(9/30/08)Math 10A. Lecture Examples.Section 2.1. How do we measure speed?†Example 1 Imagine that a pilot is flying a small airplane toward the west from anairport and that the plane is s(t) = t3+ 30t + 100 miles from the airport thours after noon (Figure 1 ). (a) What is the plane’s average velocity fromt = 1 to t = 5? (b) Draw the secant line whose slope is this average value..t1 2 3 4 5s (miles)100200300400s = t3+ 30t + 100(hours)FIGURE 1Answer: (a) [Average velocity for 1 ≤ t ≤ 5] = 61 miles per h our (b) Figu re A1t1 2 3 4 5s (miles)131375s = t3+ 30t + 100(hours)4 hours244 milesFigure A1†Lecture notes to accompany Section 2.1 of Calculu s by Hughes-Hallett et al.1Math 10A. Lecture Examples. (9/30/08) Section 2.1, p. 2Example 2 What is the average rate of change of the width w(V) =3√V (meters) of acube with respect to its volume V (cubic meters) fro m V = 0 to V = 8?Answer: [Average rate of change] =14meters per cubic meter • Figure A2V4 8w (meters)12w =3√V(cubic meters)Figure A2Example 3 Bats are warm-blooded mammals whose body temperatures keep fairlyconstant when they are awake and active. When a bat is asleep in a coldplace, however, it goe s into a sort of hybernation and its metabolism (rateof energy expenditure) drops. The next table gives the met abolism r ofa sleeping brown bat as a function of the air temperature T around it.(1)(a) What is the average rate of change with respect to temperature of thebat’s metabolism f or 20 ≤ T ≤ 30? (b) What is the average rate o f changewith respect to temperature of the bat’s metabolism for 0.5 ≤ T ≤ 2.0?Interpret its sign.T = Temperature (◦C) 0.5 2.0 10.0 20.0 30.0 37.0 41.5r = Metabolism (Calories/hour) 5.4 1.4 3.4 19.0 96.0 134.0 200.0Answer: (a) [Average rate of change for 20 ≤ T ≤ 30] = 7.7 Calories per hour per degree(b) [Average rate of change for 0.5 ≤ T ≤ 2.0] = −83Calories per hour per degree • This average rate of changeis negative because the bat’s metabolism increases as the temperature drops from 2◦C toward freezing (0◦C).(1)Data adapted from Listening in the Dark, The Acoustic Orientation of Bats and Men, Donald R. Griffin, CornellUniversity Press, Ithaca, N.Y., 1986, p. 40.Section 2.1, p. 3 Math 10A. Lecture Examples. (9/30/08)Example 4 Figure 2 shows the hours of sunshine H = H(t) in Ft. Vermillion, Alberta,Canada as a function of the time o f year t measured in mont hs with t = 0at the beginning of the year.(2)What is the approximate average rate ofchange o f the hours of sunshine with respect to time from the beginning ofApril at t = 3 to the beginning of July at t = 6?t3 6 9 12H (hours)48121620H = H(t)(months)FIGURE 2Answer: Figure A4 • [Average rate of change] ≈ 2 hours per montht3 6 9 12H (hours)48121620H = H(t)(months)Figure A4(2)Data adapted from Introduction to Physical Geography by A. Strahler, Second Edition, New Yo rk NY: John Wiley& Sons, Inc., 1970, p. 172.Math 10A. Lecture Examples. (9/30/08) Section 2.1, p. 4Example 5 Predict the derivative f0(1) of f(x) = x2by calculating its average rateof change(1 + h)2− 1hfor h = 1, h = 0.5, h = 0.25, h = 0.1, h = 0.001, andh = 0.00001.Answer: The average rates of change are in the table below an d the correspond ing secant lines are in Figures A5athrough A5f. • Prediction : f0(1) = 2h1 0.5 0.25 0.1 0.001 0.00001(1 + h)2− 1h=3 2.5 2.25 2.1 2.001 2.00001x1y426y = x21 + h x1y426y = x21 + h x1y426y = x21 + hh = 1 h = 0.5 h = 0.25Figure A5a Figure A5b Figure A5cx1y426y = x2x1y426y = x2x1y426y = x2h = 0.1 h = 0.001 h = 0.00001Figure A5d Figure A5e Figure A5fExample 6 Use the definition to find the derivative of f (x) = x2at x = 1.Answer: f0(1) = 2Interactive ExamplesWork the following Interactive Examples on Shenk’s web page, http//www.math.ucsd.edu/˜ashenk/:‡Section 2.2: Examples 1 through 4Section 2.3: Example 5 (∆x = h )‡The chapter and section numbers on Shenk’s web site r efer to his calculus manuscript and not to the chapters and sectionsof the textbook for the