Posted by farouk on Tuesday, November 15, 2011 at 3:53pm.
Show that the equation x+3sinx=2 has a root between 0.4 and 0.6.express the root to four significant figures

calculus  MathMate, Tuesday, November 15, 2011 at 4:43pm
First part:
let
f(x)=x+3sin(x)2
f(0.4)=0.432
f(0.6)=0.294
By the intermediate value theorem for a continuous function, there must be a value of x=x0 between 0.4 and 0.6 such that f(0.4)≤f(x0)≤f(0.6), specifically where f(x0)=0.0.
There are many ways to find the root of f(x)=0, for example, arrange the equation as:
x=arcsin((2x0)/3)
and iterate starting with x0=0.5
then x=arcsin((20.5)/3)=0.5236
Let x0=0.5236 and continue until the value is stable.
Alternatively, use Newton's method for the iteration:
x1=x0(x0+3sin(x0)2)/(1+3cos(x0))
Starting from x0=0.5, you will get 9 digits of accuracy with 2 iterations.
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