Use Taylor's formula to find the requested approximation of f(x, y) near the origin.
-Quadratic approximation to $\mathrm { f } ( \mathrm { x } , \mathrm { y } ) = \mathrm { e } ^ { \mathrm { x } + 3 \mathrm { y } }$
A) $x + 3 y + \frac { 1 } { 2 } x ^ { 2 } + \frac { 3 } { 2 } x y + \frac { 9 } { 2 } y ^ { 2 }$
B) $1 + x + 3 y + \frac { 1 } { 2 } x ^ { 2 } + 3 x y + \frac { 9 } { 2 } y ^ { 2 }$
C) $x + 3 y + \frac { 1 } { 2 } x ^ { 2 } + \frac { 9 } { 2 } y ^ { 2 }$
D) $1 + x + 3 y + \frac { 1 } { 2 } x ^ { 2 } + \frac { 9 } { 2 } y ^ { 2 }$

Question

Quizplus · Accepted Answer

The quadratic approximation of $f(x, y) = e^{x + 3y}$ near the origin using Taylor's formula includes the function value at the origin, which is $f(0,0) = e^0 = 1$, the first-order partial derivatives evaluated at the origin, and the second-order partial derivatives evaluated at the origin. The correct approximation includes terms for $1$, $x$, $3y$, $\frac{1}{2}x^2$, $3xy$, and $\frac{9}{2}y^2$, which matches option B.

Use Taylor's Formula to Find the Requested Approximation of F(x f(x,y)=ex+3y\mathrm { f } ( \mathrm { x } , \mathrm { y } ) = \mathrm { e } ^ { \mathrm { x } + 3 \mathrm { y } }f(x,y)=ex+3y

Use Taylor's Formula to Find the Requested Approximation of F(x $\mathrm { f } ( \mathrm { x } , \mathrm { y } ) = \mathrm { e } ^ { \mathrm { x } + 3 \mathrm { y } }$