Chapter 1 Limits, Derivatives, Integrals, and Integrals

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1 Chaper 1 Limis, Derivaives, Inegrals, and Inegrals Problem Se a. 9 cm b. From o.1: average rae cm/s From o.01: average rae 7. 1 cm/s From o.001: average rae 7. 0 cm/s So he insananeous rae of change of d a = is abou 7.0 cm/s. c. Insananeous rae would involve division by zero. d. For = 1. o 1.01, rae 31.4 cm/s. The pendulum is approaching he wall: The rae of change is negaive, so he disance is decreasing. e. The insananeous rae of change is he limi of he average raes as he ime inerval approaches zero. I is called he derivaive. f. Before = 0, he pendulum was no ye moving. For large values of, he pendulum s moion will die ou because of fricion.. a. = : y = 30, price is $3.0 = 10: y = 0, price is $.0 = 0: y = 180, price is $1.80 b. =.1, rae /f =.01, rae /f =.001, rae /f c. 47 /f. I is called he derivaive. d. = 10: 44 /f. = 0: 18 /f e. The 0-f board coss more per foo han he 10-f board. The reason is ha longer boards require aller rees, which are harder o find. Problem Se 1- Q1. Power funcion, or polynomial funcion Q. f() = 8 Q3. Eponenial funcion Q4. g() = 9 Q. Q6. h() = Q7. y = a + b + c, a 0 Q8. y = Q9. y = Q10. Derivaive 1. a. Increasing slowly b. Increasing fas. a. Increasing fas b. Decreasing slowly 1 h() 1 3. a. Decreasing fas b. Decreasing slowly 4. a. Decreasing slowly b. Increasing slowly. a. Increasing fas b. Increasing slowly c. Decreasing slowly d. Increasing fas 6. a. Decreasing fas b. Increasing slowly c. Increasing fas d. Decreasing fas 7. a. Increasing slowly b. Increasing slowly c. Increasing slowly 8. a. Decreasing fas b. Decreasing fas c. Decreasing fas 9. a. Increasing fas b. Neiher increasing nor decreasing c. Increasing fas d. Increasing slowly 10. a. Decreasing slowly b. Decreasing fas c. Decreasing fas d. Neiher increasing nor decreasing 11. a T() ( C) (s) = 40: rae 11. /s = 100: rae = 0 /s = 140: rae 08. /s b. Beween 0 and 80 s he waer is warming up, bu a a decreasing rae. Beween 80 and 10 s he waer is boiling, hus saying a a consan emperaure. Beyond 10 s he waer is cooling down, rapidly a firs, hen more slowly. 1. a v() (f/s) (s) SOLUTIONS MANUAL Calculus Soluions Manual Problem Se Key Curriculum Press SOLUTIONS MANUAL 39

2 = : rae 18 (f/s)/s = : rae = 0 (f/s)/s = 6: rae 11 (f/s)/s b. Unis are (f/s)/s, someimes wrien as f/s. The physical quaniy is acceleraion. 13. a. 18 h() Increasing a = 3 Decreasing a = 7 b. h(3) = 17, h(3.1) = Average rae = = 1.9 f/s 01. c. From 3 o 3.01: average rae = = 199. f/s 001. From 3 o 3.001: average rae = = 1.99 f/s The limi appears o be f/s. d. h(7) = 9, h(7.001) = Average rae = = f/s The derivaive a = 7 appears o be 6 f/s. The derivaive is negaive because h() is decreasing a = a f() Decrease b. Ener y No much Increase y1( ) y1( = 1 ) 1 10 r() = y (foes/year) undefined c. Subsiuing 1 for causes division by zero, so r(1) is undefined. Esimae: r approaches he average of r(0.99) and r(1.01), foes/year. (Acual is ) The insananeous rae is called he derivaive. f( 401. ) f( 4) d. = K 001. f( 4) f( 3. 99) = K 001. Insananeous rae = ( )/ = foes/year (acual: ) The answer is negaive because he number of foes is decreasing. a( 1. ) a( ) 1. a. Average rae = = baceria/h 00(. 1 ) 00(. 1 ) b. r () = r() (mm /hr) (mm) r() is undefined. c. r(.01) = = Use he solver o find when r() = = =.00088, so keep wihin 0.00 uni of a. v ( ) = π v( 6) = 88π b. 6 o 6.1: average rae = 3 π( ) = π o 6: average rae = 3 π( 6 9. ) = π Esimae of insananeous rae is ( π π)/ = π = cm 3 /cm c. r ( )= 3 π 3 π π 48π 3 r ( ) (cm /cm) r(6) is undefined. 6 Problem Se 1- Calculus Soluions Manual 00 Key Curriculum Press 40 CALCULUS COURSE SAMPLER

3 d. r(6.1) = π = r(6.1) is unis from he derivaive. Use he solver feaure o find if r() = 144π = , so keep wihin uni of a. i. 1.0 in./s ii. 0.0 in./s iii. 1.1 in./s b. 1.7 s, because y = 0 a ha ime 18. a. i in./min ii in./min iii in./min b. The rae is negaive, because y is decreasing as he ire goes down. 19. a. Quadraic (or polynomial) b. f(3) = 30 c. Increasing a abou 11.0 (.99 o 3.01) 0. a. Quadraic (or polynomial) b. f(1) = 1 c. Increasing a abou 6.0 (0.99 o 1.01) 1. a. Eponenial b. Increasing, because he rae of change from 1.99 o.01 is posiive.. a. Eponenial b. Increasing, because he rae of change from 3.01 o.99 is posiive. 3. a. Raional algebraic b. Decreasing, because he rae of change from 3.99 o 4.01 is negaive. 4. a. Raional algebraic b. Increasing, because he rae of change from.01 o 1.99 is posiive.. a. Linear (or polynomial) b. Decreasing, because he rae of change from 4.99 o.01 is negaive. 6. a. Linear (or polynomial) b. Increasing, because he rae of change from 7.99 o 8.01 is posiive. 7. a. Circular (or rigonomeric) b. Decreasing, because he rae of change from 1.99 o.01 is negaive. 8. a. Circular (or rigonomeric) b. Decreasing, because he rae of change from 0.99 o 1.01 is negaive. 9. Physical meaning of a derivaive: insananeous rae of change To esimae a derivaive graphically: Draw a angen line a he poin on he graph and measure is slope. To esimae a derivaive numerically: Take a small change in, find he corresponding change in f(), hen divide. Repea, using a smaller change in. See wha number hese average raes approach as he change in approaches zero. The numerical mehod illusraes he fac ha he derivaive is a limi. 30. See he e for he definiion of limi. Problem Se 1-3 Q1. 7 f Q. y = cos Q3. y = Q4. y = 1/ Q. y = Q6. f() = 4 Q7. Q8. y Q9. Q10. = 3 y 1. f() = f() = a. Approimaely 30.8 a. Approimaely. b. Approimaely 41.8 b. Approimaely 47.1 f() y f() h() = sin 4. g ( )= + a. Approimaely.0 a. Approimaely 7.9 b. Approimaely 1.0 b. Approimaely 1. 1 h() 3 6 g() 1 1 Calculus Soluions Manual Problem Se Key Curriculum Press SOLUTIONS MANUAL 41

4 . There are approimaely 6.8 squares beween he curve and he -ais. Each square represens ()(0) = 100 fee. So he disance is abou (6.8)(100) = 680 fee. 6. There are approimaely 3.3 squares beween he curve and he -ais. Each square represens (0.)(10) = miles. So he disance is abou (3.3)() = 66. miles. an 101. an Derivaive = 34K Derivaive = 7 (eacly, because ha is he slope of he linear funcion) 9. a. v() v( 301. ) v( 99. ) d. Rae = K Abou 1.86 (f/s)/s The derivaive represens he acceleraion. 11. From = 0 o =, he objec ravels abou 11.4 cm. From = o = 9, he objec ravels back abou 4.3 cm. So he objec is locaed abou = 7.1 cm from is saring poin. 1. See he e for he meaning of derivaive. 13. See he e for he meaning of definie inegral. 14. See he e for he meaning of limi. Problem Se 1-4 Q1. y changes a 30 Q. Derivaive 00 Q3. Q4. f(3) = 9 y The range is 0 y b. Using he solver, = s. c. By couning squares, disance 10 f. The concep used is he definie inegral. v( 01. ) v( 499. ) d. Rae = K Abou 3.1 (f/s)/s The concep is he derivaive. The rae of change of velociy is called acceleraion. 10. a. v() b. v(4) = f/s Domain: 0 4 Range: 0 v() c. By couning squares, he inegral from = 0 o = 4 is abou 1.3 f. The unis of he inegral are (f/s) s = f. The inegral ells he lengh of he slide. Q. 100 Q6. sin (π/) = 1 Q days Q8. Derivaive Q9. Definie inegral Q10. f() = 0 a = 4 1. a. v() 0,000 b. Inegral 0 (. v( 0) + v( ) + v( 10) + v( 1) + v(0) + v() + 0.v(30)) = (669.4 ) = , 000 f The sum overesimaes he inegral because he rapezoids are circumscribed abou he region and hus include more area. c. The unis are (f/s)(s), which equals fee, so he inegral represens he disance he spaceship has raveled. d. Yes, i will be going fas enough, because v(30) = 7,919.04, which is greaer han 7,000.. a. v() = 4 + sin 1.4 v() Problem Se 1-4 Calculus Soluions Manual 00 Key Curriculum Press 4 CALCULUS COURSE SAMPLER

5 b. A definie inegral has he unis of he -variable imes he y-variable. Disance = rae ime. Because v() is disance/ime and is ime, heir produc is epressed in unis of disance. c. See graph in par a. Disance 00.(. v() 0 + v(.) 0 + v() 1 + v(1.) + v() + v(.) + 0.v(3)) = 0.(6.041 ) = f d. v(3) = mi/h Maimum speed was mi/h a abou 1.1 h. 3. Disance 0.( ) = 396 f 4. Volume 3( ) = 04,000 f 3. Programs will vary depending on calculaor. See he program TRAPRULE in he Insrucor s Resource Book for an eample. The program gives T 0 = See he program TRAPDATA in he Insrucor s Resource Book for an eample. The program gives T 7 = 33, as in Eample. 7. a. 7 f() 1 4 b. T 10 = T 0 = T 0 = These values underesimae he inegral, because he rapezoids are inscribed in he region. c. T 10 : uni from he eac answer T 0 : uni from he eac answer T 0 : uni from he eac answer T n is firs wihin 0.01 uni of 18.9 when n = 7. T 7 = , which is uni from Because T n is geing closer o 18.9 as n increases, T n is wihin 0.01 uni of 18.9 for all n a. g() b. T 10 = T 0 = Τ 0 = These values overesimae he inegral, because he rapezoids are circumscribed abou he region. c. T 10 : uni from answer T 0 : uni from answer T 0 : uni from answer T n is firs wihin 0.01 of when n = 1. T 1 = , which is uni from Because T n is geing closer o he eac answer as n increases, T n is wihin 0.01 uni of he answer for all n From he given equaion, y=±( 40/ 110) 110. Using he rapezoidal rule program on he posiive branch wih n = 100 incremens gives for he op half of he ellipse. Doubling his gives an area of 13, cm. The esimae is oo low because he rapezoids are inscribed wihin he ellipse. The area of an ellipse is πab, where a and b are he - and y-radii, respecively. So he eac area is π (110)(40) = 4400π = 13, cm, which agrees boh wih he answer and wih he conclusion ha he rapezoidal rule underesimaes he area. 10. Inegral = 1( ) = The inegral will have he unis (in. )(in.) = in. 3, represening he volume of he fooball. 11. n = 10: inegral 1.04 n = 100: inegral n = 1000: inegral Conjecure: inegral = 1 The word is limi. 1. The rapezoidal rule wih n = 100 gives inegral Conjecure: inegral = If he rapezoids are inscribed (graph concave down), he rule underesimaes he inegral. If he rapezoids are circumscribed (graph concave up), he rule overesimaes he inegral Concave down Inscribed rapezoids Underesimaes inegral Concave up Circumscribed rapezoids Overesimaes inegral Calculus Soluions Manual Problem Se Key Curriculum Press SOLUTIONS MANUAL 43

6 Problem Se 1-1. Answers will vary. Problem Se 1-6 Review Problems R1. a. When = 4, d = sin [1, (4 3)] 1.4 f. b. From 3.9 o 4: average rae 40.1 f/s From 4 o 4.1: average rae 9. 3 f/s Insananeous rae 34.7 f/s The disance from waer is decreasing, so he is going down. d( 01. ) d( 499. ) c. Insananeous rae d. Going up a abou 70.8 f/s e. Derivaive R. a. Physical meaning: insananeous rae of change of a funcion Graphical meaning: slope of a angen line o a funcion a a given poin b. = 4: decreasing fas = 1: increasing slowly = 3: increasing fas = : neiher increasing nor decreasing c. From o.1: 1. average rae = = K 01. From o.01: 01. average rae = = K 001. From o.001:. 001 average rae = = K Differences beween average raes and insananeous raes, respecively: = = = The average raes are approaching he insananeous rae as approaches. The concep is he derivaive. The concep used is he limi. d. = : 3. m/s = 18: 8.7 m/s = 4: 11. m/s Her velociy says consan, 7 m/s, from 6 s o 16 s. A = 4, Mary is in her final sprin oward he finish line. R3. By couning squares, he inegral is approimaely 3.. Disance 3. f (eac answer: 3.4 ) Concep: definie inegral R4. a. f () 1 4 The graph agrees wih Figure 1-6c. b. By couning squares, inegral 1.0. (Eac answer is 1.) c. T 6 = 0.( ) = The rapezoidal sum underesimaes he inegral because he rapezoids are inscribed in he region. d. T 0 = ; Difference = T 100 = ; Difference = The rapezoidal sums are geing closer o 1. Concep: limi R. Answers will vary. Concep Problems C1. a. f(3) = b. f() f(3) = = c. f( ) f( 3) 7+ 1 ( 4)( 3) = = = , if 3 d. The limi is found by subsiuing 3 for in ( 4). Limi = eac rae = 3 4 = 1 C. The line hrough (3, f(3)) wih slope 1 is y = +. f() 3 The line is angen o he graph. Zooming in by a facor of 10 on he poin (3, ) shows ha he graph becomes sraigher and looks almos like he angen line. (Soon sudens will learn ha his propery is called local lineariy.) 6 Problem Se 1-6 Calculus Soluions Manual 00 Key Curriculum Press 44 CALCULUS COURSE SAMPLER

7 ( 4 7)( 3) C3. a. f( ) = = = , 3 When = 3, 4 7 = =. b. f() (f) (s) c..8 = 4(3 + δ ) 7 4. = 4(3 δ ) 7.8 = 1 + 4δ 7 4. = 1 4δ 7 4δ = 0.8 4δ = 0.8 δ = 0. δ = 0. d. 4(3 + δ ) 7 = + ε 1 + 4δ 7 = + ε 4δ = ε δ ε = 1 4 There is a posiive value of δ, namely 1 4 ε, for each posiive value of ε, no maer how small ε is. e. L =, c = bu no equal o 3 is needed so ha you can cancel he ( 3) facors wihou dividing by zero. Chaper Tes T1. Limi, derivaive, definie inegral, indefinie inegral T. See he e for he definiion of limi. T3. Physical meaning: insananeous rae T y T. Concep: definie inegral By couning squares, disance 466. (Eac answer is ) T Speed (f/s) Time (s) T 7 = ( ) = 46. Trapezoidal rule probably underesimaes he inegral, bu some rapezoids are inscribed and some circumscribed. T7. Concep: derivaive Speed (f/s) Time (s) Slope 1.8 (f/s)/s (Eac answer is ) Name: acceleraion T8. The roller coaser is a he boom of he hill a s because ha s where i is going he fases. The graph is horizonal beween 0 and 10 seconds because he velociy says consan, f/s, as he roller coaser climbs he ramp. T9. Disance = (rae)(ime) = (10) = 0 f T10. T = 41.; T 0 = ; T 100 = T11. The differences beween he rapezoidal sum and he eac sum are: For T : difference = For T 0 : difference = For T 100 : difference = The differences are geing smaller, so T n is geing closer o Calculus Soluions Manual Problem Se Key Curriculum Press SOLUTIONS MANUAL 4

8 T1. From 30 o 31: y( 31) y( 30) average rae = = K 1 From 30 o 30.1: y( 30. 1) y( 30) average rae = = K 01. From 30 o 30.01: y( ) y( 30) average rae = = K 001. T13. The raes are negaive because he roller coaser is slowing down. T14. The differences beween he average raes and insananeous rae are: For 30 o 31: difference = For 30 o 31.1: difference = For 30 o 30.01: difference = The differences are geing smaller, so he average raes are geing closer o he insananeous rae. y ( ) y( 30) T1. Solve = , geing 30 = So keep wihin 0.09 uni of 30, on he posiive side. T16. Concep: derivaive f( 43. ) f( 37. ) 3 9 T17. f ( 4) = = T18. Answers will vary. 8 Problem Se 1-6 Calculus Soluions Manual 00 Key Curriculum Press 46 CALCULUS COURSE SAMPLER

9 Name: Group Members: Eploraion 1-1a: Insananeous Rae of Change of a Funcion Dae: Objecive: Eplore he insananeous rae of change of a funcion. Door 4. A ime H 1 s, does he door appear o be opening or closing? How do you ell? d The diagram shows a door wih an auomaic closer. A ime H 0 s, someone pushes he door. I swings open, slows down, sops, sars closing, hen slams shu a ime H 7 s. As he door is in moion, he number of degrees, d, i is from is closed posiion depends on. 1. Skech a reasonable graph of d versus.. Suppose ha d is given by he equaion d H 00 D Plo his graph on your grapher. Skech he resuls here. 3. Make a able of values of d for each second from H 0 hrough H 10. Round o he neares d. Wha is he average rae a which he door is moving for he ime inerval [1, 1.1]? Based on your answer, does he door seem o be opening or closing a ime H 1? Eplain. 6. By finding average raes using he ime inervals [1, 1.01], [1, 1.001], and so on, make a conjecure abou he insananeous rae a which he door is moving a ime H 1 s. 7. In calculus you will learn by four mehods: algebraically, numerically, graphically, verbally (alking and wriing). Wha did you learn as a resul of doing his Eploraion ha you did no know before? 8. Read Secion 1-1. Wha do you noice? INSTRUCTOR S RESOURCE BOOK Calculus: Conceps and Applicaions Insrucor s Resource Book Eploraion Masers / 3 00 Key Curriculum Press INSTRUCTOR S RESOURCE BOOK 47

10 Name: Group Members: Eploraion 1-a: Graphs of Familiar Funcions Dae: Objecive: Recall he graphs of familiar funcions, and ell how fas he funcion is changing a a paricular value of. For each funcion: a. Wihou using your grapher, skech he graph on he aes provided. b. Confirm by grapher ha your skech is correc. c. Tell wheher he funcion is increasing, decreasing, or no changing when H 1. If i is increasing or decreasing, ell wheher he rae of change is slow or fas. 1. f() H 3 D f() 3. f() H C D f() 4. f() H sec f(). f() H sin π f(). f() H 1 f() 4 / Eploraion Masers Calculus: Conceps and Applicaions Insrucor s Resource Book 00 Key Curriculum Press 48 CALCULUS COURSE SAMPLER

11 Name: Group Members: Eploraion 1-3a: Inroducion o Definie Inegrals Dae: Objecive: Find ou wha a definie inegral is by working a real-world problem ha involves he speed of a car. As you drive on he highway you accelerae o 100 f/s o pass a ruck. Afer you have passed, you slow down o a more moderae 60 f/s. The diagram shows he graph of your velociy, v(), as a funcion of he number of seconds,, since you sared slowing. 4. How many fee does each small square on he graph represen? How far, herefore, did you go in he ime inerval [0, 0]? v ( ) Problems 3 and 4 involve finding he produc of he -value and he y-value for a funcion where y may vary wih. Such a produc is called he definie inegral of y wih respec o. Based on he unis of and v(), eplain why he definie inegral of v() wih respec o in Problem 4 has fee for is unis The graph shows he cross-secional area, y, in in., of a fooball as a funcion of he disance,, in in., from one of is ends. Esimae he definie inegral of y wih respec o. y 1. Wha does your velociy seem o be beween H 30 and H 0 s? How far do you ravel in he ime inerval [30, 0]? Eplain why he answer o Problem 1 can be represened as he area of a recangular region of he graph. Shade his region Wha are he unis of he definie inegral in Problem 6? Wha, herefore, do you suppose he definie inegral represens? 3. The disance you ravel beween H 0 and H 0 can also be represened as he area of a region bounded by he (curved) graph. Coun he number of squares in his region. Esimae he area of pars of squares o he neares 0.1 square space. For insance, how would you coun his parial square? 8. Wha did you learn as a resul of doing his Eploraion ha you did no know before? Calculus: Conceps and Applicaions Insrucor s Resource Book Eploraion Masers / 00 Key Curriculum Press INSTRUCTOR S RESOURCE BOOK 49

12 Name: Group Members: Eploraion 1-4a: Definie Inegrals by Trapezoidal Rule Dae: Objecive: Esimae he definie inegral of a funcion numerically raher han graphically by couning squares. Rocke Problem: Ella Vader (Darh s daugher) is driving in her rocke ship. A ime H 0 min, she fires her rocke engine. The ship speeds up for a while, hen slows down as Alderaan s graviy akes is effec. The graph of her velociy, v(), in miles per minue, is shown below. v() 4. Divide he region under he graph from H 0 o H 8, which represens he disance, ino four verical srips of equal widh. Draw four rapezoids whose areas approimae he areas of hese srips and whose parallel sides eend from he -ais o he graph. By finding he areas of hese rapezoids, esimae he disance Ella goes. Does he answer agree wih he answer o Problem? The echnique in Problem 4 is he rapezoidal rule. Pu a program ino your grapher o use his rule. The funcion equaion may be sored as y 1. The inpu should be he saring ime, he ending ime, and he number of rapezoids. The oupu should be he value of he definie inegral. Tes your program by using i o answer Problem Use he program from Problem o esimae he definie inegral using 0 rapezoids Wha mahemaical concep would you use o esimae he disance Ella goes beween H 0 and H 8? 7. The eac value of he definie inegral is he limi of he esimaes by rapezoids as he widh of each rapezoid approaches zero. By using he program from Problem, make a conjecure abou he eac value of he definie inegral. 8. Wha is he fases Ella wen? A wha ime was ha?. Esimae he disance in Problem 1 graphically. 9. Approimaely wha was Ella s rae of change of velociy when H? Was she speeding up or slowing down a ha ime? 3. Ella figures ha her velociy is given by v() H 3 D 1 C 100 C 110 Plo his graph on your grapher. Does he graph confirm or refue wha Ella figures? Tell how you arrive a your conclusion. 10. Based on he equaion in Problem 3, here are posiive values of ime a which Ella is sopped. Wha is he firs such ime? How did you find your answer? 11. Wha did you learn as a resul of doing his Eploraion ha you did no know before? 6 / Eploraion Masers Calculus: Conceps and Applicaions Insrucor s Resource Book 00 Key Curriculum Press 0 CALCULUS COURSE SAMPLER

13 Name: Tes 1, Chaper 1 Dae: Objecive: Show wha you have learned abou limis, derivaives, and definie inegrals so far. Par 1: No calculaors allowed (1 ) 1. One concep of calculus is limi. Wrie he verbal definiion of limi. 4. Wha is he physical meaning of he answer in Problem 3? 8. Below are he graphs of hree funcions, f, g, and h. Eplain why f() and g() have limis as approaches 7, bu h() does no have a limi. f( ) 8 g() h() 8. A hird concep of calculus is derivaive, which is an insananeous rae of change. A H 0 s, approimaely wha is he insananeous rae of change of velociy? Is he velociy increasing or decreasing? Anoher concep of calculus is definie inegral, which is a means for finding he produc of and y in a funcion where y varies. Definie inegrals can be calculaed graphically. The following graph shows he velociy, v(), in fee per second, of a moving objec. Find an esimae of he definie inegral of v() wih respec o from H 10 o H 30 s. 1 v () (Hand in his page o ge he res of he es.) Calculus: Conceps and Applicaions Insrucor s Resource Book Secion, Chaper, and Cumulaive Tess / 3 00 Key Curriculum Press INSTRUCTOR S RESOURCE BOOK 1

14 Name: Dae: Tes 1, Chaper 1 coninued Par : Graphing calculaors allowed (6 1) For Problems 6 8, derivaives and definie inegrals can be esimaed from ables of daa. The following daa show he hea capaciy, C(T ), in Briish hermal unis (Bu) per degree, of a pound of seam a various Fahrenhei emperaures, T. T C(T ) Use he rapezoidal rule o esimae he definie inegral of C(T ) wih respec o T from T H 1000 o T H Wha are he unis of he inegral in Problem 6? Wha does i represen in he real world? 8. Approimaely wha is he rae of change of hea capaciy in Problem 6 a T H 000? For Problems 9 14, Calvin and Phoebe sar up a hill in heir car. When hey are 0 f from he boom of he hill, he car runs ou of gas. They coas o a sop, hen sar rolling backward. Phoebe figures ha heir displacemen, D(), in fee from he boom of he hill, is given by D() HD C 0 C 0 9. How far are hey from he boom of he hill when H 3 s? 10. Find Calvin and Phoebe s average velociy for he inerval H 3 o H 3.1 s. 11. By using imes closer and closer o H 3, make a conjecure abou he limi he average velociy approaches as approaches The average velociy, v av (), from 3 o s can be wrien v av () H D( ) D D(3) D 3 Subsiue appropriae epressions for D() and D(3). Then plo he graph of v av (). Use a friendly window ha includes H 3. Skech he graph, showing he behavior of he funcion a H The insananeous velociy a H 3 would be he value of v av () a H 3. Show wha happens when you subsiue H 3, and hus ell why he insananeous velociy canno be found by direc subsiuion. 14. By appropriae algebra, simplify he raional epression in Problem 1. Show ha he simplified epression gives you eacly he value of insananeous velociy ha you conjecured in Problem Wha did you learn as a resul of aking his es ha you did no know before? 4 / Secion, Chaper, and Cumulaive Tess Calculus: Conceps and Applicaions Insrucor s Resource Book 00 Key Curriculum Press CALCULUS COURSE SAMPLER

15 Name: Tes, Chaper 1 Dae: Objecive: Show wha you have learned abou limis, derivaives, and definie inegrals so far. Par 1: No calculaors allowed (1 7) 1. Wrie he verbal definiion of he limi of f() as approaches c. For Problems 7, he figure shows he graph of he velociy, v, in fee per second, of a roller coaser as a funcion of he ime,, in seconds, since i sared. v (f/s) Wrie he four conceps of calculus (s). Which concep of calculus is used o find he disance he roller coaser ravels from H 0 o H 3? Esimae his disance graphically. 3. Skech he graph of a funcion f for which f() H 7, bu lim f() H. 6. Which concep of calculus is used o find he rae of change of speed a he insan when H 30? Esimae his rae graphically. Tell he unis of his rae of change and he physical name for his quaniy. 4. Skech a graph showing he graphical meaning of definie inegral. 7. Does he speed have a limi as ime approaches 1? If so, wrie he limi. If no, ell why no. (Hand in his page o ge he res of he es.) Calculus: Conceps and Applicaions Insrucor s Resource Book Secion, Chaper, and Cumulaive Tess / 00 Key Curriculum Press INSTRUCTOR S RESOURCE BOOK 3

16 Name: Dae: Tes, Chaper 1 coninued Par : Graphing calculaors allowed (8 17) Compound Ineres Problem (8 14): Calculus involves finding insananeous raes of change. For insance, if $1000 invesed in a savings accoun has he ineres compounded coninuously, hen M(), he amoun of money in he accoun afer years, could be given by he equaion M() H 1000(1.1). 8. How much is in he accoun a he end of yr? 9. Find he average rae of change of he money for he -yr period. 10. Esimae he insananeous rae of change of M() a ime H by using M() and M(.1). Is his rae slower or faser han he average rae in Problem 9? Can you hink of a real-world reason his should be rue? 11. Ge a beer esimae of he insananeous rae in Problem 10 by using M(4.9) and M(.1). 1. Ge a beer esimae of he insananeous rae in Problem 10 by using M() and M(.01), hen by using M() and M(.001). 13. Tell why he insananeous rae in Problem 10 canno be found eacly by direc subsiuion using he equaion for M(). 14. Three hundred years ago, Newon and Leibniz found algebraic ways o calculae ( calculus ) insananeous raes. Using hese mehods, he insananeous rae in Problem 10 is Show ha he average rae of change you have calculaed in Problems 10 and 1 are approaching his number as ges closer o. 1. Definie inegrals are producs of he - and y-variables of a funcion. For insance, a definie inegral is used o calculae disance (which equals rae imes ime) when he rae varies. Suppose ha you record your speed in fee per second as you drive from a parking lo. Use rapezoids o esimae he disance you ravel beween H 0 and H 0 s. Show your mehod. speed As you laer cruise down he highway, your speed in fee per second is given by v() H 30 C 0.3 Use your rapezoidal rule program o esimae he disance you go beween H 0 and H s using n H 10 incremens and using n H 100 incremens. Wha ineger value does your disance seem o be approaching as a limi as n ges larger? 17. Wha did you learn as a resul of aking his es ha you did no know before? 6 / Secion, Chaper, and Cumulaive Tess Calculus: Conceps and Applicaions Insrucor s Resource Book 00 Key Curriculum Press 4 CALCULUS COURSE SAMPLER

17 Name: Tes 3, Chaper 1 Dae: Objecive: Show wha you have learned abou limis, derivaives, and definie inegrals so far. Par 1: No calculaors allowed (1 9) 1. Wrie he verbal definiion of limi. For Problems 9, he velociy of a car, v(), in fee per second, is given as a funcion of by his graph. 70 v() For Problems 4, he figure shows he graph of funcion f for which 1 is he limi of f() as approaches, bu f() H f(). Draw a line angen o he graph a he poin on he graph where H 60. Use i o esimae he insananeous rae of change of v() a ha ime. Which concep of calculus is his insananeous rae? The funcion value will be wihin 0.3 uni of 1 if f() is beween 0.7 and 1.3. Draw horizonal lines a y H 0.7 and y H 1.3. Shade he par of he -ais for which f() will be wihin 0.3 uni of 1 if is wihin his shaded inerval. 6. How far did he car go beween H 0 and H 0 s? 7. By couning squares on he graph, esimae he disance he car wen beween H 30 and H 70 s. Which of he four conceps of calculus do you use o find his disance? 3. Eplain how he resricion... bu no equal o c in he definiion of limi applies o funcion f. 4. How could you keep f() arbirarily close o 1? 8. Draw four rapezoids on he given figure ha you could use o esimae he disance in Problem Tell wha migh have happened in he real world o make he car s velociy-ime graph look his way. (Hand in his page o ge he res of he es.) Calculus: Conceps and Applicaions Insrucor s Resource Book Secion, Chaper, and Cumulaive Tess / 7 00 Key Curriculum Press INSTRUCTOR S RESOURCE BOOK

18 Name: Dae: Tes 3, Chaper 1 coninued Par : Graphing calculaors allowed (10 18) Problems 10 1 refer o f() H C 3si n ( D 3) D Plo he graph of f. Use a friendly -window of abou [0, 10] and a suiable y-window. (Use radian mode.) Skech he resul on your paper. Show paricularly wha he graph looks like in a neighborhood of H Funcion f has a disconinuiy a H 3. Wha kind of disconinuiy? Wha is he limi of f() as approaches 3? Eplain why f() can be said o have his number as is limi as approaches Abou how far away from 3 on he righ side can go and sill have f() be wihin 0. uni of he limi? Abou how far on he lef side? Wha one number could you use for δ in he definiion of limi if ε is equal o 0.? Problems 13 1 refer o h() H ( D. 1) 13. Funcion h has a disconinuiy a H. Skech he graph of h showing wha happens a H. Wha kind of disconinuiy is i? 14. The derivaive of a funcion is he insananeous rae of change of he y-value wih respec o. Esimae he derivaive of funcion h a H.3. Show your mehod. 1. Is h() increasing or decreasing a H.3? How does he answer o Problem 14 ell his? 16. The able shows he force (in pounds) needed o srech a bungee cord o a cerain lengh (in fee). The amoun of work done in sreching a bungee cord equals he force eered on he cord imes he disance i sreches. Fee Pounds Perform an appropriae compuaion o find he amoun of work done in sreching he cord from 10 f o 0 f. Name he mehod used. Tell he unis of work in his insance. 17. Use your rapezoidal rule program o esimae he definie inegral of y H 10 sin from H 0 o H π using, 10, and 0 rapezoids. Make a conjecure abou he eac value of he inegral, he limi of he rapezoidal approimaions as he number of rapezoids increases wihou bound. 18. Wha did you learn as a resul of aking his es ha you did no know before? 8 / Secion, Chaper, and Cumulaive Tess Calculus: Conceps and Applicaions Insrucor s Resource Book 00 Key Curriculum Press 6 CALCULUS COURSE SAMPLER

19 Insananeous Rae Consider a door equipped wih an auomaic closer. When you push i, i opens quickly, and hen he closer closes i again, more and more slowly unil i finally closes compleely. 1. Open he skech Insananeous Rae.gsp in he GSP Skeches folder. Press he Open Door buon o operae he door. Observe he door opening and closing, and he graph showing he angle of he door as a funcion of ime.. Drag Poin 1 back and forh along he ime ais, and wach how he angle of he door changes and how he poin on he graph corresponds o he door s angle. Observe he values of he 1 and d 1 measuremens as you drag. Q1 For wha values of 1 is he angle increasing? How can you ell? Q Wha is he maimum angle he door reaches? A wha ime does his occur? d (angle) (ime) 10 The value is he separaion beween he wo values of ime (1 and ). 3. To find he rae of change of he angle of he door, you need o look a he door s posiion a wo differen imes. Press he Show buon o see a second poin on he graph, slighly separaed from Poin 1. Drag Poin 1 back and forh, and observe he behavior of he new poins on he graph. To change he separaion of he wo imes, press he buon labeled 1.0 and hen he buon labeled Make he separaion of he wo poins smaller han 0.1. Can you sill see wo disinc poins on he graph? Can you see he values of and d change as you make h smaller? Eperimen wih dragging he slider o change he separaion of he wo values of ime direcly. Q3 Wha is he larges separaion you can ge by moving he slider? Wha s he smalles separaion you can acually observe on he graph? Q4 As you make smaller, can you observe changes in he numeric values of and d even when you can no longer observe any changes on he graph? Hin: Divide he change in he angle by he change in he ime.. Se o 0.1, and hen use he numeric values of 1, d 1,, and d o calculae he rae of change of he door s angle a any paricular ime. (Use Skechpad s calculaor o do his calculaion.) INSTRUCTOR S RESOURCE BOOK 7

20 When you press his buon a doed line appears connecing he wo poins on he graph. You should now have wo rows of numbers in he able, wih he firs row permanen and he second row changing as he measuremens hemselves change. Q Wha are he unis of he rae of change? Wha does he rae of change ell you abou he door s moion? 6. Press he Show Rae buon o check your resul. Q6 Wha is he relaionship beween he doed line and he rae of change you calculaed? Q7 Move 1 back and forh. How can you ell from he rae of change wheher he door is opening or closing? How can you ell wheher is rae is fas or slow? Q8 Use he buons o se 1 o 1.0 and o 0.1. Wha s he rae of change? 7. Selec he numeric values of 1, d 1,, d,, and he rae of change. Wih hese si measuremens seleced, choose Graph Tabulae. Double-click he able o make he curren enries permanen. You may wan o press he 0.1 buon and hen he 0.01 buon again o check he moion of he doed line. 8. Se he ime inerval ( ) o eacly Noe he new value of he rae of change. Could you see he doed line move as you reduced he ime inerval? Wih he inerval se o 0.01, double-click he able o permanenly record hese new values. Q9 How does his rae of change compare o he value when was 0.1? 9. Similarly, record in he able he values for inervals of 0.001, , , and Q10 Wha do you noice abou he value of he rae of change as he ime inerval becomes smaller and smaller? Wha value does he rae of change seem o be approaching? Q11 Can you see he doed line move as changes from o ? 10. Se he value of 1 o 3 seconds (by pressing he 3 buon), and collec more daa on he rae of change of he door s angle. Collec one row of daa for each ime inerval from 0.1 second o second. The average rae of change is he rae of change beween wo differen values of. The insananeous rae of change is he eac rae of change a one specific value of. Because you mus have wo differen values o calculae he rae of change, one way of measuring he insananeous rae of change is by making he second value closer and closer o he firs, and finding he limi of he average rae of change as he inerval ges very small. The insananeous rae of change of a funcion ha is, he limi of he average rae of change as he inerval ges close o zero is called he derivaive of he funcion. Q1 Wha is he derivaive of he door s angle when 1 is 3 seconds? 8 CALCULUS COURSE SAMPLER