There were two slightly different tests. The answers below should apply equally well to both of them.
1. We've got a bank account, compounding continuously at a rate r. We're to be paid $D somehow... (D was either 10 or 20)
a) If we're paid the $D for N years, spread out, what's the present value?
= integral of D/N e^{-rt} dt, from 0 to N
= D/N integral of e^{-rt} dt, from 0 to N
= D/N e^{-rt}/(-r), from 0 to N
= D/N (e^{-rN} - 1)/(-r)
= D/N (1 - e^{-rN})/r if you prefer
b) If we were instead paid a lump sum, then in T years, it would grow to $100. So what's r?
D dollars now grows to D * e^{rT} in T years. So our equation is
D e^{rT} = 100
e^{rT} = 100/D
rT = log(100/D)
r = log(100/D)/T
You can break up the log, too, if you like.
c) Combine these two and get rid of e's.
D/N (1-e^{-rN})/r = D/N (1-e^{-log(100/D) N/T}) / (log(100/D)/T)
= D/N (1-e^{log(D/100) N/T}) / (log(100/D)/T)
= D/N (1-(D/100)^{N/T}) / (log(100/D)/T)
So what was really going on here? a) was just a repeat-the-formula-for- present-value. b) was work the equation for present-vs-future-value in reverse. c) was testing your ability to manipulate e's and logs, largely.
2. Supply and demand graph, lots of labeled regions, answers will be the areas of certain regions that are those little regions put together. (I told you that answers would look like e.g. A+C.)
a) How much money would the producers be paid for the quantity produced at a certain price?
Answer: the rectangle below that price (price paid) and to the left of the quantity produced at that price (quantity sold). Total money = p*q. In the gov't price case, this was the two regions in the corner; in the equilibrium price, this was six regions in the corner put together.
b) What's the _least_ amount they would have accepted for that quantity?
Now, just take the region below the supply curve (which says point by point how much the producers would have accepted) and to the left of the relevant quantity. In the gov't price case, CATHY this is just the lower left region (alone); in the equilibrium it's three regions put together.
c) Does the gov't cause a shortage, or a surplus? How do you tell this from the graph?
A shortage. Because at the price the goverment mandates, the amount produced (the quantity-coordinate of where the horizontal line at the gov't price intersects the supply curve) is less than the amount demanded (the quantity-coordinate of where the horizontal line at the gov't price intersects the demand curve).
3. Say we know about a function on [10,20] that it's continuous and (even better) twice differentiable. Also f(10)=20, f(20)=30, f' >= 0, f'' >= 0.
Basic idea of this question: how big or small can the average of f get, subject to these? Just knowing that f' >= 0, f can't go below 20 or above 30, and so the average can't. But the f'' >= 0 means it can't curve up and back down. So in fact it can't go above the straight line in between the two.
It's a little tricky to make this all precise, which is why I didn't ask you to prove any such thing; but if you play with the graphs a bit you can notice that anything that goes above the line has to have a negative second derivative somewhere.
i,ii) Draw a graph whose average is as big as possible, subject to these.
Answer: the straight line between the two. And its average is 25.
iii,iv) Draw a graph who average is close to the smallest possible.
Answer: stay at 20 until almost the last minute, then curve up very sharply to 30. The average will be very close to 20.