Chapter 13.6, Problem 49E

(a)

To determine

To graph: The three-dimensional function $f\left(x,y\right)=x^2-y^2$ at the point $\left(4,-3,7\right)$.

(b)

To determine

To calculate: The directional derivative $D_{u}f\left(-4,3\right)$ as a function of $\theta$ and graphically depict it.

(c)

To determine

To calculate: The zeros of the directional derivative of $D_{u}f\left(4,-3\right)=8\cos \theta +6\sin \theta$.

(d)

To determine

To calculate: The critical numbers of the directional derivative obtained.

(e)

To determine

To calculate: The value of $\Vert \nabla f\left(4,-3\right)\Vert$ and its interpretation.

(f)

To determine

To graph: The level curve for the function f when c equals 7 and to graph a vector in the direction of the gradient $\nabla f\left(4,-3\right)$.