UCSD MATH 10A - Lecture Examples (4 pages)

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Lecture Examples

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Lecture Examples

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4
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University of California, San Diego
Course:
Math 10a - Calculus I
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9 30 08 Math 10A Lecture Examples Sections 2 2 and 2 3 The derivative at a point and as a function The function A A t whose graph is shown in Figure 1 gives the percentage of alcohol in a person s blood t hours after he has consumed three fluid ounces of alcohol 1 The tangent line at t 1 in Figure 2 passes through the points P 1 0 18 and Q 2 0 28 What is the instantaneous rate of change of the person s blood alcohol level with respect to time one hour after consuming the alcohol Example 1 0 3 A percent of alcohol 0 3 A percent of alcohol Q A A t 0 2 0 2 0 1 0 1 2 4 6 8 10 12 t hours A A t P 1 2 FIGURE 1 4 6 8 10 12 t hours FIGURE 2 Answer A0 1 0 1 percent per hour Figure 3 shows the graph of the rate of oxygen consumption r r v liters per minute of a swimmer as a function of his velocity v meters per second and the tangent line to the graph at v 1 2 a Why is r 1 4 greater than r 1 b What is the rate of change of his oxygen consumption with respect to velocity at v 1 Give the units Example 2 r liters per minute 4 3 1 2 2 66 2 r r v 1 2 1 1 FIGURE 3 0 8 1 1 2 1 4 v meters per second Answer a r 1 4 is greater than r 1 because the swimmer uses more oxygen when he swims faster b r 0 1 Slope of the tangent line 2 66 2 1 1 2 1 Lecture 2 8 liters per minute per meters per second notes to accompany Sections 2 2 and 2 3 of Calculus by Hughes Hallett et al Data adapted from Encyclop dia Britannica Vol 1 Chicago Encyclop dia Britannica Inc 1965 p 548 2 Data adapted from The Human Machine by R Alexander New York NY Columbia University Press 1992 p 117 1 1 Math 10A Lecture Examples 9 30 08 Sections 2 2 and 2 3 p 2 Figure 4 shows the graph of an object s position s s t on an s axis as a function of the time t What is the object s approximate velocity in the positive s direction at t 6 Example 3 s s t s yards 80 60 40 20 FIGURE 4 Answer One answer Figure A3 2 4 6 8 t minutes Velocity at t 6 s0 6 10 5 yards per minute s s t s yards 80 60 40 20 Figure A3 2 4 6 8 t minutes Sections 2 2 and 2 3 p 3 Example 4 Math 10A Lecture Examples 9 30 08 Vertebrae in the human spine are separated by fibrous elastic disks that compress when the load on them increases The load on the disk is called the stress and the percent of compression is called the strain The function S S L in Figure 5 gives the strain S of lower back lumbar vertebral disks as a function of the stress L on them 3 Sketch the graph of the derivative r S0 L of the stress function S S L S percent S S L 20 10 FIGURE 5 100 Answer Figure A4a Figure A4b S percent 200 300 400 500 L kilograms S 0 50 0 11 percent per kilogram S 0 400 0 02 percent per kilogram r percent kilogram S S L r S 0 L 0 15 20 0 1 10 0 05 100 200 300 Figure A4a 3 400 500 L kilograms 100 200 300 400 500 L kilograms Figure A4b Data adapted from Physics with Illustrative Examples from Medicine and Biology Vol 1 G Benedeck and F Villars Reading Massachusetts Addison Wesley 1973 p 343 Math 10A Lecture Examples 9 30 08 Example 5 Sections 2 2 and 2 3 p 4 The next table lists the wind speed at three hour intervals one day in Dodge City Kansas 4 The time t is measured in hours with t 0 at midnight Use this data to estimate the rate of change W0 t of the wind s velocity with respect to time at 9 00 AM t 9 Wind velocity W t miles per hour in Dodge City Kansas t W t 0 3 6 9 12 15 18 21 24 12 6 12 4 12 6 15 3 16 2 15 7 15 0 11 3 12 9 Answer One approach Use the secant line in Figure A5a a left difference quotient W 0 9 0 3 miles per hour per hour Another approach Use the secant line in Figure A5b a right difference quotient W 0 9 0 9 miles per hour per hour A third approach Use the secant line in Figure A5c a centered difference quotient W 0 9 0 6 miles per hour W miles per hour W miles per hour 16 16 14 14 12 12 3 6 9 12 15 18 21 24 t 3 Figure A5a 6 9 12 15 18 21 24 t Figure A5b W miles per hour 16 14 12 3 6 9 12 15 18 21 24 t Figure A5c Interactive Examples Work the following Interactive Examples on Shenk s web page http www math ucsd edu ashenk Section 2 5 Examples 1 through 5 4 Data from Wind Energy Systems G L Johnson Englewood Cliffs N J Prentice Hall International Inc 1985 p 47 chapter and section numbers on Shenk s web site refer to his calculus manuscript and not to the chapters and sections of the textbook for the course The

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