Question

Use taylors formula to find quadratic and cubic approximations of f(x, y) =exsin y at origin. Estimate the error of the approximations if modx less than equal to 0.1 and mod y less than equal to 0.1

Answer 1

Solution:

• The given function is $f(x,y)=e^xsin(y)$
• The objective is to find the quadratic approximation of a given function using the Taylor formula and also estimate the error in the approximation if $| x | ≤ 0.1$ and $| y | ≤ 0.1 .$

Calculate First partial derivatives at $(0,0)$ .

$f x =e^xsin(y)$

$f x ( 0 , 0 ) = 0$

$f y = e^xcos(y)$

$f y ( 0 , 0 ) = 1$

Calculate second derivatives at $(0,0)$

$f x x = e^xsin(y)$

$f x ( 0 , 0 ) = 0$

$f y = e^xcos(y)$

$f y ( 0 , 0 ) = 1$

$f x y = -e^xsin(y)$

$f x y ( 0 , 0 ) = 0$

Therefore quadratic approximation of $f(x,y)$ at origin using Taylor formula is:

$T ( x , y ) ≈ 9 + \frac{1}{2} [ − 9 x^2 − 9 y^2 ] ≈ 9 − \frac{9}{2} [ x^2 + y^2 ]$

Next Calculating error at the origin using $| x | ≤ 0.1$ and $| y | ≤ 0.1$ as:

$E ( x , y ) ≤ \frac{1}{2} × 9 [ ( 0.1 )^2 + ( 0.1 ) ^2 ] ≤ 0.08685$

Thus, the Quadratic polynomial using the Taylor formula is $9 − \frac{9}{2} [ x ^2 + y ^2 ]$ and error in the approximation is less than 0.08685.