Before me is a letter from Sherlock Holmes' great-granddaughter, Ms. Sandra Holmes. It seems that the version of events I gave in Sherlock Holmes' Train Adventure didn't have all the facts right. According to the letter:

"...the real Sherlock Holmes knew many techniques of numerical integration, including the more advanced ones. He would not have used L_n or any other technique quite so elementary, my dear..."

To prepare for this reconstruction, you first need to brush up on your more advanced numerical integration techniques. Recall first the definitions of the midpoint sum (M_n) and Trapezoid sum (T_n), namely:

	n
Mn=		f(____________)(x)
	i=1

	n
Tn=		(______________)(x)
	i=1

Recall now that Simpson's rule states that:

					m
Sn	=	S2m	=	x / 3		f(x2i-2) + 4f(x2i-1) + f(x2i)
					i=1

When n=4, this formula becomes:

In the case of the integral above, that specific S₄ becomes:

Good job! It's now time to roll up our collective sleeves and help reconstruct Sherlock's mathematical reasoning as he awaits the right moment for him and Dr. Watson to escape from the train. Recall their speed readings during the first hour:

Ms. Holmes' also consulted Sherlock's dusty old notebooks and found that after the first hour, he had, in fact, managed to take readings not every 10 minutes, but every five minutes, so the actual readings were:

But how good are these estimates anyway? Remember, if Sherlock Holmes and Dr. Watson estape from the train, they estimate that they only have a limited amount of time they can survive while looking for and then heading to town, before they will be spotted and recaptured. So they want to be absolutely sure they won't be father than 25 miles from the town where Sherlock knows a trusted friend lives.

Recall that if I represents the true value of the integral of f(x) over the interval [a,b], an upper bound for
| I - T_n |   is (1/12n²) K₂(b-a)³ while an upper bound for | I - M_n |   is (1/24n²)K₂(b-a)³ where in each case K₂ is a constant giving an upper bound for __________ on [a,b]. What is the f(x) (or f(t)) which Sherlock is using?

Sherlock and Watson estimated that this K₂ whatever it was, could certainly be no greater than 6,000. Is this reasonable? What would this mean about how many minutes it would take their late 19th century train to go from no acceleration, to its maximum (engine-propelled) acceleration?

In this case, had Sherlock used the Trapezoid approximation method,

• What would his approximation have been for the first 1:20 hours of the train trip?_______________.

• What would be his upper bound for the error of this approximation?

Had Sherlock used the the Midpoint approximation method,

• What his approximation have been for the first 1:20 hours of the train trip?_______________.

• What would be his upper bound for the error of this approximation?

The real Sherlock, of course, used Simpson's Rule, Ms. Holmes assured me in her letter.

• What his approximation have been for the first 1:20 hours of the train trip?_______________.

  Recall that if I represents the true value of the integral of f(x) over the interval [a,b], an upper bound for | I - S_n |   is (1/180n⁴) K₄(b-a)⁵ where K₄ is a constant giving an upper bound for __________ on [a,b]. What would a reasonable value for K₄ be? Justify a reasonable value which you will use below.

• Given the above, what would Sherlock's upper bound be for the error of his Simpson's approximation? Can Sherlock and Watson make a fairly precise, and hence successful escape?