Chapter 13.6, Problem 60E

Using a Function Consider the function $f(x,y)=9-x^2-y^2$.

(a) Sketch the graph of f in the first octant and plot the point (1, 2, 4) on the surface.

(b) Find $D_{u}f(1,2)$, where $u=\cos \theta i+\sin \theta j$, using each given value of 0.

(i) $\theta =-\frac{\pi }{4}$ (ii) $\theta =\frac{\pi }{3}$ $\theta =\frac{3\pi }{4}$ (iv) $\theta =-\frac{\pi }{2}$

(c) Find $D_{u}f(1,2)$, where $u=\frac{v}{\Vert v\Vert }$ licintr pqpH i

(i) $v=3i+j$ (ii) $v=-8i-6j$

(iii) v is the vector from $(-1,-1)$ to (3, 5).

(iv) v is the vector from $(-2,0)$ to (1, 3).

(d) Find $\nabla f(1,2)$.

(e) Find the maximum value of the directional derivative at (1, 2).

(f) Find a unit vector u orthogonal to $\nabla f(1,2)$ and calculate $D_{u}f(1,2)$. Discuss the geometric meaning of the result.