Please solve the following integral question by finding the answer from a.) to d

Question

Please solve the following integral question by finding the answer from a.) to d.) (show all work)

The sine integral (unction Si(x) = is often used in engineering. The function is not defined n . but its limit is 1 as 1 rightarrow 0 Therefore, if we define f(0) = 1, then is continuous everywhere. If x R Make a graph of Si(x). Find all values of x where Si(x) has a local maximum. Find the approximate x and y values that represent the coordinate of the first inflection point where x>0. (Hint: A numerical method may be good here; approximate to four decimal places). Does Si(x) have any asymptotes? If so. then find the value(s).

Explanation / Answer

b) for local maxima /minima, derivative=o=> f(x)=0 => x= n(pi)

Double derivative= sin(t)/t - cos(t)/(t^2).
For local maxima, double derivative has to be -ve.at the above points.hence of the point (2n+1)(pi).( i.e odd values of n*pi )

c)   varying x in steps of 0.1, first inflection i.e double derivative= 0 occurs at x=3

d) f(t) abolute value reduces with increasing t
as t tends to infinity f(t)=0 => Si(t) does not change.
Hence fluctuations in value of Si(x) becomes negligble with increasnfg x.   from the graph it is clear value is greater than 1st local minimum and tending to 1.5

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