91Ó°ÊÓ

Use the following formulas with \(h=0.01\) to approximate \(f_{x}(0.5,0.2)\) and \(f_{y}(0.5,0.2),\) and compare the results with the values obtained from \(f_{x}(x, y)\) and \(f_{x}(x, y)\) $$ \begin{array}{l} f_{x}(x, y) \approx \frac{f(x-2 h, y)-4 f(x-h, y)+3 f(x, y)}{2 h} \\ f_{y}(x, y) \approx \frac{f(x, y-2 h)-4 f(x, y-h)+3 f(x, y)}{2 h} \end{array} $$ $$ f(x, y)=y^{2} \sin (x y) $$

Use the following formulas with \(h=0.01\) to approximate \(f_{x}(0.5,0.2)\) and \(f_{y}(0.5,0.2),\) and compare the results with the values obtained from \(f_{x}(x, y)\) and \(f_{x}(x, y)\) $$ \begin{array}{l} f_{x}(x, y) \approx \frac{f(x-2 h, y)-4 f(x-h, y)+3 f(x, y)}{2 h} \\ f_{y}(x, y) \approx \frac{f(x, y-2 h)-4 f(x, y-h)+3 f(x, y)}{2 h} \end{array} $$ $$ f(x, y)=x y^{3}+4 x^{3} y^{2} $$

Use the following formulas with \(h=0.01\) to approximate \(f_{x x}(0.6,0.8)\) and \(f_{x y}(0.6,0.8)\) $$ \begin{array}{l} f_{x x}(x, y) \approx \frac{f(x+h, y)-2 f(x, y)+f(x-h, y)}{h^{2}} \\ f_{y y}(x, y) \approx \frac{f(x, y+h)-2 f(x, y)+f(x, y-h)}{h^{2}} \end{array} $$ \(f(x, y)=\sec ^{2}\left[\tan \left(x y^{2}\right)\right] \sin (x y)\)

Use the following formulas with \(h=0.01\) to approximate \(f_{x x}(0.6,0.8)\) and \(f_{x y}(0.6,0.8)\) $$ \begin{array}{l} f_{x x}(x, y) \approx \frac{f(x+h, y)-2 f(x, y)+f(x-h, y)}{h^{2}} \\ f_{y y}(x, y) \approx \frac{f(x, y+h)-2 f(x, y)+f(x, y-h)}{h^{2}} \end{array} $$ $$ f(x, y)=\frac{x^{3}+x y^{2}}{\tan (x y)+4 x^{2} y^{3}} $$