Activity 47.
Guido is romping on the surface is \(f(x,y)=\sin x + \sin y.\) He has a special compass that always points in the direction of the gradient which he knows always points in the direction of steepest ascent.
Use software to efficiently calculate these results.
(a)
Suppose Guido starts his romp at \((\pi/2,-\pi/4)\text{.}\) He takes steps of length \(\pi/8\text{.}\)
(i)
What direction does he go for his first step?
(ii)
What is his his second position?
(iii)
Repeat this process of finding a direction and then the new point.
(iv)
Where does Guido’s romp stop? It does stop.
(b)
Guido starts a romp at \((0,-\pi/2)\text{.}\) Perform the same process with step size \(\pi/8\text{.}\)
(c)
Guido starts a romp at \((9\pi/8,3\pi/2)\text{.}\) Perform the same process with step size \(\pi/8\text{.}\)
(d)
Guido starts a romp at \((11\pi/8,11\pi/8)\text{.}\) Perform the same process with step size \(\frac{\sqrt{2}\pi}{8}\text{.}\)
(e)
Describe the types of locations at which Guido’s romps can end.
(f)
Guido starts a romp at \((11\pi/8,\pi)\text{.}\) Perform the same process with step size 0.1.
