\(\begin{array}{l}
1)\quad \text{Xét hàm số}\ f(x,y) = \sqrt{x^2 + y^2}\ \text{tại lân cận điểm $(4;3)$}\\
\text{Ta có:}\\
+)\quad f(4,3) = 5\\
+)\quad f'_x(x,y)=\dfrac{x}{\sqrt{x^2 +y^2}} \Rightarrow f'_x(4;3) = \dfrac45\\
+)\quad f'_y(x,y)=\dfrac{y}{\sqrt{x^2 +y^2}} \Rightarrow f'_y(4;3) = \dfrac35\\
\text{Ta được:}\\
\sqrt{(4,05)^2 + (3,07)^2} \approx f(4,3) + f'_x(4,3)(4,05 - 4) + f'_y(4,3)(3,07 - 3)\\
\Leftrightarrow \sqrt{(4,05)^2 + (3,07)^2} \approx 5 + \dfrac45\cdot 0,05 + \dfrac35\cdot 0,07\\
\Leftrightarrow \sqrt{(4,05)^2 + (3,07)^2} \approx 5,082\\
3)\quad \text{Xét hàm số}\ f(x,y) = \sin x\cos y\ \text{tại lân cận điểm $\left(\dfrac{\pi}{6},\dfrac{\pi}{3}\right)$}\\
\text{Ta có:}\\
+)\quad f\left(\dfrac{\pi}{6},\dfrac{\pi}{3}\right)=0,25\\
+)\quad f'_x(x,y) = \cos x\cos y \Rightarrow f'_x\left(\dfrac{\pi}{6},\dfrac{\pi}{3}\right)=0,435\\
+)\quad f'_y(x,y) = -\sin x\sin y \Rightarrow f'_y\left(\dfrac{\pi}{6},\dfrac{\pi}{3}\right) =-0,435\\
\text{Ta được:}\\
\sin28^\circ\cos61^\circ \approx f\left(\dfrac{\pi}{6},\dfrac{\pi}{3}\right) + f'_x\left(\dfrac{\pi}{6},\dfrac{\pi}{3}\right)\cdot\left(- \dfrac{2\pi}{180}\right) + f'_y\left(\dfrac{\pi}{6},\dfrac{\pi}{3}\right)\cdot \dfrac{\pi}{180}\\
\Leftrightarrow \sin28^\circ\cos61^\circ \approx 0,25 - 0,435\cdot 0,017\cdot 2 - 0,435\cdot0,017\\
\Leftrightarrow \sin28^\circ\cos61^\circ \approx 0,227815
\end{array}\)
