(9/30/08)Math 10A. Lecture Examples.Sections 2.2 and 2.3. The derivative at a point and as a function†Example 1 The function A = A(t) whose graph is shown in Figure 1 gives the percent-age of alcohol in a person’s blood t hours after he has consumed three fluidounces of alcohol.(1)The tangent line at t = 1 in Figure 2 passes throughthe points P = (1 , 0.18) and Q = (2, 0.28). What is the (instantaneous) rateof change of the person’s blood alcohol level with respect to time one hourafter consuming the alcohol?t2 4 6 8 10 12A (percent of alcohol)0.20.10.3(hours)A = A(t)t2 4 6 8 10 121A (percent of alcohol)0.20.10.3(hours)A = A(t)PQFIGURE 1 FIGURE 2Answer: A0(1) = 0.1 percent per hourExample 2 Figure 3 shows the graph of the rate of oxygen consumption r = r(v) (litersper minute) of a swimmer as a function of his velocity v (meters per second)and the tangent line to the graph at v = 1.(2)(a) Why is r(1.4) greater thanr(1)? (b) What is the rate of change of his oxygen consumption with respectto velocity at v = 1? (Give the units.)v0.8 1 1.2 1.4r (liters per minute )1234(meters per second)(1.2, 2.66)(1, 2.1)r = r(v)FIGURE 3Answer: (a) r(1.4) is greater than r(1) because the swimmer uses more oxygen when he swims faster.(b) r0(1) = [Slope of the tangent line] =2.66 − 2.11.2 − 1= 2.8 liters per minute per meters per secon d.†Lecture notes to accompany Sections 2.2 and 2.3 of Calculus by Hughes-Hallett et al.(1)Data adapted from Encyclopædia Britannica, Vol. 1, Chicago: Encyclopædia Britannica, Inc., 1965, p. 548.(2)Data adapted from The Human Machine by R. Alexander, New York, NY: Columbia University Press, 1992, p. 117.1Math 10A. Lecture Examples. (9/30/08) Sections 2.2 and 2.3, p. 2Example 3 Figure 4 shows the graph of an object’s position s = s(t) on an s-axis asa function of the time t. What is the object’s approximate velocity in thepositive s-direction at t = 6?t2 4 6 8s (yards)20406080s = s(t )(minutes)FIGURE 4Answer: One answer: Figure A3 • [Velocity at t = 6] = s0(6) ≈ −10.5 yards per minutet2 4 6 8s (yards)20406080s = s(t )(minutes)Figure A3Sections 2.2 and 2.3, p. 3 Math 10A. Lecture Examples. (9/30/08)Example 4 Vertebrae in the human spine are separated by fibro us e lastic disks thatcompress when the load on them increases. The load on the disk is calledthe stress, and the percent of compre ssion is called the strain. The functionS = S(L) in Figure 5 gives the strain S of lower back (lumbar) vertebral disksas a function of the stress L on them.(3)Sketch the graph of the derivativer = S0(L) of the stress function S = S(L).L100 200 300 400 500S (percent)2010S = S(L)(kilograms)FIGURE 5Answer: Figure A4a • S0(50) ≈ 0.11 percent per kilogram • S0(400) ≈ 0.02 percent per kilogram •Figure A4bL100 200 300 400 500S (percent)1020S = S(L)(kilograms)L100 200 300 400 500r (percent/kilogram)0.050.10.15r = S0(L)(kilograms)Figure A4a Figure A4b(3)Data adapted from Physics with Illustrative Examples from Medicine and Biology, Vol. 1, G. Benedeck and F .Villars, Reading, Massachusetts: Addison-Wesley, 1973, p.343.Math 10A. Lecture Examples. (9/30/08) Sections 2.2 and 2.3, p. 4Example 5 The next table lists the wind speed at three-hour intervals one day in DodgeCity, Kansas.(4)The time t is measured in hours with t = 0 at midnight. Usethis data to estimate the rate of chang e W0(t) of the wind’s velocity withrespect to time at 9:00 AM (t = 9).Wind velocity W(t) (miles per hour) in Dodge City, Kansast 0 3 6 9 12 15 18 21 24W (t) 12.6 12.4 12.6 15.3 16.2 15.7 15.0 11.3 12.9Answer: One approach: Use the secant line in Figure A5a (a left difference quotient). •W0(9) ≈ 0.3 miles per h our per hour •Another approach: Use the secant line in Figur e A5b (a right difference q uotient). •W0(9) ≈ 0.9 miles per h our per hour •A third approach: Use the secant line in Figure A5c (a centered difference quotient). •W0(9) ≈ 0.6 miles per h ourt3 6 9 12 15 18 21 24W (miles per hour)121416t3 6 9 12 15 18 21 24W (miles per hour)121416Figure A5a Figure A5bt3 6 9 12 15 18 21 24W (miles per hour)121416Figure A5cInteractive ExamplesWork the following Interactive Examples on Shenk’s web page, http//www.math.ucsd.edu/˜ashenk/:‡Section 2.5: Examples 1 through 5(4)Data from Wind Energy Systems, G. L. Johnson, Englewood Cliffs, N. J: Prentice Hall International, Inc., 1985, p. 4 7.‡The chapter and section numbers on Shenk’s web site refer to his calculus manuscript and not to th e chapters and sectionsof the textbook for the