Question

Of the infinitely many lines that are tangent to the curve y = −6 sin x and pass through the origin, there is one that has the largest slope. Use Newton's method to find the slope of that line correct to six decimal places.

Answer 1

Naturally, the tangent with the largest slope will be the one closest to the origin.

So, we want

-6cosx = -6sinx/x
x = tanx
solve that using Newton's method, and you have x = 4.93

so, your line is

y = -6cos(4.93)(x-4.93)+6sin(4.93)
or
y-5.98 = 0.52(x-4.93)

See the graphs at

http://www.wolframalpha.com/input/?i=plot+y+%3D+-6sin%28x%29%2C+y%3D0.52%28x-4.93%29%2B5.98+where+0%3C%3Dx%3C%3D2pi

if you tighten your accuracy, you should get a better tangent.

Answer 2

Wasn't your tangent line supposed to pass through the origin?

Answer 3

Reiny, yes it does pass through the origin.

Answer 4

I sure flubbed that one. Your later solution was spot on, Reiny.