Some practice questions (now with answers)

Of course you should look again at the practice questions for the midterms. Here are some new ones. "^" means superscript, so x^2 means x squared, etc.


1. Write out the first ten terms (up to x^9) in the Taylor series of

a) f(x) = 1 / (1-x^2)

A. We did this for 1/(1-x) and got 1 + x + x^2 + x^3 + ...
So plug in x^2, and you get
1/(1-x^2) = 1 + x^2 + x^4 + x^6 + x^8 + ... omitting terms higher than x^9.

b) (1-x^4) f(x).

A. You could take the answer above and multiply by 1-x^4. There'd be a lot of cancelation.
Or you could notice that 1 - x^4 = (1+x^2)(1-x^2), so the answer to (b) is 1 + x^2 on the nose.


2. For which values of b does
\integral_{x=0}^{infinity} (e^{bx} + e^{-bx}) dx
converge?

A. None.
If b>0, then the first term goes to infinity, and the second to zero, so the integrand doesn't go to zero (it goes to infinity).
If b<0, then the second term goes to infinity, and the first to zero, so the integrand doesn't go to zero (it goes to infinity).
If b=0, then both terms are 1, so the integrand doesn't go to zero (it's always 2).

3. You've invented the Next Big Thing. It will sell increasingly well,
until the Next Next Big Thing comes out, at which point sales will collapse.
But that won't happen until N months from now.

A representative of Microhoo^TM offers you $1000 up front, plus
a pay rate of $100/month at time t. Alternately, she can offer
you a pay rate of $t/month (paid continuously), with nothing up front.

Which deal should you take? You answer should depend on N.

A. This was a lousy question (which is why I kicked it out of the final, where no question is as badly worded as this one).

What it was supposed to get across was the idea that if you take deal #1, then by time S (S < N) you've accumulated 1000 + 100*S dollars.
Whereas if you take deal #2, by time S (S < N) you've accumulated integral_{t=0}^S t dt dollars.

For S small, 1000 + 100*S > S^2/2. Whereas for S large, deal #2 is better. You can figure out when deal #2 starts to beat deal #1 by solving this quadratic equation for S. If N is less than that number, take deal #1; if N is more, take deal #2.