, b) + ƒ,(a, b)(x − a) + ƒ,(a, b)( y − b) + R₂(x, y) - (1.37) Theorem 1.7: Taylor's formula with (integral) remainder If f(x) has n+ 1 continuous derivatives on [a,b] and c is some point in [a,b]. then for all x E [a, b] f(x) = f(c) + f'(c)(x − c) + ƒ˜(c)(x − c)² + ... 2! ƒ^(c)(x − c)² + R₂+1(x) n! (1.32) Theorem 1.8: Chain rule If the function f(x,y. =) has continuous first partial derivatives with respect to each of its variables, and x = x(t), y = y(t)., = = =(t) are continuously differentiable func- tions of 1, then g(t) = f(x(t). y(t)..(t)) is also continuously dif- ferentiable, and 8′(1) = 3 x'(1) + y²(1) ++ 2/12 (1) ax
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
![Prove Theorem 1.9 from Theorems 1.7 and 1.8.
Theorem 1.9 If f(x.y) has continuous first and second partial deriva-
tives in a neighborhood D of the point (a,b) in the (x.y) plane, then
f(x, y) f(a, b) + f,(a, b)(x- a) + f,(a, b)(y- b) + R(x, y)
(1.37)
Theorem 1.7: Taylor's formula with (integral) remainder If f(x) has
n +1 continuous derivatives on fa,b] and c is some point in fa,b].
then for all x E [a, b]
(x) = f(c) + f'(c)(x - c) +
f"(c)(x – c)'
2!
()(x – c)" + R,.,(x)
(1.32)
n!
Theorem 1.8: Chain rule If the function f(x.y.
first partial derivatives with respect to each of its variables, and
x = x(1), y = y(t), , = = :(1) are continuously differentiable func-
tions of t, then g() = f(x(t). y(). ()) is also continuously dif-
ferentiable, and
=) has continuous
%3D
8'(1) = x(1) +(1)
ay](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F4277d51e-17e0-4849-ac93-846487d61953%2F7db077dd-605f-4aaf-8a81-79d8819bb2a6%2Fx3ckvcg_processed.png&w=3840&q=75)
Transcribed Image Text:Prove Theorem 1.9 from Theorems 1.7 and 1.8.
Theorem 1.9 If f(x.y) has continuous first and second partial deriva-
tives in a neighborhood D of the point (a,b) in the (x.y) plane, then
f(x, y) f(a, b) + f,(a, b)(x- a) + f,(a, b)(y- b) + R(x, y)
(1.37)
Theorem 1.7: Taylor's formula with (integral) remainder If f(x) has
n +1 continuous derivatives on fa,b] and c is some point in fa,b].
then for all x E [a, b]
(x) = f(c) + f'(c)(x - c) +
f"(c)(x – c)'
2!
()(x – c)" + R,.,(x)
(1.32)
n!
Theorem 1.8: Chain rule If the function f(x.y.
first partial derivatives with respect to each of its variables, and
x = x(1), y = y(t), , = = :(1) are continuously differentiable func-
tions of t, then g() = f(x(t). y(). ()) is also continuously dif-
ferentiable, and
=) has continuous
%3D
8'(1) = x(1) +(1)
ay
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