SOLUTION: Of the infinitely many lines that are tangent to the curve of y = -sin x and pass through the origin, there is one that has the largest slope. Use Newton's method to find the slope

Algebra ->  Test -> SOLUTION: Of the infinitely many lines that are tangent to the curve of y = -sin x and pass through the origin, there is one that has the largest slope. Use Newton's method to find the slope      Log On


   

Question 958546: Of the infinitely many lines that are tangent to the curve of y = -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 by Fombitz(32388) About Me  (Show Source):
You can put this solution on YOUR website!
graph%28300%2C300%2C-7%2C7%2C-2%2C2%2C-sin%28x%29%29
A line tangent to the curve y=-sin%28x%29 at point has a slope of m=-cos%28x%29.
Since it goes through point (x,y) and the origin then the slope also equals.
m=%28y-0%29%2F%28x-0%29=y%2Fx=-sin%28x%29%2Fx
So then,
-sin%28x%29%2Fx=-cos%28x%29
sin%28x%29-xcos%28x%29=0
So to find the root, let
f%28x%29=sin%28x%29-xcos%28x%29
Then,
df%2Fdx=xsin%28x%29
Now using Newton's method with a starting value of x=4
.
So then at x=4.493409, the slope is equal to,
m=-cos%284.493409%29
highlight%28m=0.217234%29
graph%28300%2C300%2C-7%2C7%2C-2%2C2%2C-sin%28x%29%2C0.217234x%29