10. Calculate the tangent planes of the following functions at the indicated points
(a) $f (x, y) = x^{2} + y^{2}, p = (1, 1)$
(b) $f (x, y) = \ln x - y^{2}, p = (e, 1)$
(c) $f (x, y) = x + y, p = (0, 0)$

Answer

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Answer

$z - f (0, 0) = 1 (x - 0) + 1 (y - 0) \Rightarrow z = x + y$

Steps

Step 1 ： $\frac{\partial f}{\partial x} = 2 x, \frac{\partial f}{\partial y} = 2 y \Rightarrow \frac{\partial f}{\partial x} (1, 1) = 2, \frac{\partial f}{\partial y} (1, 1) = 2$

Step 2 ： $z - f (1, 1) = 2 (x - 1) + 2 (y - 1) \Rightarrow z - 2 = 2 (x - 1) + 2 (y - 1)$

Step 3 ： $\frac{\partial f}{\partial x} = \frac{1}{x}, \frac{\partial f}{\partial y} = - 2 y \Rightarrow \frac{\partial f}{\partial x} (e, 1) = 1, \frac{\partial f}{\partial y} (e, 1) = - 2$

Step 4 ： $z - f (e, 1) = 1 (x - e) - 2 (y - 1) \Rightarrow z - 1 = (x - e) - 2 (y - 1)$

Step 5 ： $\frac{\partial f}{\partial x} = 1, \frac{\partial f}{\partial y} = 1 \Rightarrow \frac{\partial f}{\partial x} (0, 0) = 1, \frac{\partial f}{\partial y} (0, 0) = 1$

Step 6 ： $z - f (0, 0) = 1 (x - 0) + 1 (y - 0) \Rightarrow z = x + y$

10. Calculate the tangent planes of the following functions at the indicated points(a) f(x,y)=x2+y2,p=(1,1)(b) f(x,y)=ln⁡x−y2,p=(e,1)(c) f(x,y)=x+y,p=(0,0)

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10. Calculate the tangent planes of the following functions at the indicated points
(a) $f (x, y) = x^{2} + y^{2}, p = (1, 1)$
(b) $f (x, y) = \ln x - y^{2}, p = (e, 1)$
(c) $f (x, y) = x + y, p = (0, 0)$