What is the Linear Approximation Range for Sin(f) Within 10% Error?

[Giuseppe]

Homework Statement

For g=Hf = sin (f), use a Taylor expansion to determine the range of input for which the operator is approximately linear within 10 %

Homework Equations

The taylor series from 0 to 1 , the linearization, is the most appropriate equation

The Attempt at a Solution

g(f) = sin(0) + f*cos(0) = f
g1(f) = sin(f) g2 (f) =f ( at f=0, g1=g2(f) )

g1(f) = g2(f) + error

sin(f) = f + (1/10) * sin (f)

(9/10)* sin (f) =f

the value I keep getting is when f is equal to 0. I really don't think I am doing this correctly. Any advice?

 

[HallsofIvy]

Homework Statement

For g=Hf = sin (f), use a Taylor expansion to determine the range of input for which the operator is approximately linear within 10 %

Homework Equations

The taylor series from 0 to 1 , the linearization, is the most appropriate equation

The Attempt at a Solution

g(f) = sin(0) + f*cos(0) = f
g1(f) = sin(f) g2 (f) =f ( at f=0, g1=g2(f) )

g1(f) = g2(f) + error

sin(f) = f + (1/10) * sin (f)

(9/10)* sin (f) =f

the value I keep getting is when f is equal to 0. I really don't think I am doing this correctly. Any advice?

The only place sin(f)- f= (1/10)sin(f) is at f= 0 but if one is positive and the other negative you can still compare them. Your "error" is |sin(f)- f| and you want that less than |(1/10)sin(f)|. Where are those equal?