Taylor series problem

djt0607

Hi,

I am currently studying for my xmas exams starting next week. I have a problem, which I hope you can help me with. I can solve part of the problem, but theres just one bit i cant. Heres the problem

Using taylor series, derive the following difference approximation:
f '(x) = 1/12h[f(x-2h)-8f(x-h)+8f(x+h)-f(x+2h)] + (1/30)(h^4)(f^5(c)) where c € (x-2h,x+2h)

I can do the above no problems.

However, then it asks:

Use this formula to estimate f '(0.8) from the following data: (take h = 0.1)

x 0.6 0.7 0.8 0.9 1.0
f(x) 3.32 4.055 4.953 6.050 7.389


If anyone can help me with this small problem, I would be very greatful; thanx.

dan.

MathsManiac

You just sub. in those values into the formula you've derived. x=0.8, h=0.1, so, for example f(x-2h) is f(0.6), which you're given in the table.

So, you can work out the value of your expression (all except the bit: "...+ (1/30)(h^4)(f^5(c)) ", which is the error term.)

By the way, you shouldn't expect to be able to evaluate the error term, (since if you could, you'd have an exact value of the function and wouldn't need to be approximating it). An expression is usually only developed for the error term because, in certain circumstances, it's possible to put upper or lower bounds on it, thereby telling you how accurate your estimate is likely to be (or guaranteed to be) and hence telling you whether you've got an accurate enough answer for your purposes. This issue isn't part of this particular question, so it's safe to assume you're not expected to make any comment about the error term.