(9/30/08)Math 10A. Lecture Examples.Section 1.1. Functions and change†Example 1 Draw the graph of the function A = πr2that gives the area of a circle ofradius r (centimeters).Answer: Figure A1r1A (square centimeters)10A = πr2rπr2(centimeters)Figure A1Example 2 Is the area of a circle increased more by increasing its radius f rom 1 t o 5or by increasing the radius from 12 to 13?Answer: The area is increased more when the radius is increased from 12 to 13.Example 3 (a) Find a formula for the area A = A(r) of a circular ring whose innerradius is r and whose outer radius is r + 1 with r ≥ 0 and draw its graph(Figure 1). (b) A = A(r) is a linear function because its graph is a portionof a line. What are the slope and A-intercept of the graph?r1FIGURE 1Answer: (a) A(r) = 2πr + π for r ≥ 0 • Figure A3 (a) [Slop e] = 2π • [A-intercept] = π†Lecture notes to accompany Section 1 .1 of Calculus by Hughes-Hallett et al.1Math 10A. Lecture Examples. (9/30/08) Section 1.1, p. 2r1 2 3A1020A = 2πr + ππFigure A3Example 4 A rocket is fired vertically from the ground at time t = 0 (seconds) andrises at a constant velocity until the engine shuts off at t = 1. With amathematical mode l in which air resistance is ignored, the height of theball above the ground h(t) (feet) at time t is give n by the two f ormulas,h(t) =12t for 0 ≤ t ≤ 1−16t2+ 44t − 16 for 1 < t ≤ T.Here T is the t ime when the rocket hits the ground. The graph of y = h(t)is shown in Figure 2. Use the quadratic formula to find T.t1 2y (f eet)51015y = h(t)(seconds)TFIGURE 2Answer: T =18(11 +√57).= 2.31Example 5 What are the domain and range of the function y =√x2− 1 of Figure 3?x−3 −2 −1 1 2 3y1234y =√x2− 1FIGURE 3Section 1.1, p. 3 Math 10A. L ecture Examples. (9/30/08)Answer: The domain consists of the interval x ≤ −1 and the interval x ≥ 1. • The range is the interval y ≥ 0.Example 6 Figure 4 shows the graph of the average credit card debt per householdy = D(t) (thousand dollars) as a function of the year t.(1)When was theaverage credit card debt approximately half what it was at the beginningof 2000?ty (th ousand dollars)2468199019952000y = D(t)FIGURE 4Answer: The average credit card debt per household at the beginning of 1994 was approximately half what it wasat the beginning of 2000 .Example 7 The next table gives the world capacity W(t) (megawatts) for generatingelectricity from wind power in several years t.(2)(a) Write the statement“The world wind energy generating capactity increased by more than 50,000megawatt s from 1985 to 2005” as an ine quality with no English wo rds. Thendetermine whether the statement is true or false. (b) Express the inequal-ity W(2005) > 10W(1 995) as an English sentence with no mathe maticalsymbols. Then determine whether the statement is true o r false.World Wind Energy Generating Capacityt (ye ar) 1985 1990 1995 2000 2005W(t) (megawatts) 2000 2300 4100 19,000 42,000Answer: (a) W (2005) > 50,000 + W (1985) • False (b) “The world capacity for generating electricity fromwind energy increased by more than 10 times from 19 9 5 to 2005.” • TrueInteractive ExamplesWork the following Interactive Examples on Shenk’s web page, http//www.math.ucsd.edu/˜ashenk/:‡Section 0.1: Examples 1 through 4(1)Data adapted from Newsweek, August 27, 2001, p. 36(2)Data adapted from Scientific American, September, 2006, p. 87.‡The chapter and section numbers on Shenk’s web site refer to his calculus manuscript and not to the chapters and sectionsof the textbook for the