MHB Find zeros f(x)=(x/100)-sin(x)

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The function f(x) = (x/100) - sin(x) has zeros where sin(x) equals x/100. The analysis shows that the absolute value of x must be within the range of -100 to 100 due to the properties of the sine function. Within each period of sin(x), the line y = x/100 intersects the curve twice, leading to approximately 32 intersection points from 0 to 100 and another 32 from -100 to 0, totaling 63 roots when accounting for the double-counted zero. Exact solutions are not feasible, but approximate solutions can be found using a graphing calculator in Auto or Approx mode. The discussion emphasizes the utility of graphical methods and calculators in solving such equations.
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$f(x)=(x/100)-sin(x)$
Find the zeros

Thot my TI was going to melt trying to solve this
 
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$f(x)=0 \Rightarrow \sin x =\frac{x}{100}$

Since $|\sin x| \leq 1$ we conclude that $\left | \frac{x}{100} \right | \leq 1 \Rightarrow |x| \leq 100 \Rightarrow -100 \leq x \leq 100$.

At each period of $\sin x$, the line $y=\frac{x}{100}$ will intersect the curve $y=\sin x$ twice.
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From $0$ to $100$ there are $\frac{100}{2\pi}\approx 16$ periods. So, from $0$ to $100$ there are $2 \cdot 16=32$ intersection points.

Similarily, from $-100$to $0$ there are $32$ intersection points.

So, in total there are $32+32-1=63$ (we have count the point $0$ twice) intersection points.

So, the function $f(x)$ has $63$ roots.
 
karush said:
$f(x)=(x/100)-sin(x)$
Find the zeros

Thot my TI was going to melt trying to solve this

You aren't going to be able to get exact solutions, but your TI should still be able to give you approximate ones. Make sure you're in either Auto or Approx mode, and then type in

solve( x/100 = sin(x) , x)
 
That was amazing, i didn't know how to set up the periods
MHB always out does the textbooks
Much thanks
 
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