4. When is the particle speeding up? Why? 5. When is the particle slowing down? Why?

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1 AB CALCULUS: 5.3 Positio vs Distce Velocity vs. Speed Accelertio All the questios which follow refer to the grph t the right.. Whe is the prticle movig t costt speed?. Whe is the prticle movig to the right? Why? 3. Whe is the prticle movig to the left? Why?. Whe is the prticle speedig up? Why? 5. Whe is the prticle slowig dow? Why?. Whe is the velocity icresig? Why? 7. Whe is the velocity decresig? Why? 8. Are your swers to questios & 7 the sme s your swers to & 5 respectively? Expli. 9. How fst is the prticle movig t time t = d i wht directio?. Wht is the prticle s velocity t time t =?. Whe does the prticle chge directio?. How fr does the prticle move durig the first secod d which wy? 3. How fr to the right does the prticle go d whe does it rrive there?. Wht is the totl distce trveled y the prticle? 5. Wht is the prticle s positio fter 5 secs?. Are your swers to questios & 5 the sme? Expli.

2 7. Wht is the verge velocity of the prticle over the 5 secod time itervl? 8. Wht is the verge velocity over the time itervl < t <3? 9. Wht is the verge speed over the time itervl of #8?. Wht is the prticle s ccelertio over the st secod? Wht does your swer me?. Over wht time itervl(s) does the prticle hve positive ccelertio? Why?. Over wht time itervl(s) does it hve egtive ccelertio? Why? 3. Whe is the ccelertio udefied o the itervl < t < 5?. Stte itervl of time whe the prticle hs positive ccelertio ut is slowig dow. Expli. 5. Stte itervl whe the ccelertio is egtive ut the prticle is speedig up. Expli.. Wht is the prticle s ccelertio over the time itervl (3, )? 7. Wht is the prticle s verge ccelertio over the etire 5 secod time itervl? 8. Over wht time itervl is the prticle s velocity decresig t rte of 5 feet/sec every secod?

3 AB Clculus 5.3 The Cr d Truck Prolem A cr strts t oo d trvels with the velocity show i the figure. A truck strts t pm from the sme plce d trvel t costt velocity of 5 mph.. How fr wy is the cr whe the truck strts?.. How fr hs the cr goe t 3:?. Wht ws the cr s velocity t 3:? 3. Wht ws the cr s verge velocity over the first three hours of its trip?. Grph the truck s velocity fuctio. 5. At wht poit do the two velocity fuctios itersect? Wht is the relevce of this poit esides the ovious fct tht they re trvelig t the sme speed the?. Durig the period of time whe the cr is hed of the truck:. Whe is the distce etwee them the gretest?. Approximte this distce etwee them 7.. Whe does the truck overtke the cr?. How fr hve oth trveled the?

4 AP Clculus AB Geometric Uderstdig Fudmetl Theorem of Clculus Complete the tle elow. Let f F' Expressio Wht It s Clled Is it Legth, Are, or Slope o? Sketch o f ( ) f ( ) f ( ) f ( ) f ( ) f () t dt f () t dt F( ) F( ) F( ) F( ) F'( )

5 5. Properties of the Defiite Itegrl Let s use the followig grph to develop uderstdig for ottio of the Riem Sums. Let f e fuctio tht is defied o the closed itervl,. If x is prtitio of, d xi is the width of the ith itervl, i Riem Sum of f. c is y poit i the suitervl, the the sum f Furthermore, if lim f c x exists, we sy f is itegrle o, d f x i i i i c i x i is clled is clled the defiite itegrl (or Riem Itegrl) of f from to. is clled the lower limit of itegrtio d is clled the upper limit of itegrtio. The width of ech rectgle is xi d i the defiite f c d i the defiite itegrl is the itegrl is represeted y ; the height of ech rectgle is i fuctio. Exmple : Write the followig s defiite itegrls: 3 3i 3 i i i ) lim ) lim c) lim i i i i d) lim si i e) i lim f) i i i lim i

6 Properties of Itegrtio: f ( x) c ( ) c ( c is y costt) f ( x) f ( x) k f ( x) k f ( x) c ( f ( x) g( x)) f ( x) g( x) f ( x) f ( x) f ( x) c Exmple : Suppose f x d g x, fid: ) f x ) f x g x c) f x gx 3 3 d) g x 3 f x e) f x Exmple 3: Give f ( x) 5 d g( x), use the properties to evlute the followig:. f ( x). 3 f ( x) c. 3 d. (3 f ( x)) e. ( f ( x ) g ( x )) f. ( f ( x ) g ( x )) g. g ( x ) g ( x )

7 . If 5 5 ( f ( x) 3) 7, fid f ( x) 5. If f(x) is eve fuctio d you kow f ( x), the wht is the vlue of f ( x )?. If f(x) is odd fuctio d you kow f ( x), the wht is the vlue of f ( x )? 7. Give: 7 7 f ( x ) 5, f ( x ), f ( x ) evlute ech of the followig itegrls: 5 7. f ( x). f ( x) c. f ( x) d. f ( x) e. ( ). ( ) f x f f x 5

8 WS: 5. Properties of the Defiite Itegrl Prt I: Give the grph of f x o the domi 8,.. 7 f ( x). f ( x) 3. f ( x). f ( x) 5. f ( x). 7 8 f ( x) 7. f ( x ) 8. Averge vlue i f over [, 5] 9. f (-) 8. f (). True or Flse: f ( x) + f ( x) = f ( x). True or Flse: f ( x) + f ( x) = f ( x) 3. Over wht vlues of x is f ot differetile?

9 WS: 5. Properties of the Defiite Itegrl Prt II. The Worker Prolem (Hughes-Hllett, d ed., p. 5, # Revised) A two-dy evirometl cleup opertio strted t 9 m o the first dy. The umer of workers fluctuted s show i figure 3.8. Let f (t) represet the umer of workers o the jo t time t where t represets the umer of hours pst 9 m o the first dy. Complete the missig etries i the give tle. QUANTITY. f (8) ESTIMATED VALUE (iclude uits) REAL-WORLD MEANING (s complete setece). f (8) f () 3. f() f(). workers/hour 5. f (8). 3 f () t dt (show work) 7. The cost of the cleup opertio from 9 m to 5 pm o the d dy if workers were pid $/hr. 8. The verge rte t which the lor force declied from 5 pm to m o the st dy. 9. The verge umer of workers durig the time period 5 pm to m o the st dy.

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