Review Sheet for Exam 4
Applied Calculus, Spring 2008
As always, I am just giving a list below of the main concepts to understand and with which you should be capable of working effectively. The problems I give will be worked on a key that I will post by later Friday (I hope); if not I will hand it out in class Monday. This may or may not be enough problems for you to study for the test. Before working the problems, I recommend that you study your class notes and homework.
Section 3.6: Elasticity
· I will give you the formula for elasticity at price x given a demand function
p = D(x), namely E(x) = -xD’(x)/D(x). You must know how to simplify it and work with it.
· When I say “work with it,” I mean the following
o Know the interpretation and be able to explain that it means a percentage change in demand compared to (i.e., over) the percentage change in price.
o This is why a quantity is elastic at a price x if E(x) > 1: There is a significant change in demand (a drop, of course) with a comparatively small change in price. Also inelasticity makes sense in the opposite way.
o Remember how to find revenue R(x) at price x: R(x) = x D(x).
o Know the connection between R and elasticity: Revenue will increase by raising price another unit if E(x) < 1, but revenue decreases if E(x) > 1 at price x and you raise the price by one unit.
o Know that maximum revenue is attained at the price where E(x) = 1 and be able to solve for this price, then find the demand for the item when the price is x.
· Good review problems: 4,6,10,14
Section 4.1: Area under a Graph
· It is very important to understand that every question in this homework section is asked in terms of marginal quantities, even if it is not explicitly stated. Note the two ways that such a quantity A(x) (where A represents anything such as revenue, profit, cost, etc.):
1. The marginal quantity for x items is given by A’(x) = (some function); use this to find the total quantity A over the interval [a,b].
2. The quantity A per item x is given by A(x) = (some function); use this to find the total quantity A over the interval [a,b].
Why are these really the same problem despite the difference in wording? Marginal A is an approximation of the per item amount A.
· We have a better way of finding the total area under A over [a,b] now: Simply use the Fundamental Theorem of Calculus (and a definite integral) if A is a function we know how to integrate. However, if we do not have A given explicitly, we still need to add areas of rectangles to get an approximation of the total area. For example, suppose that we don’t know – on problem 23 – that the function giving these values is f(x) = 1/x2; then I could just give you y-values on the graph to find the heights of rectangles to add up areas. See “Good review problems” below for a complete explanation.
· I will not give you a problem such as 13-22 on the test.
· Good review problems: 4, 12, 23 (Also note the other way you might be asked these as explained above. For example, I could restate Pr. 23 in this way: At x=1, the per-item cost is $1; at x=2, the per-item cost is $0.25; at x=3, the per-item cost is $0.11; … at x=6, the per-item cost is $0.03. Approximate the total cost of producing items 1 to 7.)
Section 4.2: Antiderivatives and Integrals
· Be able to find antiderivatives/indefinite integrals of all types of basic functions we have studied. You have numerous examples in class notes, too.
· Be able to solve a problem (like problems 47-58) where you know (say) the marginal cost function C’(x) – and you know the cost C(0) of producing no items – and you are asked to find the cost function C.
· Good review problems: 10, 12, 14, 18, 22, 28, 36, 44, 50, 56 (On these last two, think of f as some marginal economic quantity; you are trying to find the function for the economic quantity from the marginal one.)
Section 4.3: Area and Definite Integrals; the Fundamental Theorem of Calculus
· Be able to evaluate all kinds of definite integrals now (compare to the first bullet in 4.2 above and make sure you know the difference between the two!).
· Know that – by the Fundamental Theorem of Calculus – you can find the total area under a function f over the interval [a,b] using a definite integral.
· Understand that the answer you get from a definite integral is
(area above the x-axis) – (area below the x-axis).
· Good review problems: 4, 8, 12, 46, 53, 60
Section 4.4: Properties of Definite Integrals
· Be able to find the area under a function defined piecewise.
· Given a reminder of the formulas, be able to find (a) the area bounded by two functions f and g over [a,b] (brown box on page 428) and (b) the average value of a function f over [a,b] (brown box on page 432)
· Good review problems: 6, 16, 20, 30, 46