i 0 8) If 0 < x < 1, what is the value of x n+1 as n approaches infinity? 9) Infinite Power Series Formula.

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1 Discovery Seet # Series Notatio ) Expad ad te evaluate eac series. 9 a) i i 4 4 b) i 4 e) i i 4 f) 5+5i i c) i+5 g) 4+5i d) i ) 5i i i 8 ) Rewrite te series i Σ otatio, ad te evaluate. a) b) Series Formulas Ifiite Series Wat appes if you add up ifiitely may umbers? Most people would reply tat you get a ifiitely large result. But, wat if te series ad te special caracteristic tat eac step got smaller? Let s see wat appes 6) Cosider te followig relatio: y = x. Use a calculator to fid y, give tat a) x=. ad = e) x=.9 ad = b) x=. ad =5 f) x=.9 ad =5 c) x=. ad =5 g) x=.9 ad =5 d) x=. ad = ) x=.9 ad = 7) Wat matematical law is reflected i te above problems? c) Power Series Formula! ) Multiply. a) (x )(x+) b) (x )(x +x+) c) (x )(x +x +x+) d) (x )(x 6 +x 5 +x 4 +x +x +x+) 6 e) (x ) f) (x ) g) (x ) x i x i x i 4) Power Series Formula. Use te previous problem to state a geeral formula for x i 5) Evaluate eac expressio: a) ( ) c) 6 b) ( ) d) 8 i i Te Ifiite Power Series Formula Te Power Series Formula (from above) states: x i = x+ x 8) If < x <, wat is te value of x + as approaces ifiity? 9) Ifiite Power Series Formula. Wat does te Power Series Formula become we < x < ad approaces ifiity? ) Evaluate eac expressio: a) + ( ) + ( ) + ( ) + b) Te Dicotomy ( ) + ( ) + ( ) + c) + ( ) + ( ) + ( ) + d) ( ) + ( ) + ( ) + e) ( 4) + ( 4) + ( 4) +

2 Discovery Seet # Limits & Average Speed ) A cyclist is timed over te course of a -km ride. He starts is stopwatc as soo as e starts. He reaces te 8m marker at 6.9 secods, reaces te m marker at 6.4 secods, ad reaces te fiis lie (m) at 4.8 secods. a) Wat is te average speed of te cyclist from te first marker to te secod marker? b) Wat is te average speed of te cyclist from te secod marker to te fiis lie? c) Wat is te average speed of te cyclist from te first marker to te fiis lie? d) Wat is te average speed of te cyclist trougout te wole race? e) Give a formula for average speed give d, d, t, t. ) Wat is your average speed a) if you go 6 miles i ours? b) if you go 6 miles i our? c) if you go 6 miles i a ½ our? d) if you go 6 miles i a our? e) if you go 6 miles i a our? f) if you go 6 miles i a our? g) if you go 6 miles i ours? ) if you go miles i 6 ours? i) if you go miles i ours? ) Evaluate eac it. Use a calculator i order to fid values very close to te give it. a) b) c) d) e) f) g) X X X X X 5 X X x + 5x x x 9 x x + 9 x x 8x x x + 9 x x + x x 6x x x ) Give x < x i) ( + x )x X j) ( + X x )x 4) Fuctio Practice. Give tese fuctios f(x) = 5x g(t) = t+ p() = + 5 Evaluate eac: a) f(6) e) p(4) b) g() f) f(x+) c) p() g) p( 4 ) d) g(47) ) p(t 4)

3 Discovery Seet # Galileo s Experimet Galileo wated to fid out te relatiosip betwee distace ad time we a body (e.g., a ball) is fallig. Cotrary to popular belief, e did t drop balls from te Tower of Pisa. His timig istrumets would t ave bee accurate eoug for sometig movig so fast. He eeded to slow tigs dow so e rolled balls dow iclied plaes. Altoug e may ave doe is experimets somewat differetly, we ca imagie im collectig data by rollig a ball dow a log iclied plae, ad te markig te distace tat te ball as traveled after eac secod as passed. (Tis makes time te cotrol variable.) If te iclied plae as a agle of icliatio, we get te followig data: T (sec) D (meters) Questios: ) Wat coclusios could Galileo ave reaced about te relatiosip of distace ad time by lookig at te above data? (Look for patters!) 4) Average Speed. Fid te average speed of te ball a) i te first secods. b) from secods to 7 secods. c) from secods to 8 secods. ) If a car is drivig at a costat speed durig a etire trip, te we ca say tat distace is directly proportioal to time. Wat does tis mea? ) Fill i te Blaks, referrig to te above iclied plae problem: a) Te distace is directly proportioal to. b) Te ratio of to is a costat. c) Te costat of proportioality is equal to. d) A formula tat relates distace ad time is d) from secods to 5 secods. e) from secods to 4 secods. 5) Fid te istataeous speed of te ball at secods. 6) Fuctio Practice. Give tese fuctios f(w) = w + 8 d(t) = t r(y) = y + y Evaluate eac: a) f(5) f) r(y) b) d() g) f(5w) c) r() ) f(w+5) d) d( ) i) d(t) e) f() j) d(t+)

4 Discovery Seet #4 Istataeous Speed, Part I O te previous workseet, oe of te questios asked us to fid te istataeous speed at te istat tat te ball ad bee rollig for exactly secods. Te (seemigly isurmoutable) difficulty wit calculatig istataeous speed is tat you eed to divide te cage of distace by te cage of speed, ad tat would result i dividig zero by zero, wic seems rater mid-bogglig. We will ow work toward fidig a geeral metod for determiig istataeous speed. ) If we adjust te steepess of te iclied plae to about 7.8, we get a coveiet distace formula: d(t) = t Use tis distace formula i order to calculate te average speed a) from 4 secods to 6 secods. b) from 4 secods to 5 secods. c) from 4 secods to 4.5 secods. d) from 4 secods to 4. secods. e) from 4 secods to 4. secods. f) from 4 secods to 4. secods. g) from 4 secods to 4. secods. ) Fid te istataeous speed at 4 secods. ) Now use te same metod to fid te istataeous speed at 7 secods. 4) Explai to oe aoter ow eac of te followig formulas represets average speed: r = d t r = d d t t r = d(t ) d(t ) t t r = d(t+) d(t) 5) Evaluate eac it. Use a calculator i order to fid values very close to te it. a) b) c) d) X 5 X 5 X X x 5 x + 5 e) 8x 5 x x x + 5 f) x / x 5x x x g) 5x x x ) 6) Grap eac fuctio t is te first time. is cage i time. X X X X si x x cos x x a) f(x) = x 7 c) f(x) = x b) f(x) = x d) f(x) = x

5 Discovery Seet #5 Istataeous Speed, Part II ) Usig te metod from te previous workseet, fid te istataeous speed a) At 5 secods, give tat te distace formula is d(t) = t. b) At secods, give tat te distace formula is d(t) = t. c) At secods, give tat te distace formula is d(t) = t ) Te Calculus Average Speed Formula is: d(t+) d(t) r = It is very importat tat we uderstad ow to use tis formula. It will elp us to soo arrive at te cocept of a derivative. Use tis formula (ad sow every step!) to do te below problems. a) Give d(t) = t, fid te average speed from 5 secods to 8 secods. ) New Formulas! Wit eac problem below, use te Calculus Average Speed Formula i order to derive a ew formula. Be sure to sow every step. a) Give d(t) = t, give a formula for average speed, r, from 5 secods to 5+ secods. b) Give d(t) = t, give a formula for average speed, r, from t secods to t+ secods. c) Give d(t) = t, give a formula for istataeous speed, v(t), at t secods. d) Give d(t) = k t, give a formula for istataeous speed, v(t), at t secods. b) Give d(t) = t, fid te average speed from 5 secods to secods. 4) Grap eac fuctio (o separate grap paper). a) f(x) = x x + b) f(x) = x 9x c) f(x) = x + 9x d) f(x) = x + x 6x

6 Discovery Seet #6 Te Derivative, Part I Formulas for Istataeous Speed! O te previous seet, we took a uge step. We foud formulas for calculatig average speed, r, ad istataeous speed, v(t). Wy is tis so importat? Because we ave maaged to get aroud te paradox of istataeous speed: dividig zero by zero! Tese formulas (from te ed of te previous seet) are: r = + r = 6t + v(t) = 6t v(t) = kt ) Explai, oce agai, wat eac of te above formulas ca be used for, ad wat eac of te variables represets. ) Grap eac fuctio (o separate grap paper). a) f(x) = x + 5x b) f(x) = ¼ x + ½ x c) f(x) = x 4 9x 4) Area Uder a Curve a) Grap te equatio f(x) = x. Sade i te area bouded by tis curve, te x-axis, ad te lie x =. b) Determie a metod for approximatig te area of tis saded-i regio. c) How ca you make tis approximatio as accurate as desired? ) Use oe of te above formulas to solve eac. a) Give d(t) = t, fid te average speed from 5 secods to secods. b) Give d(t) = t, fid te average speed from secods to 8 secods. c) Give d(t) = t, fid te istataeous speed at secods. d) Give d(t) =. t, fid te istataeous speed at 4 secods. e) Give d(t) =.85 t, fid te istataeous speed at 7 secods. f) Give d(t) = 4.9 t, fid te istataeous speed at 6 secods.

7 Discovery Seet #7 Te Derivative, Part II Te Defiitio of te Derivative. Tis is ot really a formula. It simply tells us wat to do i order to fid a formula for te istataeous rate of cage for a give fuctio, y = f(x). f '(x) = dy dx = f(x+) f(x) ) Use te above defiitio of te derivative i order to determie te derivative of: a) f(x) = x ) Give wat you ave leared so far, wat do you tik te derivative of eac fuctio is? a) f(x) = x 7 b) f(x) = 4x c) f(x) = x 5 + 7x 4 x + x x + 8 ) Derive te Expoet Law for Derivatives! Fid f '(x) give f(x) = k x i i Summatio Formulas i = ½ + ½ i = ⅓ + ½ + / 6 i = ¼ 4 + ½ + ¼ = ( i i i) b) f(x) = x + 7x 4) Use oe of te above formulas i order to evaluate eac series below. a) b) c) ) Slopes of Lies! Fid te slope of te lie tat a) passes troug (5,7) ad (,). b) passes troug (4,) ad (,8). c) passes troug ( 4,) ad (5,). d) passes troug (6,) ad (86,4). e) passes troug (x,y ) ad (x,y ).

8 Discovery Seet #8 Area Uder a Curve (cotiued) We will ow cotiue workig toward fidig te area of te regio bouded by te x-axis, te lie x = 4, ad te curve f(x) = x. Our strategy is to approximate tis area by drawig i a series of ti rectagles (eac oe wit its top, rigt corer sittig o te curve), ad addig up teir areas. We will arbitrarily assig (te umber of rectagles) to be 7, ad ope tat tis gives us isigt ito te ultimate solutio, wic is acieved by avig te umber of rectagles go towards ifiity. ) Give te widt of eac rectagle ( x). ) Wat is te x-coordiate of te rigt side of te 5 t rectagle (x 5 )? ) Wat is te x-coordiate of te rigt side of te k t rectagle (x k )? 4) Wat is te eigt of te k t rectagle? 5) Wat is te area of te k t rectagle? Now, we ca say tat te area of te regio is approximately equal to te sum of te 7 rectagles, wic would be: Rect + Rect + Rect + + Rect 7 ( 4 7) + 4 ( 7) + 4 ( 7) ( 7) 6) Fid te sum of te areas of te 7 rectagles by furter workig te above sequece. (Hit: you ll eed to use a summatio formula.) 7) I order to derive a geeral formula, redo te etire above process, but wit te umber of rectagles equal to, ad te rigt boudary of te area set at x = a (istead of x = 4). 8) Use te above formula to aswer te followig questios: a) Wat is te area wit x = 4 as a rigt boudary, ad usig 5 rectagles? b) Wat is te area wit x = a as a rigt boudary, ad usig ifiitely may rectagles? Te Itegral, Part I Te Itegral Te formula foud o te last problem is called te Itegral of f(x). Usig calculus otatio, te process ca be writte as: a f(x)dx = f(x k ) x k (Were te widt of te k t rectagle is x, ad te area of te k t rectagle is f(x k ) x.) We ave just sow tat for f(x) = x a f(x)dx = a Te left side sould be read: Te itegral of from x equals to a, ad it tells us te area uder te curve from x = to a. 9) Use te above itegral formula to fid te exact area uder te curve f(x) = x a) from x = to x = 4. b) from x = to x = 5. c) from x = to x = 9. Derivative Practice ) Give f(x), fid te derivative, dy dx. a) f(x) = x 4 b) f(x) = 6x c) f(x) = 5x 6 + x d) f(x) = 7 e) f(x) = 5x 4 8x + x + 9x 7 f) f(x) = x 8 + x 5 5x + Ati-Derivatives! ) Wit eac give f(x), fid its atiderivative, F(x), suc tat we you take te derivative of F(x) te result is f(x). a) f(x) = 7x 6 b) f(x) = 9 c) f(x) = x d) f(x) = x 4 e) f(x) = 5x + 8x

9 Discovery Seet #9 Te Itegral O te last workseet, we were give te curve f(x) = x, ad, troug a log process of addig togeter ifiitely may ifiitely ti rectagles, we foud a formula for calculatig te area uder te curve: a f(x)dx = a Tis is called te itegral of f(x), ad is used to fid te area uder f(x) from x = to a. For example, te area uder te curve f(x) = x from x = to 6 is ⅓(6) = 7. ) A Sort-Cut! Fortuately tere is a ice sort-cut tat eables us to derive te itegral of a polyomial. We call tis te Power Rule for Itegrals. a) Wat do you tik te itegral of f(x) = 5x is? b) Fill i te blak i order to state te Power Rule for Itegrals. If f(x) = x Te a = c) State i words, te meaig of Power Rule for Itegrals. ) Wit eac problem below, first derive a geeral formula for calculatig te area uder te curve of f(x), te use it to calculate te requested area. a) Give f(x) = x calculate te area uder te curve from x = to x =. b) Give f(x) = ½x calculate te area uder te curve from x = to x =. c) Give f(x) = 5x 4 calculate te area uder te curve from x = to x =. Te Itegral, Part II ) Give f(x) = x 4 a) Fid a Ati-Derivatives b) Fid te ati-derivative, F(x). c) Wat do te above aswers tell you? 4) Give f(x) = 8x 6 a) Fid f '(x). b) Fid a c) Fid te ati-derivative, F(x). d) Fid F '(x). Derivative Practice 5) a) Fid f '(x) give f(x) = ¼x + 8x 7 b) Calculate te istataeous rate of cage at x = 6. 6) a) Fid f '(x) give f(x) = x + ½x b) Calculate te istataeous rate of cage at x =. Comparig Graps (Callege!) For eac distace fuctio give below, do te followig: First, determie its derivative. Grap bot te fuctio ad its derivative o te same grap. Lastly, look at te two graps. Discuss wat te graps say about te movemet of te object. 7) d(t) = t 8) d(t) = ¼ t 9) d(t) = ½ t + t ) d(t) = t 6t + 9t +

10 Discovery Seet # Tagets & Areas, Part I ** idicates more callegig, but importat! Te Coectio We ave see tat, for a give fuctio, f(x), we ca fid a ew fuctio, te itegral, tat elps us fid te area uder te curve. We also saw ow we could take te ati-derivative of f(x), wic is F(x). O te previous seet, we came to te realizatio tat te itegral (wic is geometric i ature) ad te derivative (wic is a algebraic process) are closely related; tey are iverses of eac oter. Tus, te itegral ad te ati-derivative, F(x), are essetially equal. Wat does all of tis mea? Quite simply, we ca use F(x), te atiderivative, to calculate itegrals ad fid areas uder curves. ) Calculate te area uder te curve a) f(x) = x from x = to. b) f(x) = 7x + 5 from x = to 4. c) f(x) = x x + from x = to. d) f(x) = x x + from x = to. e) f(x) = x x + from x = to. f) f(x) = 5x 4 from x = to 5. ) For ay give fuctio, f(x), wat is F '(x) always equal to? Workig wit Curves ) Give f(x) = x + 8x a) Grap f(x) b) Fid f '(x). c) Wat does f() = 5 mea? d) Wat does f '() = 6 mea? e) Fid te slope of te curve at x =. f) Fid te slope of te curve at x = 5. g) Give te poits were f(x) =. ) Give te poits were f '(x) =. (te local mi ad max coordiates) i) Calculate te area of te regio bouded by te curve ad te x-axis. 4) Give f(x) = x 9x a) Make a table of values for f(x). b) Grap f(x) ligtly i pecil. c) Determie f '(x). d) Fid f ( ). e) Fid f '( ). (Is tis a surprise?) f) Give te poits were f(x) =. g) **Give te poits were f '(x) =. (te local mi ad max coordiates) ) Make improvemets to your grap of f(x) so tat it is very accurate.

11 Discovery Seet # Tagets & Areas, Part II More Grapig ) Give f(x) = x x + x + a) Grap f(x) b) Fid f '(x). c) Fid f(). Wat does tis mea? d) Fid f '(). Wat does tis mea? e) Were is x = 5? f) Give te poits were f(x) =. g) **Give te poits were f '(x) =. ) Calculate te area of te regio bouded by te curve tat sits above te x-axis. ) Give f(x) = x 6x + x a) Determie f '(x). b) Fill i te below table of values. x f(x) f '(x) 4 c) Give te poits were f(x) =. d) **Give te poits were f '(x) =. (te local mi ad max coordiates) e) Were o te grap is te slope = 74? f) Make a very accurate grap of f(x). Area Uder a Curve ) Fid te area uder te curve give a) f(x) = x +, 4 b) f(x) = x + 4x, 4 c) f(x) = x + x 6, d) f(x) = x x + x +, e) f(x) = x 6x + x, Speed to Distace Problems (Callege!) 4) Te speed (i m/s) of a object is give by te formula v(t) = 8t +. a) Wat is te formula for its distace? b) How far does te ball roll i secods? 5) A ball, startig from stilless, rolls dow a iclied plae. After 5 secods, its speed is m/s. a) Fid te formulas for distace ad istataeous speed? b) Wat is te steepess of te iclied plae (i degrees)? c) How far does te ball roll i 4 secods?

12 Review Seet ) Give te distace formula d(t) =.5 t a) Fid te average speed from t = 4 to t = 6. b) Fid te istataeous speed at t=6. c) Fid te average speed from t = to t =. d) Fid te istataeous speed at t =.8. ) Fid te derivative, f '(x), of a) f(x) = x 6 b) f(x) = x c) f(x) = 8x d) f(x) = 5 e) f(x) = x 5 x + 7 ) Fid te slope of te lie taget to te curve f(x) = x at x =. 4) Fid te ati-derivative, F(x), of a) f(x) = x 6 b) f(x) = x c) f(x) = 8x d) f(x) = 5 e) f(x) = x 5 x + 7 5) Fid te area uder te curve give a) f(x) = x, 5 b) f(x) = x + 6, 5 c) f(x) = x x, 6) Give f(x) = x 4x a) Determie f '(x). b) Fill i te below table of values. x f(x) f '(x) c) Give te poits were f(x) =. d) Give te poits were f '(x) =. (te local mi ad max coordiates) e) Fid te slope of te lie taget to te curve at x =. f) Make a very accurate grap of f(x). g) Wat does f() = mea? ) Wat does f '() = mea? i) Wat does F() = ¾ mea? Topics to Study Average Speed ad Istataeous Speed How are tey differet? Wy is istataeous speed problematic? Defiitio of te Derivative Explai wat it meas. Te Itegral ad Area uder te Curve How is it tat te itegral is able to fid te exact area uder a curve? Fudametal Teorem of Calculus Wat does it mea? Wy is it importat?

13 Callege Seet! O te Eart Te acceleratio due to gravity of a object i free fall o eart is 9.8 m/s. Tis meas tat for every secod, te speed icreases by 9.8 m/s. Te acceleratio (a) of a ball rollig dow a iclied plae is a = 9.8 si( ). Oce we kow te acceleratio (a), we ca te easily derive a distace formula by usig d(t) = ½a t. I te case of free fall, we te get d(t) = 4.9 t. Example: If te iclied plae as a slope of, te a =.7 m/s, ad te distace formula is d(t) =.85 t (wic is wat we used o seet #). Example: If te iclied plae as a slope of 7.8, te a = 6 m/s, ad te distace formula is d(t) = t (wic is wat we used o seet #). ) A ball is rolled dow a iclied plae wit a 45 icliatio. a) Give te formula for te distace tat te ball will travel after t secods. (Assume o frictio.) b) Give a formula for te istataeous speed of te ball at t secods. c) Give a formula for te istataeous speed at te istat tat te ball as rolled d meters. ) A ramp is placed from te groud up to te top of a wall tat is 4m above te groud. Fid bot te time it takes for a ball to roll dow te ramp, ad fid its edig speed, give tat te ramp a) as a icliatio of 5. b) as a icliatio of. c) as a icliatio of 6. O te Moo Te force of gravity is very early oesixt as strog o te moo. ) Redo problem #a (above) as if it were o te moo. 4) Give your above aswer, wat is te ratio (moo:eart) of te times to get to te ed of te ramp? 5) Wat is te ratio (moo:eart) of te speeds at te ed of te ramp? 6) How log does it take to freefall m o eart, startig from rest? 7) How log does it take to freefall m o te moo, startig from rest? 8) Wat is te ratio of te above two aswers? 9) Loga as a 8cm vertical leap (o eart). a) How log is se i te air? b) Wit wat speed does se take off? c) Wat is er speed after. sec? d) Wat is er speed we se is cm off te groud? e) Wat would be er vertical leap o te moo? (Assume se leaves te groud wit te same speed as o eart.) Fid te ratio (moo:eart). f) How log would se be i te air o te moo? Fid te ratio (moo:eart). ) Tere are two Olympic ski jumps: oe o te eart, ad oe o te moo. If a ball is rolled dow eac jump, ad everytig is idetical about te two jumps, wic ball will go furter before ladig? d) as a icliatio of 9 (free fall).

14 Aswers to Seet # ) a) 9 b) c) 55 d) 7 / 8 e) 6 f) 9 g) 9 ) 9 6 ) a) 5i = 5 b) i+4 = 46 c) i = 55 i 4 i i ) a) x b) x c) x 4 d) x 7 e) x 7 f) x g) x + 4) x + x 5) a) 9 b) 54,87 c) 9 d) 54,87 6) a).59 b) 7 c) 4.97 d) e).49 f).55 g). - ) 7) If x < te x = 8) X Aswers to Seet #4 ) a) m / s b) 7 m / s c) 5.5 m / s d) 4. m / s e) 4. m / s f) 4. m / s g) 4. m / s ) 4 m / s ) 4 m / s 5) a) b) o it c) d) e) 4 f) g) ) 6) a) b) c) d) 9) If x < te x i = x = x ) a) b) c) / d) ½ e) Aswers to Seet # ) a).6 m / s b).59 m / s c). m / s d).46 m / s e) r = d d t t ) a) mp b) 6 mp c) mp d) 6 mp e) 6 mp f) 6 mp g) ) mp i)? ) a) 5 b) 6 c) o it d) 4 e) 7 f) 7 g) ) i) e.78 j) e.8 4) a) 7 b) 5 c) 5 d) 7 e) f) 5x + 7 g) ) t 8t + Aswers to Seet # ) If te time goes up, te te distace goes up by te same proportio. ) a) te square of te time b) distace to time squared c).85 d) d(t) =.85 t 4) a) 8.5 m / s b) 7.66 c) 9.7 d) 6.8 e) )?? 6) a) 8 b) 9 c) 9 d) 4 e) 8 f) 9y + 6y g) w + 8 i) 9t j) t + 6t + 9 Aswers to Seet #5 ) a) m / s b) m / s c) 6 m / s ) a) 9 m / s b) 5 m / s ) a) r = + b) r = 6t + c) v(t) = 6t d) v(t) = k t 4) a) b) c) d)

15 Aswers to Seet #6 ) a) r = 5 m / s b) r = m / s c) v = 8 m / s d) v = 7.6 m / s e) v.9 m / s f) v = 58.8 m / s ) a) b) c) Aswers to Seet #7 ) a) f(x) = x We will eed f(x+) f(x+) = (x+) = x + x + x + Usig te defiitio to fid te derivative: f '(x) = f(x+) f(x) (x f '(x) = + x + x + ) (x ) x f '(x) = + x + (x f '(x) = + x + ) f '(x) = x + x + Now we ca plug i for, wic gives us Our desired aswer: f '(x) = x ) b) f(x) = x + 7x We will eed f(x+) f(x+) = (x+) + 7(x+) = x + x + + 7x + 7 Usig te defiitio to fid te derivative: f '(x) = f(x+) f(x) (x f '(x) = + x + + 7x + 7) (x + 7x) f '(x) = x (x + + 7) f '(x) = f '(x) = x Now we ca plug i for, wic gives us Our desired aswer: f '(x) = x + 7 ) a) 7x 6 b) x c) 5x 4 + 8x 6x + x ) f '(x) = k x 4) a) 465 b) 9455 c) 5,5,5 5) a) 5 / b) 7 / c) / 9 d) / 5 e) y y x x ) 4 7 Aswers to Seet #8 4) k 4 7 5) k 4 7 ) ) k k 7 6) ( 4 7) + ( 4 7) + ( 4 7) ( 4 7) = 4 7 ( ) = 7 i i Now we use te summatio formula: i i = ⅓ + ½ + / 6, wic gives us: 4 7 (⅓ 7 + ½ 7 + / 6 7) ) Followig te same process, we get: a (⅓ + ½ + / 6 ) a + a + a 6 8) a).4 b) as approaces ifiity, te secod two factioal terms approac zero. Terefore te aswer is a. 9) a) ⅓ b) 4⅔ c) 4 ) a) 4x b) 8x c) x 5 + d) e) x 4x + x + 9 f) 8x 7 + 5x 4 5 ) a) F(x) = x 7 + C b) F(x) = 9x + C c) F(x) = x + C d) F(x) = x5 5 + C e) F(x) = 5x x + C ) a) 5a ) a) a b) a Aswers to Seet #9 b) a = a+ + = a4 4 ; a = 4 area = 4 = a 6 ; a = area = 88 c) a = a 5 ; a = area = 4 ) a) a5 5 b) x5 5 + C

16 4) a) 48x 5 b) 8a7 7 c) 8x7 + C d) 8x6 7 5) a) f '(x) = ½x + 8 b) f '(6) = 6) a) f '(x) = x + x b) f '() = 4 7) d '(t) = v(t) = Te object is movig at a costat speed 8) d '(t) = v(t) = ½ t Oe possibility is to imagie a car movig toward a wall. d is te distace away from te wall. Te car steadily decelerates, touces te wall, ad te goes backwards wit steady acceleratio. 9) d '(t) = v(t) = t + Oe possibility is tat te object is trow up a iclied plae, ad d is te object s distace upill from a mark o te plae. Note tat te maximum eigt (above te mark) is we t =, wic is also we te eigt is zero. ) d '(t) = v(t) = t t + 9 = (t )(t ) As opposed to te above problem, tis case does ot ave costat acceleratio. We ca imagie tat d is te eigt above te groud of a rocket. Tere are two momets we te rocket stops movig up (or dow), ad tat s we its speed is zero. Aswers to Seet # ) a) F(x) = x4 4 ; F() = 4 b) F(x) = 7x c) F(x) = x4 4 x + 5x; F(4) = 76 + x; F() = 4⅔ d) Same as above; F() = ¼ e) F() F() = ¼ 4⅔ = 6 7 f) F(x) = x5 ; F(5) F() = 5 4 = 88 5 ) F '(x) = f(x) ) a) See grap at rigt b) f '(x) = x + 8 c) We x =, te y-value is 5 d) We x =, te slope of te curve is 6 e) f '() = f) f '(5) = g) = x + 8x (,) ; (6,) ) = x + 8 x = 4; f(4) = 4 Te local max (slope = ) is (4,4) i) F(x) = x + 4x x; roots are ad 6 F(6) F() = ( ⅔); Area = ⅔ 4) c) f '(x) = x 9 d) e) f) f(x) = gives te roots: (-,); (,); (,) g) = x 9 x = x =.7 To fid te y-values, we plug ito f(x). f ( ).4; f (- ).4 Max & mi at (.7, -.4); (-.7,.4) Aswers to Seet # ) Give f(x) = x x + x + a) See grap o ext page wit #. b) f '(x) = x 4x + c) f() =. (, ) is a poit o te curve. d) f '() = 9. At x =, te slope is 9. e) y = 68 we x = 5. It is te poit (5, 68) f) y = at (,); (,); (,). g) We set te derivative equal to zero, ad te use te quadratic formula. As: (.5,.); (.55,.6) ) = F() F( ) F(x) = x4 4 x + x + x F() = ¼ ⅔ + ½ + = 7 F( ) = ¼ + ⅔ + ½ = Area = F() F( ) = 7 ( ) =

17 ) a) f '(x) = x x+ b) x f(x) f '(x) c) Te oly place it crosses te x-axis is (,). d) Max ad Mi are (.58, 5.6); (.4, 6.8) e) We solve 74 = x x + to get x =, 7. As: (, 4) ad (7,6) ) a) F(x) = x + x; Area = F(4) F() = 8 b) F(4) = ⅔ c) 4½ ⅓ = d) 7 ( ) = ⅔ e) 5¾ ¾ = 4) a) d(t) = 4t + t b) 4m 5) a) v(t) = 4.6 t ad d(t) =.t b) si - (. 4.9) 8. c) 6.8m Aswers to Review Seet ) a) Usig oe of te average speed formulas: d(6) d(4) 54 4 r = m/s b) Here are tree ways to do tis: Plug i two time values very close to 6. Plug i t = 6 ad t = 6+ ito te calculus average speed formula, te, after te H cacels, put zero i for. Recall tat te istataeous speed formula for a iclied plae is v(t) = kt. I tis case k =.5. Terefore v(t) = t Eac metod yields te aswer: 8m/s ) c) 7.5 m / s d) 8.4 m / s ) a) f '(x) = 6x 5 b) f '(x) = x c) f '(x) = 8 d) f '(x) = e) f '(x) = x 4 6x ) f '(x) = 4x ; f '() = 4 = 8 x 4) a) F(x) = 7 7 +C b) F(x) 5x = 4 +C c) F(x)=4x +C x d) F(x) = 5x + C e) F(x) = 6 x + 7x + C 5) a) F(x) = x F(5) F() b) F(x) = x 6 + 6x 4 ( ⅓) 45½ 6 c) F(x) = x4 4 x 6) a) f '(x) = x 4 b) x f(x) f '(x) ¼ ( 8) ¼ c) It crosses te x-axis at (,); (,); (,). d) f '(x) = = x 4 x =.5 te y-coordiates are f(.5) = +.8 Terefore te Max/Mi poits are: (.5,.8); (.5,.8) e) Te slope at x = is f '() f) See grap above. g) At x = te y-coordiate is. ) At x = te slope is. i) Te area (saded) from x = to is ¾. Aswers to Callege Seet )a) d(t)=.46 t b) v(t)=6.9 t c) v(d)=.7 d ) a) v(t) =.56 t ; d(t) =.68 t d = 5.45m ; t =.49 sec ; b) t =.8 sec ; v = 8.86 m / s c) t =.4 sec ; v = 8.86 m / s d) t =.94 sec ; v = 8.86 m / s ) t = 8.5 sec ; v =.68 m / s 4) moo : eart = 6 : 5) moo : eart = : 6 or 6 : 6 6).4 sec 7).5 sec v = 8.86 m / s 8) moo : eart = 6 : 9) a).88 sec b).96 m / s c). m / s d) v(t) = t d(t) =.96t 4.9t. =.96t 4.9t t =.846 v(.846) =. m / s e) 6: ratio 4.8 m f) sec