# 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?

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1. 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

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2. There are several handy online calculators you can use to confirm your results.

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