COALITION MATH 152, SPRING 1998 (S. A. Fulling, assisted by Vera Rice) Day 18.R 1. Announcements: Review sessions, exams, l'Hospital Web page, reminder to turn back tests and projects and pick up lab reports, 2. Growth and decay problems (brief since examples were covered in recitation yesterday). A. Scenario: The rate of increase (or decrease) of Stuff is proportional to the amount of Stuff already present: y'(t) = ky(t). B. Solution: y(t) = C e^{kt}, and C = y(0). We have solved our first differential equation! (apart from indefinite integrals). C. (Oral RAT) Applications: compound interest, radioactive decay, cooling, chemical reactions, population growth, concentration of a solution. D. Dow Jones industrial average (graphs) i. Recent years, Cartesian axes: Short-term fluctuations, long-term growth. Newspaper says: "Dow makes surprising advances.... The Dow has been reaching its thousand-point milestones in ever- decreasing periods of time." 1000 - '72 2000 - '87 3000 - '91 4000 - 2/95 5000 - 11/95 6000 - 10/96 7000 - 2/97 8000 - 7/97 People say: "Yikes! The market went up 200 points today! 30 years ago, 20 points in a day was a big move." The obvious response is, "Yes, that's the way compound interest works." Once you have more Stuff, the daily changes in Stuff are bigger. This is correct as far as it goes. But -- 30 years ago people didn't go around saying "Yikes! The market went up 20 points today! 30 years ago in 1940, 2 points a day was a big move." Maybe this isn't an exponential growth curve after all. ii. 100 years, semilog plot: Over the longest term, the curve is straight (exponential) on the average. Over the past 20-odd years, the curve is straight with a steeper slope. Over the past 40 years, the curve is definitely concave up; the period 1960-1980 was a period of almost no growth, so there really has been a change of behavior. On the other hand, the point about actual vs. subjective volatility is real: The difference between 1987 and 1929 is very obvious on this graph. 3. Volunteers to explain the species-area rule? NONE. HOWEVER, EVENTUALLY 7 MAX-MIN PROBLEM SOLUTIONS WERE RECEIVED. 4. Inverse trigonometric functions A. Using square root and arcsin as examples, review the 2 possible complications in defining an inverse function for a given function u = f(w) [so w = f^{-1}(u)]. i. f^{-1}(u) may not be defined for some values of u (so we need to be careful about the domain). ii. To make f^{-1} single-valued, we may need to exclude some values of w that satisfy u = f(w); we choose just ONE w for each u. This is called "choosing a BRANCH of the inverse function". The w chosen by this arbitrary decision may not be the one that is relevant in a particular problem! (Sometimes negative square roots do matter. \sqrt{x^2} is not always x; sin^{-1} (sin x) is not x unless x lies between \pm \pi/2.) B. Why should you care about inverse trig functions? i. Problems involving angles ii. Certain antiderivatives of algebraic functions C. What should you know about inverse trig functions? i. Emphasize sin^{-1}, tan^{-1}. Ignore csc, cot. ii. Derivatives: d/dx sin^{-1} x = 1/\sqrt{1 - x^2} d/dx tan^{-1} x = 1/(1 + x^2) iii. Integrals: \int dx 1/\sqrt{a^2 - x^2} = sin^{-1}(x/a) + C \int dx 1/(a^2 + x^2) = 1/a tan^{-1}(x/a) + C (You should know how to put in the "a" properly by integration by substitution!) I DECLARED A TEAM EXERCISE TO DERIVE THESE (ODD TEAMS FOR ONE, EVEN FOR THE OTHER). PERIOD ENDED AS WE FINISHED ARCTAN, WITH ARCSIN POSTPONED. (IN FACT, I DISCUSSED SUCH PROBLEMS EXTENSIVELY AT THE TEST REVIEW SESSION THE EVENING BEFORE DAY 19.T, AND CANCELLED THE SECOND TEAM PRESENTATION.) iv. Use trig identities and Pythagoras to relate the functions. (Draw a triangle! Then draw a circle to handle the other quadrants.) Examples: a. cos^{-1} x = \pi/2 - sin^{-1} x. Thus derivative of cos^{-1} is negative of that of sin^{-1}. OMITTED b. tan(sec^{-1} x) = ? It is \sqrt{x^2 - 1} in the first quadrant. Elsewhere there is a sign ambiguity because of the unsettled domain convention. (Show graph of sec and invert it.) v. Have some understanding of the domain question. a. Know the graphs of the ORIGINAL functions and make the simplest choice of domain (easy except for sec and csc). b. Know or figure out the relation between theta and tan^{-1} (y/x) in all four quadrants. OMITTED vi. Be able to DERIVE the derivative formulas by implicit differentiation (or the inverse function formula, dw/du = (du/dw)^{-1}). Do it for sec^{-1}. (The formula with no sign or absolute value is correct if tan y > 0 for y in the range chosen for sec^{-1}. See Exercise 99 and remark p. 408.) Interruption for team RAT: Prove the formula for derivative of ln, given the formula for derivative of exp, and vice versa. INSTEAD, I DID HALF OF THIS AT THE EVENING REVIEW SESSION AND PUT THE SECOND HALF ONTO THE WRITTEN QUIZ OF DAY 19T.