Question

Of the infinitely many lines that are tangent to the curve y = −4 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

see related questions below.

Answer 2

let the equation of the tangent be y = mx, where m is the slope
Here is a picture with m = 1/2
http://www.wolframalpha.com/input/?i=plot+y+%3D+-4sinx+%2C+y+%3D+.5x+%2C++from+0+to+20
Here is a picture with m = .9
http://www.wolframalpha.com/input/?i=plot+y+%3D+-4sinx+%2C+y+%3D+.9x+%2C++from+0+to+20

Wow, that is actually very close, but .....

let's intersect them
mx = -4sinx
m = -4sinx/x

we want m to be a max, so dm/dx = 0
dm/dx = (x(-4cosx) - (-4sinx)(1) )/x^2
= 0
4sinx - 4xcosx = 0
sinx = xcosx
sinx/cosx = x
tanx = x or tanx - x = 0

let f(x) = tanx - x
f ' (x) = sec^2 x - 1

Newton said that at the solution to f(x)
betterx = x - f(x)/f ' (x)
= x - (tanx - x)/(sec^2 x - 1)

using my 2nd graph to guess at x = 4.5

betterx = 4.5 - (tan(4.5) - (4.5) )/(sec^2 (4.5) - 1)
= 4.4936..
betterx = 4.4936.. - (tan(4.4936 ......
= 4.4934096..
betterx - next round
= 4.4934056..
next round
= 4.4934095

ok, say x = 4.4934095
then y = -4sin(4.4934095) =3.904479

the first point of intersection is (4.4934095,3.904479)
using (0,0) as the other point
the maximum slope = 3.9044../4.4934..
= .868935