Find an equation of the tangent plane to the given surface the specified point:

• $z = y \cos(x-y), (2,2,2)$
• $z = e^{x^2-y^2}, (1, -1, 1)$

Linearization

Explain why $f(x,y) = \frac{x}{x+y}$ is differentiable at $(2,1)$ and find the linearization $L(x,y)$ of $f$ at $(2,1)$.

Verify the linear approximation $\sqrt{y} + \cos^2 x \approx 1 + \frac{1}{2} y$ at $(0,0)$.

Find the linear approximation of the function $f(x,y) = \ln(x-3y)$ at $(7,2)$ and use it to approximate $f(6.9,2.06)$. Illustrate by graphing $f$ and the tangent plane.