(9/30/08)Math 10A. Lecture Examples.Section 3.2. The expon ential function†Example 1 Give an equation of the tangent line to y = exat x = 2 and draw it with thecurve.Answer: Tangent line: y = e2+ e2(x − 2) • Figure A1x1 2 3y1020y = exFigure A1Example 2 Why is the x-intercept of the tangent line in Figure 1 equal to x − 1?xyy = exxexx − 1FIGURE 1Answer: The tangent line has x-intercept equal to x − 1 ecaus e the tangent line has slope ex.Example 3 What is the derivative of y = bxat x = 0 if b is a positive constant?Answer: y0(0) = ln(b)Example 4 What is the derivative of y = ex+ xe?Answer: y0(x) = ex+ exx−1†Lecture notes to accompany Section 3.2 of Calculu s by Hughes-Hallett et al.1Math 10A. L ecture Examples. (9/30/08) Section 3.2, p. 2Example 5 Based on the mathematical model P(t) = 3.93(1.03)t(millions) for thepopulation of the United States t years after 1790 for 0 ≤ t ≤ 70,(1)whatwas the rate of change of the population with re spect to time (a) at thebeginning of 1800 and (b) at the beginning o f 1860? (c) What would therate of increase of the population have been at the beginning of 1990 ifthis mathematical model was valid up to that time? Give the exact andapproximate decimal values.Answer: (a) [Rate of change at the beginning of 1800] = 3.93 ln(1.03)(1.03)10.= 0.156 million per year(b) [Rate of change at the beginning of 1860] = 3.93 ln(1.03)(1.03)70.= 0.920 million per year(c) The rate of change at the beginning of 199 0 would have been 3.93 ln(1.03)(1.03)200.= 42.9 million per year.Interactive ExamplesWork the following Interactive Examples on Shenk’s web page, http//www.math.ucsd.edu/˜ashenk/:‡Section 3.3: Examples 1 through 8(1)Data adapted from Historical Statistics of the United States, Part 1, Washington, DC: U.S. Government PrintingOffice, 1975, p. 5.‡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 cours