(9/30/08)Math 10A. Lecture Examples.Section 3.1. Powers and polynomials†Example 1 Find formulas for (a)ddx(x3), (b)ddx(x−1), and (c)ddx(x1/2).Answer: (a)ddx(x3) = 3x2(b)ddx(x−1) = −x−2(c)ddx(x1/2) =12x−1/2Example 2 Give an equation for the tangent line to y =√x at x = 9 and draw it withthe curve.Answer: Tangent line: y = 3 +16(x − 9) • Figure A2x9 18y3y =√xFigure A2Example 3 Find the derivative of y = 5x3− x−2+ 4.Answer:dydx= 15x2+ 2x−3Example 4 What is f0(x) for f(x) = 73√x +8√x?Answer: f0(x) =73x−2/3− 4 x−3/2Example 5 What is the derivative of y =ax2+ bx + c√xfor constants a, b, and c?Answer:dydx=32ax1/2+12bx−1/2−12cx−3/2Example 6 A man driving on a straight road is s = 5t2+ 20t + 40 miles from his homet hours after noon. What is his car’s velocity at 4:00 PM?Answer: [Velocity at 4:00 PM] = 60 miles per hour†Lecture notes to accompa ny Section 3.1 of Calculus by Hughes-Hallett et al.1Math 10A. Lecture Examples. (9/30/08) Section 3.1, p. 2Example 7 A precision heater is controlled by varying the current supplied to it. Itproduces Q(I) = 100I2Calories of heat in one second when the currentis I amperes. (a) What is the (instantaneous) rate of change of Q withrespect to I at I = 3? (b) Give an equation for the tangent line to thegraph Q = Q( I) at I = 3 and draw it with the curve in an IQ-plane.Answer: Q0(3) = 600 Calories per ampere. (b) Tangent line: Q = 900 + 600(I − 3) • Figure A7I1 2 3 4 5Q (Calories)100020003000Q = Q(I)(amperes)Figure A7Interactive ExamplesWork the following Interactive Examples on Shenk’s web page, http//www.math.ucsd.edu/˜ashenk/:‡Section 2.4: Examples 1 through 4‡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