Taylor\'s theorem problem: Let f(x; y) = cos(2x + 3y). a) Find the first, second

Q: Taylor's theorem problem: Let f(x; y) = cos(2x + 3y). a) Find the first, second

Taylor's theorem states that any function satisfying certain conditions may be represented by a Taylor series f(x)=f(0)+f'(0)+x^2/2! * f''(0)+ -------------- +x^(n-1)/(n-1)! *f^(n-1)(0)

ID: 2981784 • Letter: T

Question

Taylor's theorem problem: Let f(x; y) = cos(2x + 3y).

a) Find the first, second, and third order partial derivatives of f.

b) Find the equation of the best quadratic surface (Q) approximating f at

the origin. [i.e. the second order Taylor polynomial P2(x, y)]

c) Write the expression for the error in f [ the remainder].

d) What is the order of the error: O(||(x; y)||^2) or O(||(x; y)||^3)? Hint: |x| <= ||(x, y)|| and |y| <= ||(x,y)||.

e) Find a bound on max{| f(x, y)- P2(x, y) |} on the set where ||(x, y)|| <= 0:1.

Explanation / Answer

Taylor's theorem states that any function satisfying certain conditions may be represented by a Taylor series

f(x)=f(0)+f'(0)+x^2/2! * f''(0)+ -------------- +x^(n-1)/(n-1)! *f^(n-1)(0)

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Taylor\'s theorem problem: Let f(x; y) = cos(2x + 3y). a) Find the first, second

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