Use Newton's method to find all solutions of the equation correct to six decimal places. (Enter your answers as a comma-separated list.)

6 cos(x) = x + 1
can somebody explain to me step by step on how to solve this problem?

2 answers

I assume x is in radians
we want 0 = x + 1 - 6 cos x
let f(x) = x + 1 - 6 cos x
find where f(x) = 0

pick a point, say x = 1
then
f(1) = 1 + 1 - 6 cos 1 = 2 - 6*0.54
= -1.24

plot that point (1, -1.24) on a graph
what is the slope of f(x) at that point?
f'(x) = 1 + 6 sin x
= 6*.84 = 5.05

draw that as tangent to curve of f(x) at (1,-1.24). It is aimed up toward the x axis. Where does it hit the x axis?

slope = 5.05 = 1.24/delta x
delta x = 1.24/5.05 = .246

so new x = 1 + .246 = 1.246
new f(x) = 1.246 + 1 - 6 cos 1.246
= .331

so now we are at (1.246, .331)
well .331 is not zero so repeat using that point
keep repeating until you get close to zero
There are several handy online calculators you can use to confirm your results.