Chapter 2 Describing Change: Rates

Transcription

1 Capter Describing Cange: Rates Section.1 Cange, Percentage Cange, and Average Rates of Cange $.30 $0.46 per day 5 days = Te stock price rose an average of 46 cents per day during te 5-day period. 3 beats.133 beats per second 15 seconds Te nurse counted an average of approximately.1 beats per second during te 15-second interval. $5, 000 $ per mont 3 monts Te company lost an average of $ per mont during te past tree monts pounds.833 pounds per week 6 weeks Se lost an average of.8 pounds per week during te six-week period. 5. Cange: 56.5 million 55.4 million = 1.1 million From 004 to 005, te number of passengers flown by Nortwest Airlines increased by 1.1 million million 55.4 million Percentage cange: 100% 1.986% 55.4 million From 004 to 005, te number of passengers flown by Nortwest Airlines increased by approximately %. Average rate of cange: 56.5 million 55.4 million 1.1 million = = 1.1 million passengers per year year Between 004 and 005, te number of passengers flown by Nortwest Airlines increased an average of 1.1 million passengers per year. 6. Cange: = 0. Te ACT composite average for females increased by 0. point between 00 and points points Percentage cange: 100% 0.966% 0.7 points Te ACT composite average for females increased by approximately 1% between 00 and 005. Average rate of cange: 0.9 points points 0. point = point per year years Te average female score increased by an average of point per year between 00 and

2 38 Capter : Describing Cange: Rates Calculus Concepts 7. Cange: = 07 tousand people Between 1930 and 000, te American Indian, Eskimo, and Aleut population in te U.S. increased by,07,000 people. 07 tousand people Percentage cange: 100% 67% 36 tousand people Between 1930 and 000, te American Indian, Eskimo, and Aleut population in te U.S. increased by approximately 67%. In oter words, te population in 000 was approximately six and a alf times wat it was in Average rate of cange: 07 tousand people 434 tousand people = = 9.6 tousand people per year years Between 1930 and 000, te American Indian, Eskimo and Aleut population in te U.S. increased, on average, by 9,600 people per year. 8. Cange: = 91 million users From 000 to 005, te number of Internet users in Cina increased by 91 million users. 91 million users Percentage cange: 100% 758% 1 million users From 000 to 005, te number of Internet users grew by approximately 758%. Average rate of cange: 103 million users - 1 million users 91 million users = = 18. million users per year years Te number of Internet users in Cina increased at an average rate of 18. million users per year between 000 and a. From te grap we see tat October 1987 ad trading days. (We must remember tat te first day gives us te initial value for te interval but cannot be numbered in te days over wic our average rate of cange is calculated.) million 193. million 110. million sares Average rate of cange = = 1 1 trading days 5.3 million sares per trading day Note: te answer given in te text is calculated using te number of days between October 1 and October 30 (9 calendar days instead of 1 trading days) giving 110. million sares 3.8 million sares per day. 9 days During October 1987, te number of sares traded eac day was increased at an average rate of approximately 5 million sares per trading day. Percentage cange = million sares 100% 57% 193. million sares Te number of sares traded on te last trading day of October 1987 was 57% more tan te number traded on te first trading day of tat mont.

3 Calculus Concepts Section.1: Cange, Percentage Cange, and Average Rates of Cange 39 b. c. Te volume of sares traded on a given day in October 1987 stayed near (or below) 00 million sares until mid-october wen it spiked near 600 million sares. It ten declined to near 300 million sares for te remainder of te mont. Te average rate of cange ignores te spike during October Madness. 10. a. Slope of secant line: foot = years foot / year b. In te 14-year period from 198 troug 1996, te lake level rose an average of foot per year. c. Te lake level dropped below te natural rim because of drougt conditions in te early 1990s but rose again to normal elevation by Te average rate of cange tells us tat te level of te lake in 1996 was close to te 198 level. Altoug te average rate of cange is nearly zero, te grap sows tat te lake level canged dramatically during te 14-year period. 11. a. Average rate of cange = = $0. billion per year Between 1996 and 001, sales at Kelly Services, Inc. increased by an average of $0. billion per year. b. Percentage cange = % = 30.7% 3.5 Between 1996 and 001, sales at Kelly Services, Inc. increased by 30.7%.

4 40 Capter : Describing Cange: Rates Calculus Concepts 1. a. How muc: Cange = 1.45 years How rapidly: Average rate of cange from 1980 to 1990 = 1.45 years of age = year of age / year 10 years Te median marriage age of men in te U.S. increased on average by approximately 0.15 year per year between 1980 and b. Te average rate of cange between any two points on a line will be te slope of te line. c. Average rate of cange from 000 to 003 = years of age = 0.1 year of age / year 3 years Te median age of first marriage for men grew at a sligtly slower rate during te beginning of te twenty-first century as opposed to te 1980s. 13. a. Because a scatter plot of te data is concave down, a quadratic model is best. P( x) = 0.037x x tousand dollars montly profit for an airline from a roundtrip fligt from Denver to Cicago, were x dollars is te cost of a roundtrip ticket and 00 x 450. b. P(00) tousand dollars; P(350) tousand dollars Average rate tousand dollars of profit of cange dollars of ticket price tousand dollars of profit per dollar of ticket price c. P(350) tousand dollars; P(450) tousand dollars Average rate tousand dollars of profit of cange dollars of ticket price tousand dollars of profit per dollar of ticket price 14. Note tat te table gives cumulative data. Te average rates in tis activity refer to te average number of tickets sold per week (as opposed to te rate of cange in te number of weekly ticket sales). a. Weeks First difference Average rate of cange 1 to = 16 tickets 16 tickets 3 weeks 4 to = 37 tickets 37 tickets 3 weeks 7 to = 79 tickets 79 tickets 3 weeks 10 to = 1190 tickets 1190 tickets 3 weeks 13 to = 117 tickets 117 tickets 3 weeks 16 to = 77 tickets 77 tickets 3 weeks 19 to = 353 tickets 353 tickets 3 weeks to = 138 tickets 138 tickets 3 weeks = 4 tickets per week = 109 tickets per week = 43 tickets per week 397 tickets per week 406 tickets per week 57 tickets per week 118 tickets per week = 46 tickets per week b. Between weeks 13 and 16, ticket sales grew at an average rate of approximately 406 tickets/week.

5 Calculus Concepts Section.1: Cange, Percentage Cange, and Average Rates of Cange 41 c. Because sales increased at an average rate of 43 tickets per week, te agent s commission revenue increased at an average rate of 43($5) = $6075 per week years 68.3 years 15. a. Average rate of cange: year per year (year of life 70 expectancy per year of age) 50 years 59.6 years b. Average rate of cange between ages 10 and 0: year per 10 year (year of life expectancy per year of age) 41.5 years 50 years Average rate of cange between ages 0 and 30: year per 10 year (year of life expectancy per year of age) Life expectancy decreases wit increasing age, but te magnitude of te rate of decrease gets smaller as a male gets older. 16. a. Cange: tousand dollars Average rate of cange: tousand dollars per percentage point Percentage Cange: percent b. Cange: = -$3.4 tousand Average rate of cange: = 5.85 tousand dollars per percentage point 1 8 Percentage Cange: % 8.641% 81.7 c. Te inverse relationsip would ave input units of tousand dollars and output units of percent. Cange: percent Average rate of cange: percent per tousand dollars Percentage cange: percent d. Cange: 8% - 1% = -4% 8% - 1% Average rate of cange: % per tousand $81.7 tousand - $58.3 tousand dollars 8% - 1% 1 Percentage Cange: 100% = % 1% a. p(55) p(40) million people million people b. (55) (40) million people Percentage cange: p p 100% 100% 47% p(40) 1.18 million people p(85) p(83) million people million people per year years

6 4 Capter : Describing Cange: Rates Calculus Concepts 18. a. 1995: C (5) = ln(5) undred tousand cases 1997: C (7) = ln(7).674 undred tousand cases Cange = C(7) C(5) = 0.73 undred tousand cases Tere were 7.3 tousand more cases were diagnosed in 1997 tan in tousand cases b. Percent cange: 100% 37.1% tousand cases Average rate of cange: tousand cases 0.36 undred tousand cases per year : s(6) 16.44% 1999: s(11) % Percentage cange = 100% 488.5% : R (4) = ln : R (1) = ln R(1) R(4) index points Average rate of cange = index points years per year Between 199 and 000, te CPI for refuse collection increased on an average of approximately 7.5 index points per year. 1. a. i. ii. iii. b. i. ii. y(3) y(1) = = = y( 6) y( 3) = 6 3 y( 10) y( 6) = = = = = 3 4 y(3) y(1) 6 100% = 100% 85.7% y(1) 7 y(5) y(3) 6 100% = 100% 46.% y(3) 13 iii. y(7) y(5) 6 100% = 100% 31.6% y(5) 19 c. Te average rate of cange of any linear function over any interval will be constant because te slope of a line (and, terefore, of any secant line) is constant. Te percentage cange is not constant.

7 Calculus Concepts Section.1: Cange, Percentage Cange, and Average Rates of Cange 43. a. i. Average rate of cange: y( 3) y( 1) = = = Percentage cange: ii. Average rate of cange: Percentage cange: iii. Average rate of cange: Percentage cange: y( 3) y( 1) % = 100% 84% y( 1) 1. y( 5) y( 3) = = = y( 5) y( 3) % = 100% 84% y( 3) 019. y(7) y(5) = = = y( 7) y( 5) % = 100% 84% y( 5) b. Te percentage cange of an exponential function is always constant, but te average rate of cange is not always constant. 3. a. Te balance increased by $ $ = $ b. Average rate of cange: $419.5 $ per year 4 years Between te end of year 1 and te end of year 5, te balance increased at an average rate of $104.81/ year. c. Tere are no data available for te balance in te account at te middle of te fourt year. Witout a model, tere is no way to estimate te average rate of cange in te balance from te middle to te end of te fourt year. x d. Te amount in te account can be modeled as B( x ) = 1400(1.064 ) dollars, were x is te number of years since te initial investment, 1 x 5. Amount in middle of year 4: B(3.5) = $ Amount at end of year 4: B(4) = $ $ $ Average rate of cange = = $109.5 per year year 15.6 x 4. a. Solve = x 15.6 x Solve = x 1 for x to obtain x birts per 100,000 in for x to obtain x birts per 100,000 in b. M ( y) = + 9 birts per 100,000 gives te multiple-birt rate y years y e after 1970 for te years between 1971 and 000.

8 44 Capter : Describing Cange: Rates Calculus Concepts c. M ( 5) birts per 100,000 in 1995 M ( 10) 3.3 birts per 100,000 in 1980 Bot estimates are interpolations because 1980 and 1995 are witin te given input interval. d. One possible answer: Te increased use of fertility drugs is te primary factor in te rise of te multiple-birt rate. Women aving cildren later in life also contributes to tis rise. % 5. a. APR = monts = 18% mont 1t % 19.56% t b. A( t) = P 1+ P( ) 6. a. APR = 7. a. APY = ( ) % monts = 8.868% mont 1t A( t) = % b. APR = monts = 8.868% mont c. APY = % 9.37 % 1 r P = P 1+ n 1t = ln = 1t ln 1+ 1 ln t = ln 1 nt ( ) 1 It will take just over 11 years (tat is, 11 years 1 mont). b. P = Pe = e rt 0.08t ln = 0.08t ln t = It will take approximately 8.66 years (8 years 8 monts).

9 Calculus Concepts Section.1: Cange, Percentage Cange, and Average Rates of Cange 45 c. r P = P 1+ n 4t = ln = 4t ln 1+ 4 ln t = ln 1 nt ( ) It will take 10 years 3 monts a. Option 1: 4.75% compounded semiannually A( t) = APY % 4.781% ( ) Option : 4.675% compounded continuously.04675t A( t) = 1000e t t ( ) APY ( ) 100% 4.786% t ( ) Option is te better option because te annual percentage yield is iger. b. Years Option 1 Option $ $ $163.0 $ $ $ $ $ $ $ $ $ c. Te two options differ by approximately $0.79 after 10 years. 9. a. A = ( ) n n b,c. Compounding n Amount yearly 1 $.00 semiannually $.5 quarterly 4 $.44 montly 1 $.61 weekly 5 $.69 daily 365 $.71 every our 8760 $.7 every minute 55,600 $.7 every second 31,636,000 $.7

10 46 Capter : Describing Cange: Rates Calculus Concepts d. $.7 e. lim ( ). 7 n n n 30. a. ( r ) n A = 1 + dollars n Amount Amount b. Compounding n r = 0.1 r = 0.5 yearly semiannually quarterly montly weekly daily every our every minute 55, every second 31,636, lim. 1 0 n n c. ( 1 + ) n (0.1)(1) d. = 1e ; Te answers to parts c and d are te same wen rounding is taken into consideration. Pe rt lim. 5 0 n n e. From te table ( 1 + ) n (0.5)(1) and = 1e ; Te answers are Pe rt te same wen rounding is taken into consideration. 31. One possible answer: Cange is simply a report of te difference in two output values so tat te magnitude of te cange may be considered. Percentage cange is a report of te difference in two output values so tat te relative magnitude of te cange may be considered. Average rate of cange is a report of te difference in two output values in a way tat considers te associated spread of te input values. 3. One possible answer: Grapically, te average rate of cange is te slope of te secant line. Tat is, te average rate of cange between two points is te rise divided by te run. Te cange between two points is te rise between te two points.

11 Calculus Concepts Section.: Instantaneous Rates of Cange 47 Section. Instantaneous Rates of Cange 1. One possible set of answers: a. A continuous grap or model is defined for all possible input values on an interval. A continuous model wit discrete interpretation as meaning for only certain input values on an interval. A continuous grap can be drawn witout lifting te pencil from te paper. A discrete grap is a scatter plot. A continuous model or grap can be used to find average or instantaneous rates of cange. Discrete data or a scatter plot can be used to find average rates of cange. b. An average rate of cange is a slope between two points. An instantaneous rate of cange is te slope at a single point on a grap. c. A secant line connects two points on a grap. A tangent line touces te grap at a point and is tilted te same way te grap is tilted at tat point.. One possible answer: Advantages: It is possible to find instantaneous rates of cange. In some cases a continuous model is useful for estimating output values between known data points. Disadvantage: A model usually gives only estimates instead of exact values. 3. Average rates of cange are slopes of secant lines. Instantaneous rates of cange are slopes of tangent lines. 4. One possible answer: A line is tangent to a curve if its slope matces te slope of te curve at te point of tangency. Te space (angle formed) between te curve and te line sould be approximately te same on eiter side of te point of tangency. Tangent lines lie above concave down curves and below concave up curves. At inflection points tangent lines cut troug te curve, lying above te concave down portion and below te concave up portion. A line is tangent if it intersects te grap in suc a way tat, wen viewed very closely, it appears to coincide (i.e., it almost coincides) wit te grap on some small interval. 5. Average speed = 19 0 miles 60 minutes 67.1 mp 17 minutes our 6. a. Average speed = miles 60 minutes 38. mp minutes 1 our b. Average speed = minutes = 75.0 mp 0 minutes 1 our c. Two possible answers: Tere may ave been a traffic jam or road construction between milepost 66 and milepost 80. Te driver may ave stopped at some point between tose mileposts. 7. a. Te slope is positive at A, negative at B and E, and zero at C and D. b. Te grap is sligtly steeper at point B tan at point A. 8. Te slope is positive at B and E, negative at A, and zero at C and D.

12 48 Capter : Describing Cange: Rates Calculus Concepts Te grap sows a decreasing, linear function. Te slope is a constant, negative number. 1. Te grap sows a decreasing, exponential function. Te slope is negative, but increases toward zero as one moves from left to rigt. 13. Te grap sows an increasing, logistic function. Te slope is always positive. Moving from left to rigt, te slope starts close to zero, increases, and ten decreases toward zero. 14. Te grap sows a concave down, quadratic function. Moving from left to rigt, te slope starts positive and ten decreases to zero (at te relative maximum) and ten becomes more and more negative. 15. Te lines drawn at A and C are not tangent lines because tey don t follow te tilt of te curve. 16. Te line drawn at B is not a tangent line because it doesn t follow te tilt of te curve on eiter side of te inflection point. 17.

13 Calculus Concepts Section.: Instantaneous Rates of Cange a,b. A: concave down, tangent line lies above te curve B: inflection point, tangent line lies below te curve on te left, above te curve on te rigt C: inflection point, tangent line lies above te curve on te left, below te curve on te rigt. Note: te answer key in te text is in error concerning point C. D: concave up, tangent line lies below te curve. c. d. A, D: Positive slope C: Negative slope (inflection point) B: Zero slope (inflection point) 0. a,b. A: Concave up, tangent line lies below curve. B: Neiter, tangent line lies below curve on te left, above te curve on te rigt. C: Concave down, tangent line lies above curve. D: Neiter (but not an inflection point because tis portion of te grap is linear), tangent line lies on te curve.

14 50 Capter : Describing Cange: Rates Calculus Concepts c. 1. d. A: Positive slope B: Positive slope (inflection point) C: Zero slope D: Negative slope. rise 1.75 Slope = 0.58 (Estimates may vary sligtly depending on coice of points.) run 30 Note: te answer calculations in te back of te text used a different coice of points but arrived at exactly te same estimate. (Estimates may vary sligtly depending on coice of points.)

15 Calculus Concepts Section.: Instantaneous Rates of Cange rise Slope at C = run 80 0 = 4 rise Slope at D = run = 15. (Estimates may vary depending on coice of points.) 4. rise 1.7 Slope = = 0.85 (Estimates may vary depending on coice of points.) run 5. a. Million subscribers per year b. In 000, te number of subscribers was increasing by 3.1 million subscribers per year. c. 3.1 million subscribers per year d. 3.1 million subscribers per year 6. a. Dollars per year b. Te dollar amount was decreasing by $0.3 per year c. $0.3 per year d. $0.3 per year

16 5 Capter : Describing Cange: Rates Calculus Concepts 7. a, b. A: 1.3 mm per day per C B: 5.9 mm per day per C C: 4. mm per day per C c. Te growt rate is increasing by 5.9 mm per day per C. d. Te slope of te tangent line at 3 C is 4. mm per day per C. e. At 17 C, te instantaneous rate of cange is 1.3 mm per day per C. 8. a. Eggs or larvae b. Larvae; pupae c. Eggs; pupae d. Pupae e. Larvae f. Eggs and pupae g. Eggs and larvae 9. a. Te slope at te solstices is zero. b. Te steepest points on te grap are tose were te grap crosses te orizontal axis. Te slopes are estimated as 4 degrees 4 degrees 0.4 degree per day and 0.4 degree per day 61 days 61 days A negative slope indicates tat te sun is moving from nort to sout. 30. a. Te optimum atcing temperature is were te emergence is maximized, at approximately 14 C. b. At te optimum temperature, te tangent line is orizontal. Its slope is 0 percentage points per C.

17 Calculus Concepts Section.: Instantaneous Rates of Cange 53 c. rise 65 percentage points 10 C: Slope = 7. percentage points per C run 9 C 17 C: Slope = rise run 59 percentage points = 11.8 percentage points per C 5 C C: Slope = rise 36 percentage points = 4 percentage points per C run 9 C (Estimates may vary sligtly depending on te coice of points.) d. Te inflection point appears to be at approximately 18 C. 31. a. Because te model is linear, te line to be sketced is te same as te model itself. From te equation, its slope is approximately.37 million people per year. b. Any line tangent to te grap of p(t) coincides wit te grap itself. c. Any line tangent to tis grap as a slope of approximately.37 million people per year. d. Te slope of te grap at every point will be.37 million people per year. e. Te instantaneous rate of cange is.37 million people per year.

18 54 Capter : Describing Cange: Rates Calculus Concepts 3. Note: wen comparing te two problems, different alignment of inputs must be taken into consideration. 33. a. Because a grap of te function given in tis activity lies below te one given in Activity 31, te population projections were adjusted down. b. Because te slope of te function given in tis activity is greater tan te slope of te function given in Activity 31, te growt rate was adjusted up. c. Te slope of a grap of P at t = 0 is.56 million people per year. d. Te line tangent to a grap of P at t = 0 is te same as te grap of P. e. In 00 te population will be increasing at a rate of.56 million people per year. a = 5: In 1994, te number of employees was increasing by approximately employees per year. b. Te slope of te grap cannot be found at x = 5 because a tangent cannot be drawn at a sarp point on a grap. c = 375 : In 1998, te number of employees was increasing by approximately employees per year.

19 Calculus Concepts Section.: Instantaneous Rates of Cange a. rise 60 tousand Slope = = 5,000 employed persons per year run 5 years (Te line sown is a tangent in te sense tat, if a smoot curve is visualized troug te indicated data points, te line could be tangent to te curve. Tecnically, te tangent line does not exist because te point is a sarp point. Estimates may vary sligtly depending on te coice of points.) b. Slope = rise run 140 tousand 11,700 employed persons per year (Estimates may vary.) 1 years c. Te slope to te left of 1974 does not matc te slope to te rigt of (Te difference between te slopes may be too great to visualize a smoot curve as was done in part a. It is also possible to estimate a rate of cange of industry employment in 1974 as zero because tat year appears to be a relative maximum. Estimates may vary.) 35. One possible answer: Te line tangent to a grap at a point P is te limiting position of secant lines troug P and nearby points on te grap. 36. One possible answer: It is possible to draw a tangent line at te break point for a piecewise continuous function if te function is continuous and smoot at te break point. Te function f is not smoot at x = ; terefore, a tangent line can not be drawn. Te function g is smoot and continuous at x = 3; terefore, a tangent line could be drawn.

20 56 Capter : Describing Cange: Rates Calculus Concepts Section.3 Derivative Notation and Numerical Estimates 1. a. Because P(t) is measured in miles and t is measured in ours, te units are miles per our. b. Speed or velocity. a. Because B(t) is measured in dollars and t is measured in years, te units are dollars per year. b. How quickly te investment is growing from interest income, dividends, and/or capital gains (Note tat db is not te interest rate) dt 3. a. No, te number of words per minute cannot be negative. b. Because w(t) is measured in words per minute and t is measured in weeks, te units are words per minute per week. c. Te student s typing speed could actually be getting worse, wic would mean tat W ( t) is negative. 4. a. Because C(f) is measured in busels and f is measured in pounds, te units are busels (of corn) per pound (of fertilizer). b. Yes, too muc fertilizer can cause production to decrease. c. No, te number of busels produced cannot be negative. 5. a. Wen te ticket price is $65, te airline s weekly profit is $15,000. b. Wen te ticket price is $65, te airline s weekly profit is increasing by $1500 per dollar of ticket price. c. Wen te ticket price is $90, te profit is decreasing $000 per dollar of ticket price. 6. a. Wen te ticket price is $115, te airline sells 1750 tickets in one week. 7. b. Wen te ticket price is $115, te number of tickets sold in a week is increasing by 0 tickets per dollar of ticket price. c. Wen te ticket price is $15, te number of tickets sold in a week is increasing by tickets per dollar of ticket price.

21 Calculus Concepts Section.3: Derivative Notation and Numerical Estimates Because d [ m( t)] = for all values of t, dt te slope is constant, and m must be te linear function m(t) = 0.34t a. At te beginning of te diet you weig 167 pounds. b. After 1 weeks of dieting your weigt is 14 pounds. c. After 1 week of dieting your weigt is decreasing by pounds per week. d. After 9 weeks of dieting your weigt is decreasing by 1 pound per week. e. After 1 weeks of dieting your weigt is neiter increasing nor decreasing. f. After 15 weeks of dieting you are gaining weigt at a rate of a fourt of a pound per week. g. 10. a. Wen te car travels at 55 mp, te fuel efficiency is 3.5 mpg. b. Wen te car travels at 55 mp, te fuel efficiency is decreasing by 0.5 mpg per mp. Driving a little faster will reduce fuel efficiency. c. Wen te car travels at 45 mp, te fuel efficiency is increasing by 0.15 mpg per mp. Driving a little faster will increase fuel efficiency. d. Wen te car travels at 51 mp, te fuel efficiency is neiter increasing nor decreasing as a function of speed. Based on te information in parts b, c, d, it appears tat 51 mp is te optimal speed for te best fuel efficiency. 11. Because = 3 and (0.6) = 18, we know tat te following points are on te grap: (1940, 4), (1970, 1), (000, 33), and (1980, 18). We also know te grap is concave up between 1940 and 1990 and concave down between 1990 and One possible grap:

22 58 Capter : Describing Cange: Rates Calculus Concepts 1. We know tat te grap includes te points (40, 13.) and (68, 14.4). Because te average rate of cange between 1980 and 1990 is 0.19 million students per year, we know tat over te ten-year period between 1980 and 1990 te number of students declined by 0.19 million students (10 years) = 1.9 million students. Tus te point corresponding to 1990 is year (50, ) = (50, 11.3). Because tere is no tangent line at x = 50, we know te grap at tat input is eiter not continuous or else comes to a sarp point. We sow two possible graps: 13. a. It is possible for profit to be negative if costs are more tan revenue. b. It is possible for te derivative to be negative if profit declines as more sirts are sold (because te price is so low, te revenue is less tan te cost associated wit te sirt.) c. If P (00) = 1.5, te fraternity s profit is declining. In oter words, selling more sirts would result in less profit. Profit may still be positive (wic means te fraternity is making money), but te negative rate of cange indicates tey are not making te most profit possible (tey could make more money by selling fewer sirts). 14. Because M(t) is measured in members and t is measured in years, te units on d dt members per year. [ M( t)] are 15. a. Because D(r) is measured in years and r is measured in percentage points, te units on dd dr are years per percentage point. b. As te rate of return increases, te time it takes te investment to double decreases. c. i. Wen te interest rate is 9%, it takes 7.7 years for te investment to double.

23 Calculus Concepts Section.3: Derivative Notation and Numerical Estimates 59 ii. Wen te interest rate is 5%, te doubling time is decreasing by.77 years per percentage point. A one-percentage-point increase in te interest rate will decrease te doubling time by approximately.8 years. iii. Wen te interest rate is 1%, te doubling time is decreasing by 0.48 year per percentage point. A one-percentage-point increase in te interest rate will decrease te doubling time by approximately alf of a year. 16. a.. b. Yes, assuming tat we are referring to a particular time wen a particular president was elected in a particular country, and working wit a precise definition of wat it means to be unemployed. c. i. At te time te president was elected, 3,000,000 people were unemployed. d. ii. 1 monts after te president was elected,,800,000 people were unemployed. iii. 4 monts after te president was elected, te number of people wo were unemployed was not canging. iv. 36 monts after te president was elected, te number of people wo were unemployed was increasing at te rate of 100,000 people per mont. v. 48 monts after te president was elected, te number of people wo were unemployed was decreasing at te rate of 00,000 people per mont

24 60 Capter : Describing Cange: Rates Calculus Concepts 17. a,b. Te slope of te secant line gives te average rate of cange. Between 1 mm and 5 mm, te terminal speed of a raindrop increases an average of approximately c. rise 1.5 m/s Slope = = 0.65 m/s per mm run mm 5 m/s = 1.5 m/s per mm. 4 mm Te slope of te tangent line gives te instantaneous rate of cange of te terminal speed of a 4 mm raindrop. A 4 mm raindrop s terminal speed is increasing by approximately 0.6 m/s per mm. d. By sketcing a tangent line at mm and estimating its slope, we find tat te terminal speed of a raindrop is increasing by approximately 1.8 m/s per mm. e. Percentage rate of cange = 1.8 m/s per mm 100% 8% per mm 6.4 m/s Te terminal speed of a mm raindrop is increasing by approximately 8% per mm as te diameter increases. 18. a. rise 47 customers Slope = run 4 ours 1 customers per our Between 7 a.m. and 11 a.m., te number of customers served eac our increased at an average rate of approximately 1 customers per our.

25 Calculus Concepts Section.3: Derivative Notation and Numerical Estimates 61 b. rise 50 customers Slope = run ours 5.5 customers per our At 4:00 p.m., te number of ourly customers was increasing at a rate of approximately 5.0 customers per our. c. Possible answers include accuracy of te drawn line and accuracy of eac estimated point. d. Approximately = 75 customers at 5:00 p.m. 19. a, Slope at 11 points 4 ours ours = 5.5 points per our Slope at 8 points 11 ours 3 ours =.7 points per our (Estimates may vary sligtly.) b. Estimate te two points on te grap: (4, 50) and (10, 86). 86 points 50 points 36 points Average rate of cange = = 6 points per our 10 ours 4 ours 6 ours As te number of ours tat you study increases from 4 to 10 ours, your expected grade on te calculus test increases by an average of 6 points per our. 6 points per our c. Percentage rate of cange 100% = 1% per our 50 points Wen you ave studied for 4 ours, your expected grade on te calculus test is increasing by 1% per our. d. G(4.6) G(4) G '(4) = = 53.3 points

26 6 Capter : Describing Cange: Rates Calculus Concepts 0. a. i. 1970: 1980: rise 34 deats Slope = 1.7 run 0 years = deats per 100,000 male population per year rise 8 deats Slope = 0.4 run 0 years = deat per 100,000 male population per year ii. iii. rise deats Slope = 0.04 run 50 years = deat per 100,000 male population per year 0.4 deat per 100,000 men per year Percentage rate of cange 100% 5 deats per 100,000 men = 8% per year rise 14 deats Slope = 0.7 run 0 years = deat per 100,000 male population, per year b. Deats from lung cancer increased dramatically from 1930 to 1990, wile deats from oter forms of cancer ave stayed approximately te same or decreased. Te increase in lung cancer deats may ave been caused by increased smoking. Te grap starts to level off in te 1980s due to increased awareness of te azards of smoking. c. Possible answers include diet, smoking, exercise, and medical tecnology.

27 Calculus Concepts Section.3: Derivative Notation and Numerical Estimates a. b. Sample calculation: Note: te answers in te back of te text are incorrect. Point at x = : (, 4) Point at x = 1.9: (1.9, 1. 9 ) (1.9, ) Secant line slope = = Input of close Slope Input of close Slope point on left point on rigt Limit.77 Limit.77 Te slope of te line tangent to y = x at x = is approximately.77.. a. Slope = 1 (Estimates may vary.) b. Sample calculation: Point at x = 3: (3, 3) Point at x =.9: (.9, 3.19) Secant line slope = =

28 64 Capter : Describing Cange: Rates Calculus Concepts Input of close Slope Input of close Slope point on left point on rigt Limit Limit Te slope of te line tangent to y = x + 4 x at x = 3 is. 3. a. Sketcing a tangent line at t = 4 and finding te slope of te tangent line, we estimate tat dp 3 million passengers per year. dt b. Sample calculation: Point at x = 4: (4, 83.16) Point at x = 3.9: (3.9, ) Secant line slope = = Input of close Slope Input of close Slope point on left point on rigt Limit 3.34 Limit 3.34 P'(4) 3.34 million passengers per year In 004, te number of passengers going troug te Atlanta International Airport eac year was increasing by approximately 3.34 million passengers per year. P'(4) 3.34 million passengers per year c. PROC = 100% 100% 4.016% per year P(4) million passengers In 004, te number of passengers going troug te Atlanta International Airport eac year was increasing by approximately 4% per year. d. One possible answer: Using numbers based on an equation is more accurate, but tere are many situations wen te equation of a grap is not provided.

29 Calculus Concepts Section.3: Derivative Notation and Numerical Estimates a. $350 Slope = $87.50 per year 4 years (Estimates may vary.) b. Sample calculation (output values are rounded): 10 Point at t = 10: (10, 1500( ) ) (10, ) Point at t = 10.1: (10.1, ( ) ) (10.1, 44.90) Secant line slope $89.35 per year Input of close Slope Input of close Slope point on left point on rigt Limit from eiter direction $89.18 per year c. Te numerical estimation is te more accurate metod. In part a, te tangent line was estimated, and te rise and run were estimated. In part b, te Limit is te only estimate. B'(10) $89.18 per year d. PROC = 100% 100% 3.99% per year B(10) $ Te balance in a savings account is increasing by 3.99% per year after 10 years. 5. a. Sample calculation: Point at x = 13: (13, Point at x = 1.9: (1.9, 0.181(13) 8.463(13) ) = (13, ) 0.181(1.9) 8.463(1.9) ) = (1.9, ) Secant line slope = seconds per year of age

30 66 Capter : Describing Cange: Rates Calculus Concepts b. PROC = Input of close Slope Input of close Slope point on left point on rigt Limit from eiter direction seconds per year of age T '(13) seconds per year 100% 100% 5.59 percent per year T (13) seconds c. Te swimmer s time is decreasing. Tus te swimmer is swimming faster, and te time is improving. 6. a. Input of close Slope Input of close Slope point on left point on rigt Limit from eiter direction $7.44 billion per year S '(10) 7.44 billion dollars per year b. In 000 te annual total U.S. factory sales of electronics was increasing by approximately $7.44 billion per year. 7. a. P(x) and C(P) can be combined by function composition to create te profit function. A( x) = C ( P( x) ) = 1.0 x is te profit in American dollars from te sale of x mountain bikes. b. Canadian: C(400) $ American: A(400) $ c. We can estimate A'(400) $34.56 per mountain bike sold

31 Calculus Concepts Section.3: Derivative Notation and Numerical Estimates 67 Input of close Slope Input of close Slope point on left point on rigt Limit from eiter direction $34.56 per mountain bike 8. a. x P( x) 1.0 F( x) = x = x is te average profit per mountain bike in Canadian dollars for te sale of x mountain bikes. = = 1.0 x is te profit in American dollars from te sale of x mountain C ( P( x) ) bikes. So, M ( x) = = 1.0 x is te average profit per mountain bike in x x American dollars for te sale of x mountain bikes. b. A( x) C ( P( x) ) c. We can estimate M '(400) $0.075 per mountain bike sold Input of close Slope Input of close Slope point on left point on rigt Limit from eiter direction $0.075 per mountain bike 9. One possible answer: Te percentage cange gives te relative amount of cange in te output from an initial input value to a second input value. Te percentage rate of cange is a relative measure of te rate of cange at a particular input value in comparison to te output value at tat point. 30. One possible answer: Te slope of a tangent line can be used to estimate te function values at nearby input values. If te tangent line lies below te grap of te function, te estimate will underestimate te function value. If te tangent line lies above te grap of te function, te estimate will overestimate te function value. For te functions studied in tis text, tis metod is most accurate for input values close to te input value at te point of tangency and least accurate for input values tat are far away from te input value at te point of tangency. 31. One possible answer: Finding te rate of cange numerically using numbers found wit an equation is bot more accurate and more time-consuming tan drawing a tangent line to a grap and estimating te slope of tat tangent line. However, tere are many situations wen te equation of a grap is not provided and te rate of cange must be estimated grapically. 3. One possible answer: Because a numerical estimate of a rate of cange relies on a continuous, smoot function, it may not exactly reflect te actual rate of cange in te

32 68 Capter : Describing Cange: Rates Calculus Concepts underlying situation, especially wen te function used to model te situation did not fit te data well near te input value under consideration. Section.4 Algebraically Finding Slopes 1. x 3 x x x + 3 x x x x x lim ; lim ; lim + x x x x x x does not exist. x 3 5x 3x 9 x 3 + 5x 3x x 5x 5x lim ; lim ; lim + x 3 3x 9 x 3 3x 9 x 3 3 x 9 does not exist 3. x x + 7x x 0 x + x 3 x x + 7x x + 7x x + 7x lim = 7; lim = 7; lim = 7 x 0 x x 0 x x 0 x 4. x 0 1 x e + e x 0 x + e 1 + x x 1+ x 1+ x e e e e e e lim =.7183; lim =.7183; lim =.7183 x 0 x x 0 x x 0 x Note: Te real limit in tis case is e. x e x

33 Calculus Concepts Section.4: Algebraically Finding Slopes ( ) () 4 16 f x f x f '() = lim = lim = 16 x x x x x 4x 16 x x + 4x 16 x ( ) (1.5).3 ( 5.175) s t s t t s'(1.5) = lim = lim = lim = 6.9 t 1.5 t 1.5 t 1.5 t 1.5 t 1.5 t 1.5 t t 1.5.3t t t 1.5.3t ( 6t + 7) ( 89) g( t) g(4) 6t + 96 g '(4) = lim = lim = lim = 48 t 4 t 4 t 4 t 4 t 4 t 4 t 4 + t 4 6t 96 t t 4 6t Or using direct substitution: 6t ( t 4)( t + 4) lim = lim = lim 6( t + 4) = 6(4 + 4) = 48 t 4 t 4 4 t 4 4 ( 4 p + p ) ( 4) m( p) m( ) 4 p + p + 4 m'( ) = lim = lim = lim = 0 p p ( ) p p + p p + p 4 p + p + 4 p + p + 4 p + p + 4 p

34 70 Capter : Describing Cange: Rates Calculus Concepts Or using direct substitution: 4 p + p + 4 ( p + )( p + ) lim = lim = lim ( p + ) = + = 0 p p + p p + p 9. Point: (, ( )) x f x ( x, 3x ) Close Point: ( x +, f ( x + ) ) ( x +, 3( x + ) ) ( x +, 3x + 3 ) Slope of te Secant: Limit of te Slope of te Secant: 10. Point: ( x, f ( x) ) ( x, 15x + 3) 3 x + 3 (3 x ) 3 x x + 3 = = x + x dy 3 = lim = lim 3 = 3 dx 0 0 Close Point: ( x +, f ( x + ) ) ( x +, 15( x + ) + 3) ( x +, 15x ) (15 3) Slope of te Secant: x + + x + x + + x = = x + x Limit of te Slope of te Secant: 11. Point: (, ( )) x f x ( x, 3x ) Close Point: ( x, f ( x ) ) dy 15 = lim = lim15 = 15 dx ( x, 3( x ) ) ( x +, 3 ( x + x + )) ( x +, 3x + 6x + 3 ) Slope of te Secant: Limit of te Slope of te Secant: 1. Point: (, ( )) x f x ( x, -3x 5x) Close Point: ( x, f ( x ) ) + + ( x, 3( x )( x ) ) (3x + 6x + 3 ) 3x 6x + 3 (6x + 3 ) = = x + x (6x + 3 ) f '( x) = lim = lim(6x + 3 ) = 6x + 3(0) = 6x ( x +, -3( x + ) 5( x + ) ) ( x +, -3 ( x + x + ) 5 x 5 ) ( x +, -3x 6x 3 5x 5) Slope of te Secant: ( ) ( 3x 6x 3 5x 5 ) 3x 5 x ( 6x 3 5) = x + x Limit of te Slope of te Secant: ( 6x 3 5) f '( x) = lim = lim( 6x 3 5) = 6x 3(0) 5 = 6x 5 0 0

35 Calculus Concepts Section.4: Algebraically Finding Slopes Point: (, ( )) 3 x f x ( x, x ) Close Point: ( x, f ( x ) ) Slope of te Secant: ( x +, ( x + ) ) ( x +, x 3 + 3x + 3x + 3 ) ( + + ) ( x + 3x + 3 x + ) x 3x + 3x + 3x 3x = = x + x Limit of te Slope of te Secant: dy ( 3x + 3x + ) = lim = lim(3x dx x + ) = 3x + 3 x(0) + 0 = 3x 14. Point: (, ( )) x f x ( x, x 3 5x + 7) Close Point: ( x, f ( x ) ) + + ( x, ( x ) 3 5( x ) 7) ( x, ( x 3 3x 3 x 3 ) 5x 5 7) ( x +, x 3 + 6x + 6x + 3 5x 5 + 7) Slope of te Secant: ( ) x x x x x x x x = x + x Limit of te Slope of te Secant: ( ) df x x = = x + x + dx lim lim(6 6 5) 0 0 ( ) = x + x + = x 6 6 (0) ( ) x x = 15. a. T( 13) = seconds b. c. d. T (13 + ) = 0.181(13 + ) 8.463(13 + ) = T( 13 + ) T( 13) = = lim = lim ( ) = 0.181(0) = seconds per year of age Te swim time for a 13-year-old is decreasing by seconds per year of age. Tis tells us tat as a 13-year-old atlete gets older, te atlete s swim time improves.

36 7 Capter : Describing Cange: Rates Calculus Concepts a. S (10) = 0.009(10) 0.11(10) (10) billion dollars 3 b. S (10 + ) = 0.009(10 + ) 0.11(10 + ) (10 + ) = 0.009( (10 + ) ) 0.11( = billion dollars c. 3 S(10 + ) S(10) = = billion dollars per year d. lim = In 000, annual U.S. factory sales of consumer electronic goods to dealers was increasing by 4.97 billion dollars per year. ) 17. a. f ( 3) =.05 billion gallons b. f (3 ) 0.04(3 ) 0.18(3 ) = = billion gallons c. ( ) d. f (3 + ) f (3) = = ( ) = billion gallons per year ( ) lim = lim( ) = (0) = In 001, te amount of fuel Nortwest Airlines consumed eac year was decreasing by 7 million gallons per year. 18. a. c(8) = 0.498(8) (8) = index points b. c (8 + ) = 0.498(8 + ) (8 + ) = 0.498( ) (8 + ) c. d. = index points ( ) c(8 + ) c(8) = = ( ) index points per year lim = lim( ) = index points per year 0 0 Te consumer price index for college tuition was increasing by index points per year in 1998.

37 Calculus Concepts Section.4: Algebraically Finding Slopes a. Point: ( t, 16t + 100) Close Point: ( t +, 16( t + ) + 100) ( t +, 16( t + t + ) + 100) ( t +, 16t 3t ) b. Slope of te Secant: ( 16t 3t ) ( 16t + 100) 16 3t ( 16 3t) Limit of te Slope of te Secant: dh dt = = = t + t ( 16 3 ) dh t = lim = lim( 16 3t) = ( 16(0) 3t) = 3t feet per second dt t 0 t 0 is te speed of a falling object t seconds after te object begins falling (given tat te object as not reaced te ground). t 1 = 3(1) = 3 feet per second After 1 second, te object is falling at a speed of 3 feet per second. 0. a. d (0) = 0.6 miles nort of Howe Road b. Point: ( m, 0.8 m +0.6) c. Close Point: ( m +, 0.8( m + ) + 0.6) ( m +, 0.8m ) Slope of te Secant: ( 0.8m ) ( 0.8 m ) 0.8 = m + m 0.8 Limit of te Slope of te Secant: d '( m) = lim = lim 0.8 = 0.8 miles per minute m 0 m 0 miles 60 minutes miles 0.8 = 16.8 minute 1 our our Te tractor is moving 16.8 miles per our. 1. a. Te number of drivers of age a years in 1997 can be modeled as D( a) = a a million drivers, 16 a 1.

38 74 Capter : Describing Cange: Rates Calculus Concepts b. D( a + ) D( a) D'( a) = lim 0 = lim 0 = lim 0 (.045( a + ) ( a + ) ) ( 0.045a a ) ( ) 0.045a 0.09a a a 1.774a a + = lim 0 ( 0.09a ) = lim 0 = lim.09 a ( ) = 0.09a 0.045(0) = 0.09a Tus D ( a) = 0. 09a million drivers per year of age were a is te age in years c. D ( 0) = 0. 06million drivers per year of age. For 0-year-olds, te number of licensed drivers in 1997 was decreasing by 6,000 drivers per year of age.. a. P( a) = 6.536a a percent gives te percentage of females of age a years wo were licensed drivers in 1997, 15 a 19. b. P( a + ) P( a) P'( a) = lim 0 = lim 0 = lim 0 0 ( 6.536( a + ) ( a + ) 14.68) ( 6.536a a 14.68) ( ) 6.536a 13.07a a a 39.84a ( 13.07a ) = lim 0 a = lim 0 = lim ( ) ( a ) = 13.07a 6.536(0) = 13.07a Tus P'( a) = 13.07a percentage points per year of age

39 Calculus Concepts Section.4: Algebraically Finding Slopes 75 c. P '(16) = percentage points per year of age. For 16-year-olds, te percentage of females wo were licensed drivers was increasing by approximately 30.7 percentage points per year of age in D '(16) d. 100% 100% 78% per year of age D(16) For 16-year olds, te percentage of females wo were licensed drivers was increasing by approximately 78 percent per year of age in One possible answer: Finding a slope grapically is te only approac if all tat is available is a grap of te function witout an accompanying equation. Finding a slope grapically may be appropriate if all tat is needed is a quick approximation of te rate of cange at a point. If a more precise determination of te rate of cange at one single point is needed and an equation is available, it may be appropriate to find te slope at tat one point numerically, using a table of limiting values of te slopes of increasingly close secant lines. If an equation is available and te rate of cange at several different points is needed, it migt be appropriate to use an algebraic metod to find a formula for te slope. A final consideration wen coosing between te algebraic metod and te numerical metod to find slope is te difficulty involved in using te algebraic metod. We are generally limited to using te algebraic metod for linear, quadratic, or cubic functions. 4. One possible answer: In te previous section we discovered tat te line tangent to a grap at point P line is te limiting position of secant lines troug P and nearby points on te grap. In te algebraic metod, is used to denote te distance between te point of tangency x and a close point x+. Grapically, we visualize a secant line drawn between tese two points. In te last step of te four-step metod we take te limit of te slope of tis secant line as 0. Wen we numerically estimate a rate of cange, we coose points tat are increasingly closer to te point of tangency and calculate te slope of te secant line between tese close points and te point of tangency. Finally, we look at te limit of tese slopes. Capter Concept Review 1. a. i. A, B, C ii. E iii. D b. Te grap is steeper at B tan it is at A, C, or D. c. Below: C, D, E Above: A At B: above to te left of B, below to te rigt of B d.. a. Feet per second per second or feet per second squared. Tis is acceleration. b. Te speed of te roller coaster increased after point D.