Find the quadratic approximation to the function f(x, y)= cos x cos y valid near the origin.
A) $f ( x , y ) = 1 - \frac { x ^ { 2 } } { 2 } + \frac { y ^ { 2 } } { 2 }$
B) $f ( x , y ) = 1 + \frac { x ^ { 2 } } { 2 } + \frac { y ^ { 2 } } { 2 }$
C) $f ( x , y ) = 1 - \frac { x ^ { 2 } } { 2 } - \frac { y ^ { 2 } } { 2 }$
D) $f ( x , y ) = 1 - x ^ { 2 } - y ^ { 2 }$
E) $f ( x , y ) = 1 - \frac { x ^ { 2 } } { 4 } - \frac { y ^ { 2 } } { 4 }$

Question

Quizplus · Accepted Answer

The quadratic approximation of a function $f(x, y)$ near the origin involves the function value at the origin, the first derivatives (which are zero for $\cos x$ and $\cos y$ at the origin), and the second derivatives at the origin. The second derivatives of $\cos x$ and $\cos y$ with respect to $x$ and $y$ respectively are $-\cos x$ and $-\cos y$, which evaluate to $-1$ at the origin. Thus, the quadratic approximation is $f(x, y) = 1 - \frac{x^2}{2} - \frac{y^2}{2}$.

Find the Quadratic Approximation to the Function F(x, Y)= Cos f(x,y)=1−x22+y22f ( x , y ) = 1 - \frac { x ^ { 2 } } { 2 } + \frac { y ^ { 2 } } { 2 }f(x,y)=1−2x2​+2y2​

Find the Quadratic Approximation to the Function F(x, Y)= Cos $f ( x , y ) = 1 - \frac { x ^ { 2 } } { 2 } + \frac { y ^ { 2 } } { 2 }$