AB Clculus Exm Review Sheet A. Preclculus Type prolems A1 Find the zeros of f ( x). This is wht you think of doing A2 A3 Find the intersection of f ( x) nd g( x). Show tht f ( x) is even. A4 Show tht f ( x) is odd. A5 Find domin of f ( x). A6 Find verticl symptotes of f ( x). A7 If continuous function f ( x) hs f ( ) < k nd f ( ) > k, explin why there must e vlue c such tht < c < nd f ( c) = k. B. Limit Prolems B1 Find lim f ( x). x " This is wht you think of doing B2 B3 Find lim f x x " piecewise function. ( ) where f ( x ) is Show tht f ( x) is continuous. B4 Find lim f ( x) nd lim f ( x). x "# x "$# B5 Find horizontl symptotes of f ( x). www.mstermthmentor.com - 1 - Stu Schwrtz
C. Derivtives, differentiility, nd tngent lines C1 C2 C3 C4 C5 C6 C7 C8 C9 C10 Find the derivtive of function using the derivtive definition. Find the verge rte of chnge of f on [, ]. Find the instntneous rte of chnge of f t x =. Given chrt of x nd f ( x) nd selected vlues of x etween nd, pproximte f "( c) where c 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 ( ). Find x-vlues of horizontl tngents to f. Find x-vlues of verticl tngents to f. Approximte the vlue of f ( x 1 + ) if you know the function goes through point x 1, y 1 ( ). Find the derivtive of f ( g( x) ). This is wht you think of doing C11 C12 C13 The line y = mx + is tngent to the grph of f ( x) t ( x 1,y 1 ). Find the derivtive of the inverse to f x ( ) t x =. Given piecewise function, show it is differentile t x = where the function rule splits. www.mstermthmentor.com - 2 - Stu Schwrtz
D. Applictions of Derivtives D1 Find criticl vlues of f ( x). This is wht you think of doing D2 D3 D4 Find the intervl(s) where f ( x) is incresing/decresing. Find points of reltive extrem of f x ( ). Find inflection points of f ( x). D5 D6 Find the solute mximum or minimum of f x ( ) on [, ]. Find rnge of f ( x) on ("#,#). D7 Find rnge of f ( x) on [, ] D8 D9 Show tht Rolle s Theorem holds for f x ( ) on [, ]. Show tht the Men Vlue Theorem holds for f x ( ) on [, ]. D10 Given grph of f "( x), determine intervls where f ( x) is incresing/decresing. D11 Determine whether the liner pproximtion for f ( x 1 + ) overestimtes or under-estimtes f ( x 1 + ). D12 Find intervls where the slope of f ( x) is incresing. D13 Find the minimum slope of f ( x) on [, ]. www.mstermthmentor.com - 3 - Stu Schwrtz
E. Integrl Clculus E1 E2 E3 E4 E5 E8 E9 E10 E11 E12 Approximte " f ( x) using left Riemnn sums with n rectngles. Approximte " f ( x) using right Riemnn sums with n rectngles. Approximte Riemnn sums. Approximte " f ( x) using midpoint " f ( x) using trpezoidl summtion. Find " f ( x) where <. Mening of Given x dt. " f ( x), find " [ f ( x) + k]. Given the vlue of F ntiderivtive of f is F, find F( ). Find d Find d x dt. g( x) dt. ( ) where the This is wht you think of doing F. Applictions of Integrl Clculus F1 F2 Find the re under the curve f x the intervl [, ]. ( ) on Find the re etween f ( x) nd g( x). This is wht you think of doing F3 Find the line x = c tht divides the re under f ( x) on [, ] into two equl res. www.mstermthmentor.com - 4 - Stu Schwrtz
F4 F5 F6 F7 F8 F9 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) is rotted out the x-xis. Given se ounded y f ( x) nd g( x) on [, ] the cross sections of the solid perpendiculr to the x-xis re squres. Find the volume. Solve the differentil eqution dy = f ( x )g( y). Find the verge vlue of f ( x) on [, ]. Find the verge rte of chnge of F " x [ ]. ( ) on, This is wht you think of doing F10 y is incresing proportionlly to y. F11 Given dy, drw slope field. G. Prticle Motion nd Rtes of Chnge G1 G2 G3 G4 G5 G6 Given the position function s t prticle moving long stright line, find the velocity nd ccelertion. Given the velocity function v( t) nd s( 0), find s( t). ( ) of Given the ccelertion function t prticle t rest nd s( 0), find s( t). ( ) of determine if prticle is speeding up or slowing down t t = k. Given the position function s( t), find the verge velocity on [, ]. Given the position function s t ( ), find the instntneous velocity t t = k. This is wht you think of doing www.mstermthmentor.com - 5 - Stu Schwrtz
G7 G8 G9 G10 G11 G12 G13 G14 G15 Given the velocity function v( t) on [, ], find the minimum ccelertion of prticle. find the verge velocity on [, ]. determine the difference of position of prticle on [, ]. determine the distnce prticle trvels on [, ]. Clculte " ( ) dt v t without clcultor. Given the velocity function v( t) nd s( 0), find the gretest distnce of the prticle from the strting position on [ 0, ]. The volume of solid is chnging t the rte of The mening of # R "( t) dt. 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 chnging t t = m minutes nd c) the time t when the wter in the tnk is t minimum or mximum. This is wht you think of doing www.mstermthmentor.com - 6 - Stu Schwrtz
AB Clculus Exm Review Sheet - Solutions A. Preclculus Type prolems A1 A2 A3 A4 A5 A6 A7 This is wht you think of doing Find the zeros of f ( x). Set function equl to 0. Fctor or use qudrtic eqution if qudrtic. Grph to find zeros on clcultor. Find the intersection of Set the two functions equl to ech other. Find intersection on f ( x) nd g( x). clcultor. Show tht f ( x) is even. Show tht f ("x) = f ( x). This shows tht the grph of f is symmetric to the y-xis. Show tht f ( x) is odd. Show tht f ("x) = " f ( x). This shows tht the grph of f is symmetric to the origin. Find domin of f ( x). Assume domin is ("#,#). Restrict domins: denomintors " 0, squre roots of only non-negtive numers, logrithm or nturl log of only positive numers. Find verticl symptotes of f ( x). Express f x in fctored form, nd do ny cncelltions. Set denomintor equl to 0. If continuous function f ( x) hs This is the Intermedite Vlue Theorem. f ( ) < k nd f ( ) > k, explin why there must e vlue c such tht < c < nd f ( c) = k. ( ) s frction, express numertor nd denomintor B. Limit Prolems B1 B2 B3 B4 B5 This is wht you think of doing Find lim f ( x ). Step 1: Find f ( ). If you get zero in the denomintor, x " Step 2: Fctor numertor nd denomintor of f ( x). Do ny cncelltions nd go ck to Step 1. If you still get zero in the denomintor, the nswer is either, -, or does not exist. Check the signs of lim f ( x ) nd lim f ( x) for equlity. x " # x " + Find lim f x x " piecewise function. Show tht f x ( ) where f ( x) is ( ) = lim ( ) y plugging in to Determine if lim f x f x x " # x " + f ( x),x < nd f ( x),x > for equlity. If they re not equl, the limit doesn t exist. f x ( ) is continuous. Show th) lim ( ) exists x " 2) f ( ) exists 3) lim f ( x) = f ( ) x " ( ) s frction. Determine loction of the highest Find lim f ( x) nd lim f x x "# x "$# Find horizontl symptotes of f ( x). lim f x x "# ( ). Express f x power: Denomintor: lim f ( x) = lim f ( x) = 0 x "# x "$# Both Num nd Denom: rtio of the highest power coefficients Numertor: lim f x x "# ( ) = ±# (plug in lrge numer) ( ) nd lim f x x "$# ( ) www.mstermthmentor.com - 7 - Stu Schwrtz
C. Derivtives, differentiility, nd tngent lines C1 C2 C3 C4 C5 C6 C7 C8 C9 C10 C11 C12 C13 Find the derivtive of function using the derivtive definition. Find the verge rte of chnge of f on [, ]. Find the instntneous rte of chnge of f t x =. Given chrt of x nd f ( x) nd selected vlues of x etween nd, pproximte f "( c) where c 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 ( ). Find x-vlues of horizontl tngents to f. Find x-vlues of verticl tngents to f. Approximte the vlue of f x 1 + if you know the function goes through point ( x 1, y 1 ). ( ) This is wht you think of doing f ( x + h) # f ( x) f x Find lim or lim h " 0 h x " x # Find f ( ) " f ( ) " Find f " ( ) ( ) # f ( ) Strddle c, using vlue of k c nd vlue of h c. f " c ( ) # f ( k) $ f ( h) k $ h Find slope m = f "( x i ). Then use point slope eqution: y " y 1 = m( x " x 1 ) Find slope m"= #1. Then use point slope eqution: f $ x i ( ) ( ) ( ) s frction. Set numertor of f "( x) = 0. y " y 1 = m x " x 1 Write f " x Write f " x ( ) s frction. Set denomintor of f "( x) = 0. Find slope m = f "( x i ). Then use point slope eqution: y " y 1 = m( x " x 1 ). Evlute this line for y t x = x 1 +. Note: The closer is to 0, the etter the pproximtion will e. Also note tht using concvity, it cn e determine if this vlue is n over or under-pproximtion for f ( x 1 + ). f ( g( x) )# g "( x). Two reltionships re true: 1) The function f nd the line shre the sme slope t x 1 : m = f "( x 1 ) 2) The function f nd the line shre the sme y-vlue t x 1. Follow this procedure: 1) Interchnge x nd y in f ( x). 2) Plug the x-vlue into this eqution nd solve for y (you my need clcultor to solve grphiclly) 3) Using the eqution in 1) find dy implicitly. Find the derivtive of f ( g( x) ). This is the chin rule. You re finding " The line y = mx + is tngent to the grph of f ( x) t ( x 1,y 1 ). Find the derivtive of the inverse to f x ( ) t x =. Given piecewise function, show it is differentile t x = where the function rule splits. 4) Plug the y-vlue you found in 2) to dy First, e sure tht f x derivtive of ech piece nd show tht lim f x ( ) is continuous t x =. Then tke the ( ) = lim f x x " # $ x " + $ ( ). www.mstermthmentor.com - 8 - Stu Schwrtz
D. Applictions of Derivtives D1 D2 D3 D4 D5 D6 D7 D8 D9 This is wht you think of doing Find criticl vlues of f ( x). Find nd express f "( x) s frction. Set oth numertor nd denomintor equl to zero nd solve. Find the intervl(s) where f ( x) is Find criticl vlues of f "( x). Mke sign chrt to find sign incresing/decresing. of f "( x) in the intervls ounded y criticl vlues. Positive mens incresing, negtive mens decresing. Find points of reltive extrem of Mke sign chrt of f "( x). At x = c where the derivtive f ( x). switches from negtive to positive, there is reltive minimum. When the derivtive switches from positive to negtive, there is reltive mximum. To ctully find the point, evlute f ( c). OR if f "( c) = 0, then if f " ( c) > 0, there is reltive minimum t x = c. If f " ( c) < 0, there is reltive mximum t x = c. (2 nd Derivtive test). Find inflection points of f ( x). Find nd express f " ( x) s frction. Set oth numertor nd denomintor equl to zero nd solve. Mke sign chrt of f " ( x). Inflection points occur when f " ( x) witches from Find the solute mximum or minimum of f x ( ) on [, ]. Find rnge of f ( x) on "#,# Find rnge of f x positive to negtive or negtive to positive. Use reltive extrem techniques 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 techniques to find reltive mx/mins. Evlute f t these vlues. Then exmine f ( ) nd f ( ). Then exmine lim f ( x ) nd lim f ( x). x "# x "$# ( ) on [, ] Use reltive extrem techniques 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 f x ( ) on [, ]. Show tht the Men Vlue Theorem holds for f ( x) on [, ]. D10 Given grph of f "( x), determine intervls where f ( x) is incresing/decresing. D11 Determine whether the liner pproximtion for f ( x 1 + ) overestimtes or under-estimtes f ( x 1 + ). Show tht f is continuous nd differentile on [, ]. If f ( ) = f ( ), then find some c on [, ] such tht f " c Show tht f is continuous nd differentile on [, ]. If f ( ) = f ( ), then find some c on [, ] such tht ( ) = f ( ) # f ( ) f " c # Mke sign chrt of f " x where f " x ( ) nd determine the intervls ( ) is positive nd negtive. ( ) = 0. Find slope m = f "( x i ). Then use point slope eqution: y " y 1 = m( x " x 1 ). Evlute this line for y t x = x 1 +. If f " ( x 1 ) > 0, f is concve up t x 1 nd the liner pproximtion is n underestimtion for f ( x 1 + ). f " ( x 1 ) < 0, f is concve down t x 1 nd the liner pproximtion is n overestimtion for f ( x 1 + ). www.mstermthmentor.com - 9 - Stu Schwrtz
D12 D13 Find intervls where the slope of f ( x) is incresing. Find the minimum slope of f ( x) on [, ]. Find the derivtive of f "( x) which is f " ( x). Find criticl vlues of f " ( x) nd mke sign chrt of f " x positive intervls. Find the derivtive of f "( x) which is f " ( x). Find criticl vlues of f " ( x) nd mke sign chrt of f " ( x). Vlues of x where f " ( x) switches from negtive to positive re potentil loctions for the minimum slope. Evlute f "( x) t those vlues nd lso f "( ) nd f "( ) nd choose the lest of these vlues. ( ) looking for E. Integrl Clculus E1 E2 E3 E4 E5 E8 E9 E10 E11 E12 Approximte " f ( x) using left Riemnn sums with n rectngles. Approximte " f ( x) using right Riemnn sums with n rectngles. Approximte " f ( x) using midpoint Riemnn sums. Approximte " f ( x) using trpezoidl summtion. Find This is wht you think of doing # A = " & % ( f ( x $ n ' 0 ) + f ( x 1 ) + f x 2 # A = " & % ( f x $ n ' 1 " f ( x) where <. " f ( x) = # " f ( x) Mening of Given x dt. " f ( x), find " [ f ( x) + k]. ( ) where the Given the vlue of F ntiderivtive of f is F, find F( ). Find d Find d x dt. g( x) dt. [ ( ) +...+ f ( x n"1 )] [ ( ) + f ( x 2 ) + f ( x 3 ) +...+ f ( x n )] Typiclly done with tle of points. Be sure to use only vlues tht re given. If you re given 7 points, you cn only clculte 3 midpoint rectngles. # A = " & % ([ f ( x $ 2n ' 0 ) + 2 f ( x 1 ) + 2 f ( x 2 ) +...+ 2 f ( x n"1 ) + f ( x n )] This formul only works when the se of ech trpezoid is the sme. If not, clculte the res of individul trpezoids. The ccumultion function ccumulted re under function f strting t some constnt nd ending t some vrile x. " [ f ( x) + k] = " f ( x) + " k Use the fct tht F( ) = F( ) + f ( x) " f ( x) = F ( ) # F( ) so ". Use the clcultor to find the definite integrl. x d dt = f ( x). The 2nd Fundmentl Theorem. d g( x) dt = f g x ( ( )) # $ g ( x). The 2nd Fundmentl Theorem. www.mstermthmentor.com - 10 - Stu Schwrtz
F. Applictions of Integrl Clculus F1 F2 F3 F4 F5 This is wht you think of doing Find the re under the curve f ( x) on f ( x) the intervl [, ]. " Find the re etween f ( x) nd g( x). Find the intersections, nd of f x Find the line x = c 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, c ( ) nd g x ( ). If [ ], then re A = # [ f ( x) " g( x) ]. " f ( x) = " f ( x) or " f ( x) = 2 " f x c Disks: Rdius = f ( x): V = " f x # c [ ( )] 2 ( ) Wshers: Outside rdius = f ( x). Inside rdius = g( x). Estlish the intervl where f x nd, where f ( x) = g x ( ) " g( x) nd the vlues of $ ([ ( )] 2 # [ g ( x ) ] 2 ) ( ). V = " f x [ ( ) " g( x) ] 2. F6 Given se ounded y Bse = f ( x) " g( x). Are = se 2 = f x f ( x) nd g( x) on [, ] the cross sections of the solid perpendiculr to Volume = # [ f ( x) " g( x) ] 2 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 chnge of d 2 F "( x) on [, ]. # F "( x) dt = " ( ) $ F "( ) $ $ F10 y is incresing proportionlly to y. dy = ky which trnsltes to y = Cekt dt F11 Given dy, drw slope field. dy Use the given points nd plug them into, drwing little lines with the clculted slopes t the point. www.mstermthmentor.com - 11 - Stu Schwrtz
G. Prticle Motion nd Rtes of Chnge G1 G2 G3 G4 G5 G6 G7 G8 G9 G10 G11 G12 Given the position function s( t) of prticle moving long stright line, find the velocity nd ccelertion. Given the velocity function v( t) nd s( 0), find s( t). Given the ccelertion function t of prticle t rest nd s( 0), find s( t). determine if prticle is speeding up or slowing down t t = k. Given the position function s( t), find the verge velocity on [, ]. Given the position function s( t), find the instntneous velocity t t = k. Given the velocity function v( t) on [, ], find the minimum ccelertion of prticle. find the verge velocity on [, ]. ( ) determine the difference of position of prticle on [, ]. determine the distnce prticle trvels on [, ]. Clculte clcultor. " ( ) dt v t without ( ) nd Given the velocity function v t s( 0), find the gretest distnce of the prticle from the strting position on [ 0, ]. v t This is wht you think of doing ( ) = s "( t) ( t) = v "( t) = s " ( t) ( ) = v( t) dt s t v t " + C. Plug in s 0 " + C 1. Plug in v 0 " ( ) dt + C 2. Plug in s 0 ( ) = ( t) dt ( ) = v t s t ( ) to find C. ( ) = 0 to find C 1. ( ) to find C 2. Find v( k) nd ( k). If oth hve the sme sign, the prticle is speeding up. If they hve different signs, the prticle is slowing down. ( ) " s( ) Avg. vel. = s " Inst. vel. = s "( k). ( ) nd set "( ) = 0. Set up sign chrt nd find criticl Find t t vlues. Evlute the ccelertion t criticl vlues nd lso nd to find the minimum. Avg. vel. = " Displcement = Distnce = " v( t) dt # " v( t) dt v( t) dt Set v( t) = 0 nd mke sign chrge of v( t) = 0 on, intervls [, ] where v t On intervls [, ] where v t " = " v t ( ) > 0, v( t) dt Generte sign chrt of v t s( t) = " v( t) dt + C. Plug in s 0 Evlute s t mximum distnce from s( 0). ( ) < 0, v( t) dt ( ) dt " = " v t [ ]. On ( ) dt ( ) to find turning points. ( ) to find C. ( ) t ll turning points nd find which one gives the www.mstermthmentor.com - 12 - Stu Schwrtz
G13 G14 G15 The volume of solid is chnging t the rte of The mening of # R "( t) dt. 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 chnging t t = m minutes nd c) the time t when the wter in the tnk is t minimum or mximum. This is wht you think of doing dv dt =... This gives the ccumulted chnge of R t # R "( t ) dt = R m ) g + # [ F( t) " E( t) ] dt ) d dt 0 m ( ) on [, ]. ( ) $ R( ) or R( ) = R( ) + R "( t) # [ F( t) " E( t) ] dt = F m 0 c) set F m m ( ) " E( m) # dt ( ) " E( m) = 0, solve for m, nd evlute g + # [ F( t) " E( t) ] dt t vlues of m nd lso the endpoints. 0 www.mstermthmentor.com - 13 - Stu Schwrtz