Solving Taylor Series Problem with m-th Derivative Bound

[brad sue]

Hi ,
I have some difficulties to solve this problem. It is from my numerical methods class but the problem is about taylor series:

It is known that for 4 < x < 6, the absolute value of the m-th derivative of a certain function f(x) is bounded by m times the absolute value of the quadratic 11x-28-x2. That is, |f(x)(m)| < m|11x - 28 - x2| for m > 0. How many terms would be required in the Taylor series for f about x = 4.5 to evaluate f(5.8) accurate to 7 decimals?

Please, I need some sugestions. I am not sure if i need to use the remainder formula to find it.

Please somebody help me!

Thank you

Brad

 

[HallsofIvy]

Yes, you should use the remainder theorem. You would find the remainder after n terms for the Taylor's series in terms of (x- 4.5)ⁿ. Of course, here x- 4.5= 5.8- 4.5= 1.3. For what n is the remainder less than 0.0000001?

What is the maximum possible value of |11x- 28- x²| for 4< x< 6?

 

[brad sue]

ok,
thank I think i get it.

Yes, you should use the remainder theorem. You would find the remainder after n terms for the Taylor's series in terms of (x- 4.5)ⁿ. Of course, here x- 4.5= 5.8- 4.5= 1.3. For what n is the remainder less than 0.0000001?

What is the maximum possible value of |11x- 28- x²| for 4< x< 6?