The M1M1 treasure hunt, week 2
[The following is certainly not an extract from a famous author's new novel. Indeed, any perceived resemblance to any character, real or fictional, should be regarded as a coincidence.]
The red-headed man slumped down in the back row of the lecture theatre next to a studious young woman.
"Ron!" she hissed. "How nice to see you. I didn't know you were keen on Maths."
"I'm not really," he replied, going even redder, " I just wanted to see...um, er, who's the bald guy up front?"
"He's one of our lecturers. He's talking about functions today - some of them are amazing!"
She added, musingly, "He looks a bit like V----..."
"Don't say the name!" cried Ron.
"Sshhh!" called out several nearby students, eager not to miss a word. [I can dream.]
"Apparently there's some sort of treasure about. I'm going to find it - it might help pay my university fees."
"Well can't you just cast a "Tresoris Revelio" spell, or something?"
"That would be cheating, Ron. I'm going to use the magic of Maths instead."
"Can I help?"
"Oh Ron, this isn't really your sort of...er, yes, of course I'd love you to help. Try this one. We need to find a function f(x) which is both even and odd."
"That can't be right. Every number is either even OR odd. Not both."
"Not integers, Ron. Functions. We have to write the function in the box below and press "return":"
"Well, that wasn't very difficult. And certainly not very magical. Are all the clues so boring?"
"No of course not. This one I can hardly believe. P(x) and Q(x) are two polynomials, and the degree of Q is n which is at least two bigger than the degree of P. The n roots of Q(x)=0 are all different."
"Polly what?"
"Nomials. You remember quadratic equations? A bit like that. Anyway, suppose x=a is a solution to Q(x)=0. You form the ratio
P(a)/Q'(a) where Q' is the derivative of Q."
"But what if Q'(a)=0? Then I can't do it."
"Very good, Ron!" smiled Hermione fondly. "But remember that all the roots of Q are different. That means it can't happen. If Q'(a)=0, there would have to be a double root."
"Oh, ok, so I calculate that ratio. What then?"
"You do the same thing for all the n solutions to Q(x)=0. Then add up all the n ratios, and put the answer in the box below:"
"But it could be anything. He doesn't know what polynomials we choose."
"I did say it was magic, Ron. I don't understand how it works, but why don't you just pick a cubic whose roots you know. I'll pick a linear function for the numerator and we can work together, add up our 3 answers and see what we get. Er, would you like me to do the differentiation?"
"So what happens if we get them all right? Do we get the treasure?"
"Not yet - it'll uncover a clue. Also, he said that this week the three answers will give us a hint as to which fictional character we'll meet next week.
This one doesn't look hard. We start with the function f(x)=48/(x-1) for |x| > 1. First, we have to find its odd part, which is a function g(x)."
"You do that, Hermione. I'm exhausted."
"We then have to find the inverse of that function, g-1. Finally, we calculate the value g-1(7) and write that below."
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Week 1
Week 2
Week 3
Week 4
Week 5
Week 6
Week 7
Week 8
Week 9