91Ó°ÊÓ

Find \(d w / d t\) (a) using the appropriate Chain Rule and (b) by converting \(w\) to a function of \(t\) before differentiating. \(w=x^{2}+y^{2}+z^{2}, \quad x=e^{t} \cos t, \quad y=e^{t} \sin t, \quad z=e^{t}\)

Find both first partial derivatives. \(f(x, y)=\int_{x}^{y}\left(t^{2}-1\right) d t\)

Find an equation of the tangent plane and find symmetric equations of the normal line to the surface at the given point. $$ z=\arctan \frac{y}{x}, \quad\left(1,1, \frac{\pi}{4}\right) $$

In Exercises 27-32, use the function $$f(x, y)=3-\frac{x}{3}-\frac{y}{2}$$ Sketch the graph of \(f\) in the first octant and plot the point (3,2,1) on the surface.

Describe the domain and range of the function. $$ g(x, y)=\frac{1}{x y} $$

In Exercises 25-28, discuss the continuity of the function and evaluate the limit of \(f(x, y)\) (if it exists) as \((x, y) \rightarrow(0,0)\). \(f(x, y)=\ln \left(x^{2}+y^{2}\right)\)

Examine the function for relative extrema and saddle points. $$ f(x, y)=2 x y-\frac{1}{2}\left(x^{4}+y^{4}\right)+1 $$

In Exercises 27-32, use the function $$f(x, y)=3-\frac{x}{3}-\frac{y}{2}$$ Find \(D_{\mathrm{u}} f(3,2),\) where \(\mathbf{u}=\cos \theta \mathbf{i}+\sin \theta \mathbf{j}\) (a) \(\theta=\frac{\pi}{4}\) (b) \(\theta=\frac{2 \pi}{3}\)

Find both first partial derivatives. \(f(x, y)=\int_{x}^{y}(2 t+1) d t+\int_{y}^{x}(2 t-1) d t\)

Examine the function for relative extrema and saddle points. $$ z=\left(\frac{1}{2}-x^{2}+y^{2}\right) e^{1-x^{2}-y^{2}} $$