Review Sheet for Exam 3

Applied Calculus

Preliminary Notes: (a) We will cover 3.5 on Monday’s class; the test is Friday, April 4. (b) These problems are not intended to be all you need to review. You should review your class notes, do all problems I give you to do in the section, check all answers on the key, then go through anything extra that gives you trouble more thoroughly. (c) I will post the answers to all of the “Good Problems to Review” on the Web page by Monday afternoon.

2.5	· Given the price function p, remember that revenue is R(x) = x p(x). · After R is obtained as above (or given), remember profit is P(x) = R(x) – C(x); be able to maximize profit using P’(x) = 0, finding x, then be able to find the right price. · Be able to set up and solve completely a minimizing inventory costs problem. Remember that (total cost) = (carrying cost) + (reorder cost) and be able to set these up as done/defined on Example 6, p. 267. Be able to minimize this cost. · Good Problems to Review: 26, 30, 31 (not on key – done in class), 42
2.6	· Understand – given any economic quantity E – what the marginal economic quantity E’(x) means; it approximates E(x+1)- E(x), the latter being the exact amount of the economic quantity it costs to produce the (x + 1)st item. · By the above, we also have that E(x+1) = E(x) + E’(x). Understand this and how to use it. · Good Problems to Review: 4, 8, 17
3.1	· Be able to differentiate all kinds of exponential functions that include e^x, including those mixed with functions we’ve learned how to differentiate previously (and remembering product, quotient and chain rules!). · Good Problems to Review: 14, 18, 20, 24, 32, 34, 36, 40, 46
3.2	· Know the all-important definition of logarithms: Definition on p. 319; from this, know how to transform an exponential equation into a log one and vice-versa. · Know and be able to work with P1-P6, p. 323, so you can use them – especially for (a) differentiating an ln x-type function and (b) to solve equations involving unknown exponents. · Be able to differentiate all kinds of functions involving ln x. · Good Problems to Review: 4, 8, 12, 14, 36, 42, 44-50 even, 78
3.3	· Know how the growth/decay equation A = Pe^rt (as well as any of its alternate forms – as we’ve mentioned several times, they are all the same equation!) comes from a differential equation and the observation of proportionality of change to amount present. · Be able to use the growth/decay equation A = Pe^rt like a mathematician possessed to solve growth problems for time t. · Know that the (sometimes-called) Rule of 70 comes from 2 = e^rt, why it comes from this and its usage to solve a doubling time of an investment problem. · Good Problems to Review: 10, 22, 30
3.4	· Be able to do all the same things as in 3.3, now using A = Pe^rt as a decay equation (i.e., so r < 0). That is, especially be able to solve a half life-type problem and a present value-type problem. · Good Problems to Review: 2, 12, 18
3.5	· Know the derivatives of a^x and logax very well and be able to differentiate all kinds of functions related to these (again, remembering all the rules for differentiation). · Good Problems to Review: 2, 6, 10, …, 30.