( ) as a fraction. Determine location of the highest

Transcription

1 AB/ Clulus Exm Review Sheet Solutions A Prelulus Type prolems A1 A A3 A4 A5 A6 A7 This is wht you think of doing Find the zeros of f( x) Set funtion equl to Ftor or use qudrti eqution if qudrti Grph to find zeros on lultor Find the intersetion of Set the two funtions equl to eh other Find intersetion on f( x) nd g( x) lultor Show tht f( x) is even Show tht f( x)= f( x) This shows tht the grph of f is symmetri to the y-xis Show tht f( x) is odd Show tht f( x)=f( x) This shows tht the grph of f is symmetri to the origin Find domin of f( x) Assume domin is (,) Restrit domins: denomintors, squre roots of only non-negtive numers, logrithm or nturl log of only positive numers Find vertil symptotes of f( x) Express in ftored form, nd do ny nelltions Set denomintor equl to If ontinuous funtion f( x) hs This is the Intermedite Vlue Theorem f<k nd f>k, explin why there must e vlue suh tht << nd f( )=k s frtion, express numertor nd denomintor B Limit Prolems B1 B B3 B4 B5 This is wht you think of doing Find limf( x) Step 1: Find f If you get zero in the denomintor, x Step : Ftor numertor nd denomintor of f( x) Do ny nelltions nd go k to Step 1 If you still get zero in the denomintor, the nswer is either, -, or does not exist Chek the signs of lim f ( x ) nd lim f( x) for equlity x x + Find limf( x) where f( x) is Determine if lim f ( x )= lim f( x) y plugging in to x x x + pieewise funtion f( x),x< nd f( x),x> for equlity If they re not equl, the limit doesn t exist Show tht is ontinuous Show th) lim exists x ) f exists 3) limf( x)= f x s frtion Determine lotion of the highest Find limf( x) nd lim x x Express Find horizontl symptotes of f( x) lim x power: Denomintor: limf( x)= lim f( x)= x x Both Num nd Denom: rtio of the highest power oeffiients Numertor: lim x nd lim x f( x)=± (plug in lrge numer) wwwmstermthmentorom Stu Shwrtz

2 B6 B7 B8 B9 f( x) lim Find x g( x) if limf( x)= nd limg( x)= x x Find limf( x)g( x)= ( ±) x Find limf( x)g x x Find limf( x) g x x This is wht you think of doing Use L Hopitl s Rule: lim x g x =lim x f x g x Express g( x)= 1 nd pply L Hopitl s rule 1 g( x) = Express f( x)g( x) with ommon denomintor nd use L Hopitl s rule =1 or or Tke the nturl log of the expression nd pply L Hopitl s rule, rememering to tke the resulting nswer nd rise e to tht power C Derivtives, differentiility, nd tngent lines C1 C C3 C4 C5 C6 Find the derivtive of funtion using the derivtive definition Find the verge rte of hnge of f on [, ] Find the instntneous rte of hnge of f t x = Given hrt o nd f( x) nd seleted vlues o etween nd, pproximte f ( ) where is vlue etween nd Find the eqution of the tngent line to f t ( x 1,y 1 ) Find the eqution of the norml line to f t x 1,y 1 This is wht you think of doing f( x+h) f( x) f( x) f Find lim or lim h h x x Find f f Find f wwwmstermthmentorom Stu Shwrtz Strddle, using vlue of k nd vlue of h f ( ) f ( k) f ( h) k h Find slope m = yy 1 =m xx 1 Find slope m= f ( x i ) Then use point slope eqution: 1 Then use point slope eqution: f ( x i ) s frtion Set numertor of f ( x) = yy 1 =m xx 1 C7 Find x-vlues of horizontl Write f x tngents to f C8 Find x-vlues of vertil tngents Write f ( x) s frtion Set denomintor of f ( x) = to f C9 Approximte the vlue of f( x 1 +) Find slope m = f ( x i ) Then use point slope eqution: if you know the funtion goes yy 1 =m( xx 1 ) Evlute this line for y t x =x 1 + Note: through point ( x 1,y 1 ) The loser is to, the etter the pproximtion will e Also note tht using onvity, it n e determine if this vlue is n over or under-pproximtion for f( x 1 +) C1 Find the derivtive of f( g( x) ) This is the hin rule You re finding f ( g( x) ) g ( x) C11 The line y =mx+ is tngent to Two reltionships re true: the grph of f( x) t ( x 1,y 1 ) 1) The funtion f nd the line shre the sme slope t x 1 : m = f ( x 1 ) ) The funtion f nd the line shre the sme y-vlue t x 1

3 C1 C13 Find the derivtive of the inverse to t x = Given pieewise funtion, show it is differentile t x = where the funtion rule splits This is wht you think of doing Follow this proedure: 1) Interhnge x nd y in f( x) ) Plug the x-vlue into this eqution nd solve for y (you my need lultor to solve grphilly) 3) Using the eqution in 1) find dy impliitly 4) Plug the y-vlue you found in ) to dy First, e sure tht derivtive of eh piee nd show tht lim f x x is ontinuous t x = Then tke the = lim f x x + D Applitions of Derivtives D1 D D3 D4 D5 D6 D7 D8 This is wht you think of doing Find ritil vlues of f( x) Find nd express f ( x) s frtion Set oth numertor nd denomintor equl to zero nd solve Find the intervl(s) where f( x) is Find ritil vlues of f ( x) Mke sign hrt to find sign inresing/deresing of f ( x) in the intervls ounded y ritil vlues Positive mens inresing, negtive mens deresing Find points of reltive extrem of Mke sign hrt of f ( x) At x = where the derivtive f( x) swithes from negtive to positive, there is reltive minimum When the derivtive swithes from positive to negtive, there is reltive mximum To tully find the point, evlute f( ) OR if f ( ) =, then if f ( ) >, there is reltive minimum t x = If f ( ) <, there is reltive mximum t x = ( nd Derivtive test) Find infletion points of f( x) Find nd express f ( x) s frtion Set oth numertor nd denomintor equl to zero nd solve Mke sign hrt of f ( x) Infletion points our when f ( x) withes from Find the solute mximum or minimum of on [, ] Find rnge of f( x) on, Find rnge of positive to negtive or negtive to positive Use reltive extrem tehniques to find reltive mx/mins Evlute f t these vlues Then exmine f nd f The lrgest of these is the solute mximum nd the smllest of these is the solute minimum Use reltive extrem tehniques to find reltive mx/mins Evlute f t these vlues Then exmine f nd f Then exmine limf( x) nd lim f( x) x x on [, ] Use reltive extrem tehniques to find reltive mx/mins Evlute f t these vlues Then exmine f nd f Then exmine f nd f Show tht Rolle s Theorem holds for on [, ] Show tht f is ontinuous nd differentile on [, ] If f= f, then find some on [, ] suh tht f = wwwmstermthmentorom Stu Shwrtz

4 D9 Show tht the Men Vlue Theorem holds for on [, ] D1 Given grph of f ( x), determine intervls where f( x) is inresing/deresing D11 Determine whether the liner pproximtion for f( x 1 +) overestimtes or under-estimtes f( x 1 +) D1 D13 Find intervls where the slope of f( x) is inresing Find the minimum slope of f( x) on [, ] Show tht f is ontinuous nd differentile on [, ] If f= f, then find some on [, ] suh tht f ( ) = f f Mke sign hrt of f ( x) nd determine the intervls where f x is positive nd negtive Find slope m = f ( x i ) Then use point slope eqution: yy 1 =m( xx 1 ) Evlute this line for y t x =x 1 + If f ( x 1 ) >, f is onve up t x 1 nd the liner pproximtion is n underestimtion for f( x 1 +) f ( x 1 ) <, f is onve down t x 1 nd the liner pproximtion is n overestimtion for f( x 1 +) Find the derivtive of f ( x) whih is f ( x) Find ritil vlues of f ( x) nd mke sign hrt of f x positive intervls Find the derivtive of f ( x) whih is f ( x) Find ritil vlues of f ( x) nd mke sign hrt of f ( x) Vlues of x where f ( x) swithes from negtive to positive re potentil lotions for the minimum slope Evlute f ( x) t those vlues nd lso f nd f nd hoose the lest of these vlues looking for E Integrl Clulus E1 E E3 E4 E5 Approximte using left Riemnn sums with n retngles Approximte using right Riemnn sums with n retngles Approximte midpoint Riemnn sums Approximte trpezoidl summtion Find using using f( x) where < This is wht you think of doing A = f ( x n ) + f ( x 1 ) + A = n 1 [ ++ f ( x n1 )] [ + f ( x ) + f ( x 3 ) ++ f ( x n )] Typilly done with tle of points Be sure to use only vlues tht re given If you re given 7 points, you n only lulte 3 midpoint retngles A = [ f ( x n ) +f ( x 1 ) +f ( x ) ++f ( x n1 ) + f ( x n )] This formul only works when the se of eh trpezoid is the sme If not, lulte the res of individul trpezoids f( x) = f( x) wwwmstermthmentorom Stu Shwrtz

5 E6 E7 E8 E9 E1 E11 E1 Mening of Given x f( t) f( x), find [ f( x)+k] where the Given the vlue of F ntiderivtive of f is F, find F Find d Find d Find Find x f( t) g( x) f( t) f( x) This is wht you think of doing The umultion funtion umulted re under funtion f strting t some onstnt nd ending t some vrile x [ f( x)+k] = f( x) + k Use the ft tht f( x) =FF so F=F+ f( x) Use the lultor to find the definite integrl x d f( t) = f( x) The nd Fundmentl Theorem d g( x) f ( t) = f ( g( x) ) g ( x) The nd Fundmentl Theorem f( x) =lim h = limf h h F( ) f( x) g( x) If u-sustitution doesn t work, try integrtion y prts: udv=uv vdu h F Applitions of Integrl Clulus F1 F F3 F4 F5 This is wht you think of doing Find the re under the urve f( x) on f( x) the intervl [, ] Find the re etween f( x) nd g( x) Find the intersetions, nd of Find the line x = tht divides the re under f( x) on [, ] into two equl res Find the volume when the re under f( x) is rotted out the x-xis on the intervl [, ] Find the volume when the re etween f( x) nd g x the x-xis is rotted out f( x)g( x) on, = nd g x If [ ], then re A= [ f( x)g( x) ] f( x) or f( x) = Disks: Rdius = f( x): V = [ ] Wshers: Outside rdius = f( x) Inside rdius = g( x) Estlish the intervl where f( x) g x nd, where f( x)=g( x) V = nd the vlues of ([ ] [ g ( x ) ] ) wwwmstermthmentorom Stu Shwrtz

6 This is wht you think of doing F6 Given se ounded y Bse = f( x)g( x) Are = se = [ f( x)g( x) ] f( x) nd g( x) on [, ] the ross setions of the solid perpendiulr to Volume = [ f( x)g( x) ] the x-xis re squres Find the volume F7 Solve the differentil eqution Seprte the vriles: x on one side, y on the other with the F8 dy = f ( x nd dy in the numertors Then integrte oth sides, )g( y) rememering the +C, usully on the x-side Find the verge vlue of f( x) on [, ] f( x) F vg = t F9 Find the verge rte of hnge of d F ( x) on [, ] F ( x) = ( ) F ( ) F1 y is inresing proportionlly to y dy =ky whih trnsltes to y=cekt F11 Given dy, drw slope field dy Use the given points nd plug them into, drwing little lines with the lulted slopes t the point F1 Ftor x Find +x+ into non-repeting ftors to get x +x+ ( mx+n) ( px+q) prtil frtions nd integrte eh frtion F13 Use Euler s method to pproximte f( 1) given formul for dy= dy ( x ), y new =y old +dy dy, ( x,y ) nd x= 1 F14 Is the Euler s pproximtion n over- or under-pproximtion? Look t sign of dy nd d y in the intervl This gives inresing/deresing nd onvity informtion Drw piture to sertin the nswer F15 F16 F17 F18 A popultion P is inresing logistilly Find the rrying pity of popultion growing logistilly Find the vlue of P when popultion growing logistilly is growing the fstest Given ontinuous f( x), find the r length on [, ] dp dp dp =kp( CP) =kp( CP)=C=P =kp ( CP ) Set d P = [ ] L = 1+ f x wwwmstermthmentorom Stu Shwrtz

7 G Prtile Motion nd Rtes of Chnge G1 G G3 G4 G5 G6 G7 G8 G9 G1 G11 G1 Given the position funtion s( t) of prtile moving long stright line, find the veloity nd elertion Given the veloity funtion v( t) nd s( ), find s( t) Given the elertion funtion t of prtile t rest nd s, find s( t) Given the veloity funtion v( t), determine if prtile is speeding up or slowing down t t = k Given the position funtion s( t), find the verge veloity on [, ] Given the position funtion s( t), find the instntneous veloity t t=k Given the veloity funtion v( t) on [, ], find the minimum elertion of prtile Given the veloity funtion v( t), find the verge veloity on [, ] Given the veloity funtion v( t), determine the differene of position of prtile on [, ] Given the veloity funtion v( t), determine the distne prtile trvels on [, ] Clulte lultor v t without nd Given the veloity funtion v t s, find the gretest distne of the prtile from the strting position on [, ] v t This is wht you think of doing = s ( t) ( t) = v ( t) = s ( t) s( t)= v( t) +C Plug in s to find C v( t)= ( t) +C 1 Plug in v( )= to find C 1 s( t)= v( t) +C Plug in s( ) to find C Find v( k) nd ( k) If oth hve the sme sign, the prtile is speeding up If they hve different signs, the prtile is slowing down Avg vel = s ( )s Inst vel = s ( k) nd set = Set up sign hrt nd find ritil Find t t vlues Evlute the elertion t ritil vlues nd lso nd to find the minimum Avg vel = v( t) Displement = v t Distne = v t Set v( t)= nd mke sign hrge of v( t)= on [, ] On intervls [, ] where v( t)>, v t On intervls [, ] where v( t)<, v t Generte sign hrt of v t s( t)= v( t) +C Plug in s Evlute s t mximum distne from s = v t = v t to find turning points to find C t ll turning points nd find whih one gives the wwwmstermthmentorom Stu Shwrtz

8 G13 G14 G15 The volume of solid is hnging t the rte of The mening of R ( t ) Given wter tnk with g gllons initilly, filled t the rte of F( t) gllons/min nd emptied t the rte of E( t) gllons/min on [, ] ) The mount of wter in the tnk t t = m minutes ) the rte the wter mount is hnging t t = m minutes nd ) the time t when the wter in the tnk is t minimum or mximum This is wht you think of doing dv = This gives the umulted hnge of R t R ( t ) = R R m [ ] ) g+ F( t)e t m on [, ] or R = R + R ( t) ) d [ F( t)e( t) ] =F( m)e( m) ) set F( m)e( m)=, solve for m, nd evlute m [ ] g+ F( t)e t t vlues of m nd lso the endpoints H Prmetri nd Polr Equtions - H1 H H3 H4 H5 H6 H7 H8 H9 Given x= f( t),y=g( t), find dy dy dy = This is wht you think of doing Given x= f( t),y=g( t), find d y d dy x = f ( t),y =g( t), find d y = d dy = Given x= f( t),y=g( t), find r length on [, ] L = + dy Express polr eqution in the form x=ros= fos y=rsin= fsin of r= f in prmetri form Find the slope of the tngent line tor= f x=ros y=rsin dy dy = d d Find horizontl tngents to polr x=ros y=rsin urve r= f Find where rsin= when ros Find vertil tngents to polr x=ros y=rsin urve r= f Find where ros= when rsin Find the re ounded y the polr urve r= f on [ 1, ] A= 1 1 r d= [ f ] d Find the r length of the polr urve r= f on [ 1, ] s = [ f ] + [ f ] d = r + dr d d wwwmstermthmentorom Stu Shwrtz

9 I Vetors nd Vetor-vlued funtions - I1 Find the mgnitude of vetor v v 1,v This is wht you think of doing v = v 1 +v I Find the dot produt: u 1,u v 1,v u 1,u v 1,v =u 1 v 1 +u v I3 The position vetor of prtile moving in the plne is ) v( t) = x ( t), y ( t) r( t)= x( t),y( t) Find ) the ) ( t) = x ( t), y ( t) veloity vetor nd ) the elertion vetor I4 The position vetor of prtile moving in the plne is Speed = v( t) = [ x ( t) ] + y t r( t)= x( t),y( t) Find the speed of the prtile t time t I5 Given the veloity vetor s( t)= x( t) + y( t) +C v( t)= x( t),y( t) nd position t Use s time t =, find the position vetor I6 Given the veloity vetor v( t)= x( t),y( t), when does the prtile stop? I7 The position vetor of prtile moving in the plne is Distne = [ x ( t) ] + y ( t) r( t)= x( t),y( t) Find the distne the prtile trvels from to J Tylor Polynomil Approximtions - [ ] - slr to find C, rememering tht it is vetor v( t)=x( t)= AND y( t)= [ ] J1 J J3 Find the nth degree Mlurin polynomil to f( x) Find the nth degree Tylor polynomil to x = entered t Use the first-degree Tylor polynomil to f( x) entered t x = to pproximte f( k) nd determine whether the pproximtion is greter thn or less thn f( k) P n P n This is wht you think of doing ( x) = f ( ) + f ( )x + f ( ) x +! f ( ) 3! ( x) = f x f n n! + f ( ) ( x ) + ( x ) 3 x n f x f + + f ( n) 3! n! Write the first-degree TP nd find f( k) Use the signs of f f! ( x ) n nd to determine inresing/deresing nd onvity nd drw your line (1 st degree TP) to determine whether the line is under the urve (under-pproximtion) or over the urve (over-pproximtion) + wwwmstermthmentorom Stu Shwrtz

10 J4 J5 J6 J7 Given n nth degree Tylor polynomil for f out x =, find f ( ), f ( ), f ( ),, f ( n) ( ) Given Tylor polynomil entered t, determine if there is enough informtion to determine if there is reltive mximum or minimum t x = Given n nth degree Tylor polynomil for f out x =, find the Lgrnge error ound (reminder) Given n nth degree Mlurin polynomil P for f, find the f( k)p( k) f f f This is wht you think of doing will e the onstnt term in your Tylor polynomil (TP) will e the oeffiient of the x term in the TP! f ( n) will e the oeffiient of the x term in the TP will e the oeffiient of the x n term in the TP n! If there is no first-degree x-term in the TP, then the vlue of out whih the funtion is entered is ritil vlue Thus the oeffiient of the x term is the seond derivtive divided y! Using the seond derivtive test, we n tell whether there is reltive mximum, minimum, or neither t x = R n ( x)= f ( n+1 ) ( z) n+1 x The vlue of z is some numer ( n+1)! z represents the ( n+1) st derivtive of etween x nd f n+1 z This usully is given to you This is looking for the Lgrnge error the differene etween the vlue of the funtion t x=k nd the vlue of the TP t x=k K Infinite Series - K1 K K3 K4 K5 Given n, determine whether the sequene n onverges Given n, determine whether the series n ould onverge Determine whether series onverges Find the sum of geometri series Find the intervl of onvergene of series This is wht you think of doing n onverges if lim n exists n If lim n =, the series ould onverge If lim n, the n n series nnot onverge (nth term test) Exmine the nth term of the series Assuming it psses the nth term test, the most widely used series forms nd their rule of onvergene re: Geometri: r n - onverges if r <1 n= 1 p-series: - onverges if p > 1 n p n=1 Alternting: ( 1) n n - onverges if < n+1 < n Rtio: n=1 n - onverges if lim n+1 n n= n <1 r n = 1r n= If not given, you will hve to generte the nth term formul Use test (usully the rtio test) to find the intervl of onvergene nd then hek out the endpoints wwwmstermthmentorom - - Stu Shwrtz

11 K6 K7 K8 K9 K n f( x)=1+x+ x + x3 xn ++ 3! n! + f( x)=e x f( x)=x x3 3! + x5 x7 5! f( x)=1 x! + x4 x6 4! f( x)=1+x+x +x 3 ++x n + K11 Given formul for the nth derivtive of f( x) Write the first four terms nd the generl term for the power series for f( x) entered t x = K1 Let S 4 e the sum of the first 4 terms of n lternting series for f( x) Approximte f( x)s 4 K13 Write series for expressions like e x 7! + f( x)=sinx 6! + f( x)=osx This is wht you think of doing The hrmoni series divergent f( x)= 1 1x Convergent : ( 1,1) f ( x) = f ( ) + f ( ) ( x ) + ( x ) 3 f 3! f + + f ( n) x! ( x ) n n! + + This is the error for the 4 th term of n lternting series whih is simply the 5 th tern It will e positive sine you re looking for n solute vlue Rther thn go through generting Tylor polynomil, use the ft tht if f( x)=e x, then f( x )=e x So f( x)=e x =1+x+ x + x 3 3! + x 4 xn nd 4! n! f( x )=e x =1+x + x 4 + x 6 3! + x 8 n x + + 4! n! + wwwmstermthmentorom Stu Shwrtz