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.

Question

Reiny · Answer

let the point of contact be (x,y)
or (x, -6sinx)
Then for a tangent,
slope = m = (-6sinx - 0)/(x-0) = -6sinx/x

m = (-6sinx - 0)/(x-0) = -6sinx/x

dm/dx = (x(-6cosx - (-6sinx))/x^2 , by quotient rule
= 0 for a max/min

-6xcosx + 6sinx = 0
xcosx - sinx = 0
xcosx = sinx
x = sinx/cosx
tanx = x

let y = tanx - x
y' = sec^2 x - 1

Newton's Method"
x2 = x1 - y/y'
= x1 - (tan x1 - x1)/(sec^2 x1 - 1)
according to my sketch, x is between π and 2π, so how about
x1 = 4.5
x2 = 4.4936...

x1 = 4.4936
x2 = 4.4934096

x1 = 4.4934096
x2 = 4.4934095
how about that???

so m = -6sinx/x = 1.303402

and the tangent equation would be y = 1.303402x

check with Wolfram:
http://www.wolframalpha.com/input/?i=plot+y+%3D+-6sinx+%2C+y+%3D+1.3034x+%2C+0+%3C+x+%3C+10

Now , how good is that?

btw:
http://www.wolframalpha.com/input/?i=tanx+%3D+x
look at the 3rd solution.
we came up with that solution, had our guess been close to one of the other solutions, our Newton's Method would have zeroed in on the one closest to our guess.