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Error function An essential function in statistics and the
Chapter 8, Problem 74E(choose chapter or problem)
Error function An essential function in statistics and the study of the normal distribution is the error function
\(\operatorname{erf}(x)=\frac{2}{\sqrt{\pi}} \int_{0}^{x} e^{-t^{2}} d t\).
a. Compute the derivative of erf (x).
b. Expand \(e^{-t^{2}}\) in a Maclaurin series, then integrate to find the first four nonzero terms of the Maclaurin series for erf.
c. Use the polynomial in part (b) to approximate erf (0.15) and erf (-0.09). d. Estimate the error in the approximations of part (c).
Questions & Answers
QUESTION:
Error function An essential function in statistics and the study of the normal distribution is the error function
\(\operatorname{erf}(x)=\frac{2}{\sqrt{\pi}} \int_{0}^{x} e^{-t^{2}} d t\).
a. Compute the derivative of erf (x).
b. Expand \(e^{-t^{2}}\) in a Maclaurin series, then integrate to find the first four nonzero terms of the Maclaurin series for erf.
c. Use the polynomial in part (b) to approximate erf (0.15) and erf (-0.09). d. Estimate the error in the approximations of part (c).
ANSWER:Solution 74E
Step 1:
Error function ; erf(x) = dt
- Now we need to compute the derivative of erf(x).
Consider , erf(x) = dt ,
Derivative both sides with respect to ‘x’ , then we get ;
(erf(x)) =
(
dt )
= dt )
= (
(x) -
(0))
=
Therefore , the derivative of erf(x) is ;