Max Step Size for Error < 5*10^-4 in Linear Interpolation of e^x on [0,1]

[devilsys]

Homework Statement

Hi m8s,

determine the max. step size that can be used in the function of F(x)=e^x
∈ [0, 1] so the error in the linear interpolation less than 5*10^-4

Homework Equations

The Attempt at a Solution

i used the Taylor expansion to get an eqn. so i be able to get the general interpolation egn. as:
f(x)=1+x+x²/2+x³/6+...
and to the linear the eqn. will be:
f(x)=1+x

then i got the table with my x₀ =0.1 and my step size = 0.2 , and i got p(x)~0.18x+1.0625 ,
and i said that the e^x = P(x) to able to get the step size (h) as asked .
But the problem is i can't get it to the error less than 5*10^-4 in my step size.

thxx in advance !

 

[mighty2000]

Thank you for posting your question. To determine the maximum step size that can be used in the given function F(x)=ex ∈ [0, 1] so that the error in the linear interpolation is less than 5*10^-4, we can use the formula for the error in linear interpolation:

Error = (M2/2)*(h^(2))

Where M2 is the maximum value of the second derivative of the function over the interval [0,1] and h is the step size.

In this case, the second derivative of F(x)=ex is also equal to ex, which has a maximum value of e when x=1. Therefore, M2 = e and we can plug this into the formula:

Error = (e/2)*(h^(2))

To make the error less than 5*10^-4, we can set this formula equal to 5*10^-4 and solve for h:

5*10^-4 = (e/2)*(h^(2))
h = √(5*10^-4/(e/2)) ≈ 0.00282

Therefore, the maximum step size that can be used in this function is approximately 0.00282. I hope this helps and let me know if you have any further questions. Good luck with your calculations!