Answer: Investigation Consider the function at the point (a) Use a computer algebra

Chapter 13, Problem 49

(choose chapter or problem)

Consider the function

\(f(x, y)=x^{2}-y^{2}\)

at the point (4, -3, 7).

(a) Use a computer algebra system to graph the surface represented by the function.

(b) Determine the directional derivative \(D_{\mathrm{u}} f(4,-3)\) as a function of \(\theta\), where \(\mathbf{u}=\cos \theta \mathbf{i}+\sin \theta \mathbf{j}\). Use a computer algebra system to graph the function on the interval \([0,2 \pi)\).

(c) Approximate the zeros of the function in part (b) and interpret each in the context of the problem.

(d) Approximate the critical numbers of the function in part (b) and interpret each in the context of the problem.

(e) Find \(\|\nabla f(4,-3)\|) and explain its relationship to your answers in part (d).

(f) Use a computer algebra system to graph the level curve of the function at the level c = 7. On this curve, graph the vector in the direction of \(\nabla f(4,-3)\), and state its relationship to the level curve.

Text Transcription:

f(x,y)=x^2-y^2

D_uf(4,-3)

theta

u=costhetai+sinthetaj

[0,2pi)

|nablaf(4,-3)\|

nablaf(4,-3)