1 AB Clculus Pth to Five Problems # Topic Completed Definition of Limit One-Sided Limits 3 Horizontl Asymptotes & Limits t Infinity 4 Verticl Asymptotes & Infinite Limits 5 The Weird Limits 6 Continuity t Point 7 Averge Rte of Chnge 8 Instntneous Rte of Chnge 9 Tngent Line 0 Horizontl Tngent Lines Liner Approimtion Derivtives of Inverse Functions 3 Differentibility Implies Continuity 4 Conditions tht Destroy Differentition 5 Implicit Differentition 6 Verticl Tngent Lines 7 Strtegies for Finding Limits/L Hospitl s Rule 8 Relted Rtes 9 Position, Speed, Velocity, Accelertion 0 Intermedite Vlue Theorem Men Vlue Theorem & Rolle s Theorem Etrem on n Intervl 3 Finding Incresing/Decresing Intervls 4 Reltive Mimums nd Minimums 5 Points of Inflection 6 Finding Concve Up/Concve Down Intervls 7 U-Substitution Rule 8 Approimting Are
2 # Topic Completed 9 Fundmentl Theorem of Clculus 30 Properties of Definite Integrls 3 Averge Vlue of Function 3 Fundmentl Theorem of Clculus 33 Etension of FTC 34 Accumulting Rtes 35 Functions Defined by Integrls 36 Solving Differentil Equtions 37 Slope Fields 38 Eponentil Growth nd Decy 39 Prticle Motion Summry 40 Are Between Curves 4 Volumes of Slbs (Cross Sections) 4 Volume of Rottions (Discs & Wshers)
3 PTF #AB 0 Definition of Limit The intended height (or y vlue ) of function, f( ). (Remember tht the function doesn t ctully hve to rech tht height.) Written: lim f( ) c Red: the limit of f( ) s pproches c Methods for finding limit:. Direct substitution. Look t the grph Some resons why limit would fil to eist:. The function pproches different number from the left side thn from the right side.. The function increses or decreses without bound. 3. The function oscilltes between fied vlues.. Evlute lim Use the grph below to find the following limits. H(). Find the limit (if it eists). lim b ) lim H( ) b) lim H( ) b
4 PTF #AB 0 One-Sided Limits lim f( ) c lim f( ) c mens the limit from the right. mens the limit from the left. A curve hs limit if nd only if lim f( ) = lim f( ). (Left-hnd limit = Right-hnd limit) c A curve is continuous on closed intervl b, if it is continuous on the open intervl b, nd lim f ( ) f ( ) nd lim f ( ) f ( b) b mtch the function vlue t the endpoints.) c. (Limit t the endpoints hs to Use the grph to the right to find the following limits, if they eist. 5. lim g ( ) 6 6. lim g ( ) 6 7. lim g ( ) 0 8. lim g ( ) 3. lim g ( ) 3 9. lim g ( ) 0. lim g ( ) 3 3. lim g ( ) 0 4. g( 3)
5 PTF #AB 03 Horizontl Asymptotes & Limits t Infinity Horizontl Asymptotes:. If f ( ) c s, then y c is horizontl symptote.. A horizontl symptote describes the behvior t the fr ends of the grph. 3. It is helpful to think of n End Behvior Function tht will mimic the given function (wht will dominte s the vlues get lrge in both directions?) Limits t Infinity:. Grphiclly, limit t infinity will level off t certin vlue on one or both ends.. Anlyticlly, find n End Behvior Function to model the given function. Then use direct substitution to evlute the limit. 3. Short Cut: Top Hevy: limit DNE Bottom Hevy: limit = 0 Equl: limit = rtio of leding coefficients *Plese be creful with the shortcut. Some functions ct strnge nd require some etr thought. Also, wtch our for limits t, they cn require etr thought. Find the horizontl symptotes nd evlute the limits.. lim n n 4n 0, 000n b 6. If the grph of y hs c horizontl symptote y nd verticl symptote 3, then c? 5. lim lim For 0, the horizontl line y is n symptote for the grph of the function f. Which of the following sttements must be true? lim lim 6 6 (A) f (0) (B) f( ) for ll 0 (C) f () is undefined (D) lim f( ) (E) lim f( )
6 PTF #AB 04 Verticl Asymptotes & Infinite Limits Verticl Asymptotes:. If f( ) s c, then c is verticl symptote.. If function hs verticl symptote, then it is not continuous. 3. Verticl symptotes occur where the denomintor = 0, there is no common fctor, nd the numertor 0. Infinite Limits:. Grphiclly, n infinite limit increses/decreses without bound t verticl symptote.. Anlyticlly, direct substitution yields 0 in the denomintor only, with no common fctor or indeterminte form. 3. Numericlly, substitute deciml number pproching the limit to see if the y- vlues re pproching + or infinity. Find the verticl symptotes nd intervls where the function is continuous. Find lim f( ) nd lim f( ).. g ( ) 5 4. y 5. ht () t t 4 5t6 5. f( ) 4 ( )
7 PTF #AB 05 The Weird Limits To work these problems you need to be ble to visulize the grphs nd end behvior for most functions. lim e lim e 0 lim e 0 lim e lim ln lim tn lim ln 0 lim tn lim tn lim tn lim 0 lim 0 lim 0 lim 0 Evlute the limit of the inside functions first, nd then evlute the outside function t tht number. Evlute the following limits.. lim e 3. lim ln 5. lim tn 3 4. lim 0 e
8 PTF #AB 06 Continuity t Point To prove function is continuous t point, c, you must show the following three items re true:. f() c eists (the function hs y-vlue for the -vlue in question). lim f( ) eists c (the function hs left nd right hnd limit nd they re equl) 3. lim f ( ) f ( c) (the function s vlue is equl to the limit t tht -vlue) c Stte how continuity is destroyed t c for ech grph below.. 4. If the function f is continuous nd if 4 f( ) when, then f ( )?. 5. Let h be defined by the following, h( ) 3 9 For wht vlues of is h not continuous? Justify For wht vlue of the constnt c is the function f continuous over ll rels? c 3 f( ) c 3
9 # of bcteri The verge rte of chnge of ( ) following:. y dy d. f ( b) f ( ) b PTF #AB 07 Averge Rte of Chnge f over the intervl, b cn be written s ny of the 3. Slope of the secnt line through the points, f nd, b f b. *Averge rte of chnge is your good old slope formul from Algebr I.. In n eperiment of popultion of bcteri, find the verge rte of chnge from P to Q nd drw in the secnt line. (45, 340) Q. An eqution to model the free fll of bll dropped from 30 feet high is f ( ) Wht is the verge rte of chnge for the first 3 minutes? Stte units. P (3, 50) 3. Use the tble below to # of dys ) estimte f '(870) b) interpret the mening of the vlue you found in prt () t (yr) f() t (millions)
10 PTF #AB 08 Instntneous Rte of Chnge The instntneous rte of chnge, or the derivtive, of f( ) t point cn be written s ny of the following: f ( h) f ( ). f '( ) lim. This finds the vlue of the slope of the tngent line t the h0 h specific point.. Anlyticlly, find the difference quotient f ( ) f ( ) f ( h) f ( ) f ( ) f ( ) f '( ) lim lim lim. h0 h 0 This finds the generic eqution for the slope of the tngent line t ny given point on the curve. 3. Grphiclly, it is the slope of the tngent line to the curve through the point, f.. Set up the limit definition of the derivtive t for the function f ( )? 3. If f is differentible function, then f '( ) is given by which of the following? f ( h) f ( ) I. lim h h f ( ) f ( ) II. lim f ( h) f ( ) III. lim h. Fill in the blnks: h h The lim h0 h () I only (c) I nd II only only (e) I, II nd III (b) II only (d) I nd III finds the of the function.
11 PTF #AB 09 Tngent Line To find the eqution of tngent line to function through point, you need both point nd slope:. You my hve to find the y vlue of the point on the grph by plugging in the given vlue into the originl eqution.. Find the derivtive of f nd evlute it t the given point to get the slope of the tngent line. (Most times you will plug in just the vlue, but sometimes you need to plug in both the vlue nd the y vlue. The slope must be number nd must not contin ny vrible.) 3. Use the point nd the slope to write the eqution in point-slope form: y y m vlue vlue. Let f be the function defined by 3 f ( ) Find the eqution of the tngent line to the grph of f t the point where. 3. Find the eqution of the line tngent to 4 the grph of f ( ) t the point where f '( ). You will need to use your clcultor for this problem.. If the line tngent to the grph of the function f t the point (,7) psses through (-, -), then f '()?
12 PTF #AB 0 Horizontl Tngent Lines To find the point(s) where function hs horizontl tngent line:. Find f '( ) nd set it equl to zero. (Remember tht frction is zero only if the numertor equls zero.). Solve for. 3. Substitute the vlue(s) for into the originl function to find the y vlue of the point of tngency. 4. Not ll vlues will yield y vlue. If you cnnot find y vlue, then tht point gets thrown out. 5. Write the eqution of your tngent line. Remember tht since it is horizontl, it will hve the eqution y y. vlue. Find the point(s), if ny, where the function hs horizontl tngent lines. ) f 3 ( ) 5 4. Let h be function defined for ll 0 nd the derivtive of h is given by h'( ) for ll 0. Find ll vlues of for which the grph of h hs horizontl tngent. b) gt () 3 t 3. If function f hs derivtive f '( ) 3 sin for 0, find the -coordintes of the points where the function hs horizontl tngent lines.
13 PTF #AB Liner Approimtion Stndrd Liner Approimtion: n pproimte vlue of function t specified - coordinte. To find liner pproimtion:. Write the eqution of the tngent line t nice -vlue close to the one you wnt.. Plug in your -vlue into the tngent line nd solve for y.. Find liner pproimtion for f (.) if 6 f( )? 3. Find liner pproimtion for f (.67) if f ( ) sin?. Evlute 39 without clcultor (use liner pproimtion).
14 . Find f '( ). PTF #AB Derivtives of Inverse Functions. Mke sure tht you hve figured out which vlue is the nd y vlues for ech function ( f( ) nd f ( ) ) 3. Substitute the -vlue for f into f '( ). 4. The solution is the vlue you found in step #3. If g is the inverse function of F nd F() 3, find the vlue of g '(3) for 3 F( ). 4. Let f be the function defined by 5 f ( ). If g( ) f ( ) nd?, is on f, wht is the vlue of g '()
15 PTF #AB 3 Differentibility Implies Continuity Differentibility mens tht you cn find the slope of the tngent line t tht point or tht the derivtive eists t tht point.. If function is differentible t c, then it is continuous t c. (Remember wht is mens to be continuous t point.). It is possible for function to be continuous t c nd not differentible t c.. Let f be function such tht f h f lim 5. h0 h Which of the following must be true? I. f is continuous t? II. f is differentible t? III. The derivtive of f is continuous t?. Let f be function defined by f( ) k p For wht vlues of k nd p will f be continuous nd differentible t? () I only (c) I nd II only only (e) II nd III only (b) II only (d) I nd III
16 PTF #AB 4 Conditions tht Destroy Differentibility Remember for function to be differentible, the slopes on the right hnd side must be equl to the slopes on the left hnd side. There re four conditions tht destroy differentibility:. Discontinuities in the grph. (Function is not continuous.). Corners in the grph. (Left nd right-hnd derivtives re not equl.) 3. Cusps in the grph. (The slopes pproch on either side of the point.) 4. Verticl tngents in the grph. (The slopes pproch on either side of the point.). The grph shown below hs verticl tngent t (,0) nd horizontl tngents t (,-) nd (3,). For wht vlues of in the intervl,4 is f not differentible?. Let f be function defined by f( ) ) Show tht f is/is not continuous t 0. b) Prove tht f is/is not differentible t 0.
17 PTF #AB 5 Implicit Differentition. Differentite both sides with respect to.. Collect ll dy terms on one side nd the others on the other side. d 3. Fctor out the dy d. 4. Solve for dy by dividing by wht s left in the prenthesis. d Errors to wtch out for: Remember to use the product rule Remember to use prenthesis so tht you distribute ny negtive signs Remember tht the derivtive of constnt is zero. Find dy d for y y If y 5, wht is the vlue of t the point 4,3? d y d. Find the instntneous rte of chnge t, for y y.
18 PTF #AB 6 Verticl Tngent Lines To find the point(s) where function hs verticl tngent line:. Find f '( ) nd set the denomintor equl to zero. (Remember tht the slope of verticl line is undefined therefore must hve zero on the bottom.). Solve for. 3. Substitute the vlue(s) for into the originl function to find the y vlue of the point of tngency. 4. Not ll vlues will yield y vlue. If you cnnot find y vlue, then tht point gets thrown out. 5. Write the eqution of your tngent line. Remember tht since it is verticl, it will hve the eqution. vlue. Find the point(s), if ny, where the function hs verticl tngent lines. Then write the eqution for those tngent lines.. Consider the function defined by 3 y y 6. Find the -coordinte of ech point on the curve where the tngent line is verticl. ) g( ) 3 3 b) f ( ) 4
19 PTF #AB 7 Strtegies for Finding Limits/L Hospitl s Rule Steps to evluting limits:. Try direct substitution. (this will work unless you get n indeterminte nswer: 0/0). Try L Hospitl s Rule (tke derivtive of top nd derivtive of bottom nd evlute gin.) 3. Try L Hopitl s Rule gin (s mny times s needed.) 4. Use fctoring nd cnceling or rtionlizing the numertor. Find the following limits if they eist.. 3 lim( 5) 3 6. tn lim 0 sin. lim( cos ) 6 7. sin(5 ) lim 0 3. lim 56 cos 8. lim 0 sin lim, ( 0) limsec 0 5. g( ) g(0) 3 lim, g( )
20 PTF #AB8 Relted Rtes Set up the relted rte problem by:. Drwing digrm nd lbel.. Red the problem nd write Find =, Where =, nd Given = with the pproprite informtion. 3. Write the Relting Eqution nd if needed, substitute nother epression to get down to one vrible. 4. Find the derivtive of both sides of the eqution with respect to t. 5. Substitute the Given nd When nd then solve for Find.. The top of 5-foot ldder is sliding down verticl wll t constnt rte of 3 feet per minute. When the top of the ldder is 7 feet from the ground, wht is the rte of chnge of the distnce between the bottom of the ldder nd the wll?. An inverted cone hs height of 9 cm nd dimeter of 6 cm. It is leking wter 3 t the rte of cm min. Find the rte t which the wter level is dropping when h 3 cm. V r h 3
21 PTF #AB 9 Position, Speed, Velocity, Accelertion. Position Function: the function tht gives the position (reltive to the origin) of n object s function of time.. Velocity (Instntneous): tells how fst something is going t tht ect instnt nd in which direction (how fst position is chnging.) 3. Speed: tells how fst n object is going (not the direction.) 4. Accelertion: tells how quickly the object picks up or loses speed (how fst the velocity is chnging.) Position Function: st () or t () Velocity Function: v( t) s'( t) Speed Function: speed v() t Accelertion Function: ( t) v'( t) s''( t). A prticle moves long the -is so tht t time t (in seconds) its position is 3 ( t) t 6t 9t feet. d) Wht is the verge velocity on the intervl,3? ) Wht is the velocity of the prticle t t 0? The ccelertion t t 0? b) During wht time intervls is the prticle moving to the left? To the right? e) Wht is the verge ccelertion on the intervl 3,6? c) During wht time intervls is the ccelertion positive? Negtive? f) Wht is the totl distnce trveled by the prticle from t 0 to t 5?
22 PTF #AB 0 Intermedite Vlue Theorem If these three conditions re true for function:. f is continuous on the closed intervl b,. f ( ) f ( b) 3. k is ny number between f( ) nd f() b Then there is t lest one number c in b, for which f () c k. f(b) k hs to be in here b *As long s the function is continuous nd the endpoints don t hve the sme y-vlue, then the function must tke on every y-vlue between those of the endpoints. f() you cn find c-vlue in here tht will give you tht k-vlue. Use the Intermedite Vlue Theorem to 3 show tht f ( ) hs zero in the intervl 0,.. Let f( ) be continuous function on the intervl. Use the tble of vlues below to determine which of the following sttements must be true f( ) I. f( ) tkes on the vlue of 5 II. A zero of f( ) is between - nd - III. A zero of f( ) is 6 (A) (B) (C) (D) (E) I only II only III only I nd II only I, II, nd III
23 PTF #AB Men Vlue Theorem & Rolle s Theorem Men Vlue Theorem: Wht you need: function tht is continuous nd differentible on closed intervl f ( b) f ( ) Wht you get: f '( c) where c is n -vlue in the given intervl b Verblly it sys: The instntneous rte of chnge = verge rte of chnge Grphiclly it sys: The tngent line is prllel to the secnt line Rolle s Theorem (specil cse of Men Vlue Theorem): Wht you need: function tht is continuous nd differentible on closed intervl AND the y -vlues t the endpoints to be equl Wht you get: f '( c) 0 where c is n -vlue in the given intervl Verblly it sys: The derivtive equls zero somewhere in the intervl Grphiclly it sys: There is horizontl tngent line (m or min). Let f be the function given by 3 f ( ) 7 6. Find the number c tht stisfies the conclusion of the Men Vlue Theorem for f on,3.. Determine if Rolle s Theorem pplies. If so, find c. If not, tell why. 4 f ( ), for 3. Let f be function tht is differentible on the intervl,0. If f () 5, f (5) 5, nd f (9) 5, which of the following must be true? Choose ll tht pply. I. f hs t lest zeros. II. The grph of f hs t lest one horizontl tngent line. III. For some c, c 5, then f( c) 3
24 PTF #AB Etrem on n Intervl Etrem: the etreme vlues, i.e. the bsolute mimums nd minimums Etreme Vlue Theorem: As long s f is continuous on closed intervl, then f will hve both n bsolute mimum nd n bsolute minimum. Finding Etrem on closed intervl:. Find the criticl numbers of the function in the specified intervl.. Evlute the function to find the y -vlues t ll criticl numbers nd t ech endpoint. 3. The smllest y -vlue is the bsolute minimum nd the lrgest y -vlue is the bsolute mimum.. Find the bsolute etrem of ech function for the given intervl: c. f e ( ) on 0,3. f ( ) on, b. f ( ) cos( ) on 0,
25 PTF #AB 3 Finding Incresing/Decresing Intervls. Find the criticl numbers.. Set up test intervls on number line. 3. Find the sign of f '( ) (the derivtive) for ech intervl. 4. If f '( ) is positive then f( ) (the originl function) is incresing. If f '( ) is negtive then f( ) (the originl function) is decresing.. Find the intervls on which the function 3 3 f ( ) is incresing nd decresing. Justify. 3. The derivtive, g ', of function is continuous nd hs two zeros. Selected vlues of g ' re given in the tble below. If the domin of g is the set of ll rel numbers, then g is decresing on which intervl(s)? Incresing? g'( ) Let f be function given by 4 f ( ). On which intervls is f incresing? Justify.
26 PTF #AB 4 Reltive Mimums nd Minimums First Derivtive Test:. If f '( ) chnges from + to -, then is reltive m.. If f '( ) chnges from - to +, then is reltive min. Second Derivtive Test:. If f ''( ) is neg (the function is ccd), then is reltive m.. If f ''( ) is pos (the function is ccu), then is reltive min. * must be criticl number* To find the y-vlue or the m/min nd to see if it is n bsolute m/min:. Tke the -vlues nd plug them bck in to the originl eqution.. Compre.. The function defined by f ( ) 3 for ll rel numbers hs reltive mimum t =? Justify Wht is the minimum vlue of f ( ) ln? Justify.. Find the reltive mimum vlue for f ( ) 3 e. Justify. 4. If f hs criticl number t nd f ''( ) 3, then wht cn you conclude bout f t?
27 PTF #AB 5 Points of Inflection Points of Inflection: Points on the originl function where the concvity chnges.. Find where y '' is zero or undefined these re your possible points of inflection (PPOIs). Must test intervls to find the ctul POIs they re only where the second derivtive chnges sign!. Write the eqution of the line tngent 3 to the curve y 3 t its point of inflection.. Given f ''( ) 3, find the points of inflection of the grph of y f ( ).
28 PTF #AB 6 Finding Concve Up/Concve Down Intervls. Find the PPOIs.. Set up test intervls on number line. 3. Find the sign of f ' '( ) (the second derivtive) for ech intervl. 4. If f ' '( ) is positive then f( ) (the originl function) is concve up (ccu). If f ' '( ) is negtive then f( ) (the originl function) is concve down (ccd).. Find the intervls on which the function f ( ) 6 3 is concve up or concve down. Justify.. Let f be function given by 4 3 f ( ) On which intervls is f concve down? Justify.
29 PTF #AB 7 U-Substitution Rule. Let u inner function.. Find du, then solve for d. 3. Substitute u & du into the integrnd (it should know fit one of the integrtion rules). 4. Integrte. 5. Substitute the inner function bck for u. d 8. Integrte Integrte e tn cos d. Integrte sin 3cos3 d 5. Integrte e e d 3. Integrte e 3 d 6. Using the substitution u, d is equl to 0 (A) u du (B) 0 u du (C) 5 u du (D) 0 u du (E) 5 u du
30 PTF #AB 8 Approimting Are Finding Left or Right Riemnn Sum or Trpezoidl Sum:. Divide the intervl into the pproprite subintervls.. Find the y-vlue of the function t ech subintervl. 3. Use the formul for rectngle ( bh ) or trpezoid ( b h h ) to find the re of ech individul piece. 4. You must show work to ern credit on these! 5. Alwys justify left or right Riemnn sum s n over or under pproimtion using the fct tht the function is incresing or decresing. Left Sum Right Sum Incresing Under ppro. Over ppro. curve Decresing Over ppro. Under ppro. curve. Use left Riemnn Sum with 4 equl 4 subdivisions to pproimte d. 0. Vlues of continuous function f( ) re given below. Use trpezoidl sum with four subintervls of equl length to pproimte. f( ) f( ) Is this pproimtion n over or underestimte? Justify.
31 PTF #AB 9 Fundmentl Theorem of Clculus If f is continuous function on b, nd F is n ntiderivtive of f on b,, then b f ( ) d F( b) F( ) Grphiclly this mens the signed re bounded by, b, y f ( ), nd the -is. 3. Evlute: 4 6 d 3 4. Evlute: 5 e ln d 4. Evlute: 0 sin d 5. Evlute: 4 d 3. Evlute: 0 e 4 d 6. Wht re ll the vlues of k for k which d 0? 3
32 PTF #AB 30 Properties of Definite Integrls. If f is defined t, then f ( ) d 0 b. If f is integrble on b,, then ( ) ( ) b f d f d b c b 3. If f is integrble, then ( ) ( ) ( ) f d f d f d c 0. If f ( 3 ) d 4 nd f 0 3, then f ( ) d? ( ) d 7. Which, if ny, of the following re flse? b b b I. ( ) ( ) ( ) ( ) f g d f d g d II. b b b f ( ) g( ) d f ( ) d g( ) d b b III. ( ) ( ) cf d c f d
33 PTF #AB 3 Averge Vlue of Function If f is integrble on b,, then the verge vlue from the intervl is b f ( ) d b To find where this height occurs in the intervl:. Set f( ) nswer (verge vlue).. Solve for. 3. Check to see if the vlue in the given intervl.. Find the verge vlue of f ( ) sin over 0,. 3. The function t 7t f( t) 6 cos 3sin 0 40 is used to model the velocity of plne in miles per minute. According to this model, wht is the verge velocity of the plne for 0 t 40? (clcultor) 3. Find the verge vlue of y on the intervl 0,, then find where this vlue occurs in the intervl.
34 PTF #AB 3 nd Fundmentl Theorem of Clculus To find the derivtive of n integrl: d f ( t) dt f ( ) d d *Remember tht must be constnt. If it is not, then you must use your properties of integrls to mke it constnt.. For () F () F( ) t dt, find 4. Given f ( ) 4 g( ) f e () f '( ), find 3 t dt nd 0 (b) F '(3) (b) g ( ) in terms of n integrl d d 3 t. Evlute: e 3 dt (c) g'( ) 3. Find '( ) 3 F if F( ) sec t dt (d) g '(0) (e) Write the eqution for the tngent line to g ( ) t 0
35 PTF #AB 33 Etensions of FTC. FTC s Accumultion ( Integrte removes the rte! ): b. Chnge in Popultion: between time nd b) b b. Chnge in Amount: P'( t) dt P( b) P( ) (gives totl popultion dded R'( t) dt R( b) R( ) (gives totl mount dded of wter, snd, trffic, etc. between time nd b). FTC s Finl Position ( Integrte to find the end! ): b Prticle Position: ( ) ( ) ( ) time, b) b Totl Amount: ( ) ( ) '( ) S b S v t dt (gives prticle position t certin R b R R t dt (gives totl mount of wter, snd, trffic, etc. t given time, b). A prticle moves long the y -is so tht v( t) t sin t for t 0. Given tht st () is the position of the prticle nd tht s(0) 3, find s ().. A metl A metl of length 8 cm is heted t one end. The function T '( ) 3 gives the temperture, in C, of the wire cm from the heted end. 8 Find '( ) T d nd indicte units of 0 mesure. Eplin the mening of the temperture of the wire.
36 PTF #AB 34 Accumulting Rtes Identify the rte going in nd the rte going out. To find m or min point, set the two rtes equl to ech other nd solve. To find the totl mount Totl=Initil Amt b Rte Added b Rte Removed Remember to think of different blocks of time for piece-wise functions. Try to visulize wht is hppening in the sitution before you try to put the mth to work. A fctory produces bicycles t rte of p( w) 95 0.w w bikes per week for 0 w 5. They cn ship bicycles out t 90 0 w 3 rte of sw ( ) bikes/week w 5 3. Find when the number of bicycles in the wrehouse is t minimum.. How mny bicycles re produced in the first weeks?. How mny bicycles re in the wrehouse t the end of week 3? 4. The fctory needs to stop production if the number of bicycles stored in the wrehouse reches 0 or more. Does the fctory need to stop production t ny time during the first 5 weeks? If so, when?
37 PTF #AB 35 Functions Defined by Integrls F( ) f ( t) dt F '( ) f ( t) (The function in the integrnd is the derivtive eqution!) These problems work just like curve sketching problems you re looking t derivtive grph so nswer ccordingly. To evlute Fb, () find the re under the curve from where it tells you to strt () to the number given (b). Let f be function defined on the closed intervl 0,7. The grph of f, consisting of four line segments, is shown below. Let g be the function given by g( ) f ( t) dt. 3. Find the -coordinte of ech point of inflection of the grph of g on the intervl 0 7. Justify your nswer.. Find g (3), g '(3), nd g ''(3). 4. Let h( ) f ( t) dt 3. Find ll criticl vlues for h ( ) nd clssify them s minimum, mimum or neither.. Find the verge rte of chnge of g on the intervl 0 3.
38 PTF #AB 36 Solving Differentil Equtions. Seprte the vribles (usully worth point on free response question).. Integrte both sides, putting C on the side with the dependent vrible (found on the bottom of the differentil). (If there is no C, you lose ll points for this prt on free response question.) 3. If there is n initil condition, get to point where it is esy to substitute in the initil condition nd then solve for C. 4. Use the C you found nd then continue to solve for f( ) (if needed.). Find solution y f ( ) to the differentil eqution stisfying f (0). dy 3 d e y 3. Find y f ( ) by solving the differentil dy eqution y 6 with initil d condition f (3). 4 dy. If y nd if y when d, then when, y?
39 PTF #AB 37 Slope Fields. Substitute ordered pirs into the derivtive to compute slope vlues t those points.. Construct short line segments on the dots to pproimte the slope vlues. 3. For prticulr solution, sketch in the curve using the initil condition nd guided by the tngent lines.. Consider the differentil eqution dy y d.. On the es provided, sketch slope field for the given differentil eqution t the twelve points indicted. c. Find the prticulr solution y f ( ) to the given differentil eqution with the initil condition f (0) 3. Drw in the solution. b. Describe ll points in the y plne for which the slopes re positive.
40 PTF #AB 38 Eponentil Growth & Decy Direct Vrition is denoted by y k. k Inverse Vrition is denoted by y. * k is clled the constnt of vrition nd must be found in ech problem by using the initil conditions. If y is differentible function of t such tht dy ky dt, then kt y Ce.. If dy ky nd k is non-zero constnt, dt then y could be (A) e kty (B) e kt kt (C) e 3. The number of bcteri in culture is 5 growing t rte of 3000e t per unit of time t. At t 0, the number of bcteri present ws 7,500. find the number present t t 5. (D) kty 5 (E) ky
41 PTF #AB 39 Prticle Motion Summry Position Function: st () or t () Velocity Function: v( t) s'( t) Accelertion Function: ( t) v'( t) s''( t) b Displcement: v () t dt b Totl Distnce: v () t dt Position of the Prticle t time t b: s( b) s( ) v( t) dt b A prticle moves long the -is with velocity t time t 0 given by v( t) e t. At time t 0, s. 3. Find ll vlues of t for which the prticle chnges direction. Justify your nswer.. Find (3), v (3) nd s (3). 4. Find the displcement nd totl distnce of the prticle over the time intervl 0t 3.. Is the speed of the prticle incresing t time t 3? Give reson for your nswer.
42 PTF #AB 40 Are Between Curves If f nd g re continuous on b, nd g( ) f ( ) b, then the re between the curves is found by b bounded by the verticl lines nd A f ( ) g( ) d To find the re of region:. Sketch or drw the grphs.. Determine whether you need d or dy (going verticlly or horizontlly) 3. Find the limits from the boundries, es or intersections. 4. Set up the integrl by Top Bottom if d or Right Left if dy. 5. Integrte nd evlute the integrl.. Find the re of the region in the first qudrnt tht is enclosed by the grphs 3 of y 8 nd y Find the re of R, the region in the first qudrnt enclosed by the grphs of f ( ) sin( ) nd g( ) e. (clcultor). The re of the region bounded by the lines 0, nd y 0 nd the curve y e is.
43 PTF #AB 4 Volumes of Slbs (Cross Sections) Volume = Are Volume of Slbs (Cross Sections): If the solid does NOT revolve round n is, but insted hs cross sections of certin shpe. b d V A( ) d (perpendiculr to the is) or ( ) to the y is) A ( ) represents the re of the cross section Equilterl Tringle: 3 A s Semicircle: 4 V A y dy (perpendiculr c A s 8 Rectngle: A s( height ) Squre: A s Isos. Rt. Tri (on hyp.) : A s Isos. Rt. Tri (on leg): 4 A s. Let R be the region in the first qudrnt under y for 4 9. Find the volume of the solid whose bse is the region R nd whose cross sections cut by plnes to the -is re squres.. Find the volume of the sold whose bse is enclosed by y nd whose cross sections tken perpendiculr to the bse re semicircles. 3. Find the volume of solid whose bse is the circle y 9 nd whose cross-sections hve re formul given A( ) sin. by
44 PTF #AB 4 Volumes of Rottions (Discs & Wshers) Volume of Disks: If the solid revolves round horizontl/verticl is nd is flush up ginst the line of rottion. b d V r d (horizontl is) or V r dy (verticl is) r is the length of chord from curve to is of rottion Volume of Wsher: If the solid revolves round horizontl/verticl is nd is NOT flush up ginst the line of rottion. b d V R r d (horizontl is) or c c V R r dy (verticl is) R is the length of chord from frthest wy curve to is of rottion r is the length of chord from closest in curve to is of rottion. Find the volume of the solid generted by the grph bounded by y nd the line y 4 when it is revolved bout the -is. (clcultor) 3. Find the volume of the solid generted by revolving y with y 3 nd 0 bout the y -is.. The region enclosed by the -is, the line 3, nd the curve y is rotted bout the -is. Wht is the volume of the solid generted?