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

In summary, the problem asks to use a Taylor expansion to determine the range of input for which the operator g=Hf=sin(f) is approximately linear within 10%. The appropriate equation is g(f) = sin(0) + f*cos(0) = f. By comparing sin(f) and f, the error is found to be |sin(f)- f| and it must be less than |(1/10)sin(f)| for the operator to be approximately linear. This occurs when f = 0.
  • #1
Giuseppe
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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?
 
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  • #2
Giuseppe said:

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?
 

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

What is a Taylor Expansion and how is it used?

A Taylor Expansion is a mathematical tool used to approximate a function with a polynomial. It is useful for calculating values of a function at points where it may be difficult to evaluate directly, or for finding the behavior of a function near a particular point.

What is the error in a Taylor Expansion?

The error in a Taylor Expansion is the difference between the approximate value of the function and the actual value. It is also known as the remainder term, and can be calculated using the Lagrange form of the remainder.

How is the degree of a Taylor polynomial related to the error?

The degree of a Taylor polynomial is directly related to the error in the approximation. As the degree of the polynomial increases, the error decreases. However, it is important to note that a higher degree polynomial may not always provide a more accurate approximation, as it may introduce more complicated calculations.

Can a Taylor Expansion be used for any function?

A Taylor Expansion can be used for any function that is infinitely differentiable, meaning it has derivatives of all orders. However, for some functions, the Taylor polynomial may not converge to the function itself, and other methods may need to be used for approximation.

How can I determine the convergence of a Taylor Expansion?

The convergence of a Taylor Expansion can be determined by calculating the radius of convergence, which is the distance from the center point to the nearest point where the function is not analytic. If the distance is greater than the radius of convergence, the Taylor polynomial will not converge to the function. Additionally, the Taylor series can be checked for convergence using various tests, such as the ratio test or the alternating series test.

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