(10/28/08)Math 10C. Lecture Examples.Sections 14.1 and 14.2. Partial derivatives†Example 1 The table below is from a study of the effect of exercise on the bloodpressure of women. P = P(t, E) is the average blood pressure, measured inmillimeters of mercury (mm Hg), of women of age t years who are exercisingat the rate of E watts.(1)(One watt is 0.8 6 Calo ries per hour.) What is theapproximate rate of change with respect to age of the average blood pressureof fo rty-five-year old women who are exercising at the rate of 100 watts?P = P(t , E) (millimeters of mercury)t = 25 t = 35 t = 45 t = 55 t = 65E = 150 178 180 197 209 195E = 100 163 165 181 199 200E = 50 145 149 167 177 181E = 0 122 125 132 140 158Answer: Pt(45, 100) ≈ 1.8 millimeters of mercury per year (using a right difference quotient); orPt(45, 100) ≈ 1.6 millimeters of mercury per year (using a left difference quotient); orPt(45, 100) ≈ 1.7 millimeters of mercury per year (using a centered difference quotient)Example 2 Use the table from Example 1 to find the approximate rate of change withrespect to age of the average blood pressure of fifty-five-year-old womenwho are exercising at the rate of 75 wat ts.Answer:∂P∂E(62,75)≈ 0.44 millimeters of mercury per watt†Lecture notes to accompany Section s 14.1 and 14.2 of Calculus by Hughes-Hallett et al.(1)Data adapted from Geigy Scientific Tables, edited by C. Lentner, Vol. 5, Basel, Switzerland: CIBA-GE I GY Limited,1990, p. 29.1Math 10C. Lecture Examples. (10/28/08) Sections 14.1 and 14.2, p. 2Example 3 Figure 1 shows level curves of the temperature T = T(t, h)◦F as a f unctionof time t (hours) and the depth h (centimete rs) beneath t he surface o fthe ground at O’Neil, Nebraska, from noon one day (t = 0) until the nextmorning.(2)(a) What was the approximate rate of change of the temperature withrespect to t ime at 4:00 PM at a point 14 centime ters beneath the surfaceof the ground?(b) What was the approximate rate of change of the temperature withrespect to depth at 4 :00 PM at a point 14 centimeters beneath the surfaceof the ground?FIGURE 1Answer: (a) Figure A3 • Tt(4, 14) ≈ 0.5 degree per hour (b) Th(4, 14) ≈ −0.25 degree per centimeterFigure A3(2)Data ad apted from Fundamentals of Air Pollution by S. Williamson, Reading, MA: Addison Wesley, 1973.Sections 14.1 and 14.2, p. 3 Math 10C. Lecture Examples. ( 10/28/08)Example 4 Find the x- and y-derivatives of f (x, y) = x3y − x2y5+ x.Answer:∂f∂x= 3x2y − 2xy5+ 1 •∂f∂y= x3− 5x2y4Example 5 What are gx(2, 5) and gy(2, 5) for g(x, y) = x2e3y?Answer: gx(2, 5) = 4e15• gy(2, 5) = 12e15Example 6 The volume of a right circular cylinder of radius r and height h is equalto the product V(r, h) = πr2h of its height h and the area πr2of its base(Figure 2). What are (a) the rate of change of the volume with respect tothe radius and (b) the rate of change of the volume with respect to theheight and what are their geometric significance?rh[Area of base ] = πr2[Volume] = πr2h[Lateral surface area] = 2πrhFIGURE 2Answer: (a)∂V∂r= 2πrh is the a rea of the lateral surface ( the sides) of the cylinder.(b)∂V∂h= πr2is the area of the base.Example 7 What are the first-order partial derivatives of f = ln(xy)?Answer: fx=1x• fy=1yExample 8 What are (a) hxyand (b) hyxfor h(x, y) = cos x + sin y + 100xy?Answer: (a) hxy= 100 (b) hyx= 100Interactive ExamplesWork the following Interactive Examples on S h enk’s web page, http//www.math.ucsd.edu/˜ashenk/:‡Section 14.3: Examples 1 through 5Section 14.7: Example 2Section 14.8: Example 2‡The chapter and section numbers on Shenk’s web site refer to his calculus manuscript an d not to the chapters and sectionsof the textbook for the