0

If you are using piecewise quadratic polynomials to approximate the function $f (x) = \ln x$ on the interval $[1, 2]$ and expect the maximum error to be smaller than $10^{-6}$, how many subintervals do you need?

nagu
  • 9
  • Why not just write smaller than $10$? Or did you in fact mean smaller than $10-6$, as in "smaller than $4$? Or should it be "smaller than $10$ but bigger than $6$"? – Bobson Dugnutt Mar 14 '16 at 14:41
  • I guess the (absolute) error should be $<10^{-6}$ – gammatester Mar 14 '16 at 14:42
  • @ gammtester ...yes u are right –  Mar 14 '16 at 14:50
  • is any body give hint or solve this prob please – nagu Mar 14 '16 at 14:58
  • We can't really answer the question because there is not enough information. How are you choosing the piecewise quadratic polynomials? There are many interpolation methods which are slightly different and have slightly different error bounds. If this is something you are going over in a class, then it is likely you proved some sort of error bound like $\text{Err} \le \tfrac{K(b-a)^2}{N^2}$ where $N$ is the number of subintervals and $K$ is some constant depending on the particular $f$ you are approximating and its derivatives. – User8128 Mar 14 '16 at 15:05
  • IMO the problem description needs more details, e.g.: Should the piecewise quadratic approximation function be continous at the sub-inverval end points? Are intervals of any length allowed or should they have all the same length etc? My guess is that you need to more than 40 intervals, because the error of the quadratic Chebyshev approximation of $\ln(x)$ on $[1, 1+1/40]$ is less that $10^{-6}.$ – gammatester Mar 14 '16 at 15:07

0 Answers0