a) The fixed cost is the cost at q=0. In this case, zero.
b) profit = revenue - cost = price*quantity - C = 7*q - C(q). Take the derivative w.r.t. q and set to zero, you know how this goes from here.
c) What's q, as a function of p? If p=7, q=34; for each increase in p by 1, q decreases by 2. Let's write p = 7+c. Then q = 34 - 2*c = 34 - 2*(p-7). Now take profit = p*q - C(q), look at d(profit)/dp at p=7, and see if it's positive or negative.
5.3#26
profit = revenue - cost = price*q - cost(q) = p c K^alpha L^beta - w L. We're told that p is constant (Soviet central planning decision), and c, K, alpha, beta, and w are constant. Take d(profit)/dL and find out what level of L makes it zero.
5.5#2
E=.5. 3% increase in price --> 1.5% decrease in demand. Vice versa for decrease, increase.
5.5#21
E_cross measures how much more people will buy chicken if the price of beef goes up. If this is low (<1), it means people don't care about the price of beef - it doesn't affect their chicken decisions. If this is high (near, or more than, 1), it means people do indeed care.
In short, this measures the interchangeability of chicken and beef. If the example had been beef vs. bicycles, you'd expect it to be very low; if milk vs. powdered milk, you'd expect it a lot higher.
5.5#22
This measures how much more you buy of something depending on your income. For example, when I left school and got a real salary for the first time, I started buying a lot more CDs. But I continued drinking the same amount of milk. So milk's elasticity w.r.t. income was, for me, low, but CDs-elasticity w.r.t. income was high.
5.6#16
The vertical line through dose = 10mg cuts %effective at about 99%. So by that much we're helping most people. The vertical line through dose = 20mg cuts %lethal at about 1%. Surely that's an acceptable level of death.
Incidentally, the vertical line through dose = 33mg cuts at about 50% dead. This is called the "LD-50" of the drug, for "lethal dose for 50%", and is the usual way one refers to unreasonably high quantities of the drug.
One way of measuring the dangerousness of a drug is the ratio of the lethal dose to the effective dose - that says something about how careful you have to be to administer it.
5.6#17
On the 50mg vertical line, we cut %lethal at about 5%, and %effective at 85%. So, we kill 5% of the people, and leave 15% of the people not sufficiently affected. Pretty bad, huh? This is the sort of drug that the FDA would only approve for really life-threatening conditions, I guess.
Review section:
#20
"For every 1/4 decrease in price [from $4], demand increases by 200 units." Let price = p = 4- d / (1/4), where d is that decrease measured in 1/4's. Then we're told that q = quantity demanded = 4000 + 200 d.
So d/(1/4) = 4-p, so 200d = 200-50p, therefore q = 4200-50p. Now maximize revenue = p*q as usual.
(Actually, if you know parabolas well, you'll avoid taking this derivative and just say p*q = p(4200-50p) = 50 p (84-p). Since that crosses the x-axis at 0 and 84, its max has to happen halfway in between, at p=42. So q=2100, p*q = 44100.)
#37
a) If C = a t e^{-bt}, the max is reached at time 1/b, and equals a/eb - we did this in class. Anyway you can use product & chain rules to take that derivative (which was how we did it).
b) is just "plug in t=15,60".
c) This is only possible to give an approximate answer to, by looking for the solutions to 20 t e^{-.03t} = 10 by hunting with a calculator. Yuck. I'm not going to ask such questions on the test.
#38
This is in some sense a ridiculous question - polynomials are a general enough class to look like whatever you want. So if you want to misread the question you can say "it looks like a polynomial" about every one. But that's not right, because the qestion is "which is it most likely to be?" and it's unlikely that they'd give you functions that didn't have better descriptions than just "polynomial" every time.
So basically, see if it looks like one of the others, and if not, _then_ say polynomial. Also, the only ones with critical points are polynomial & surge.