AB Calculus Review Sheet

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1 AB Clculus Review Sheet Legend: A Preclculus, B Limits, C Differentil Clculus, D Applictions of Differentil Clculus, E Integrl Clculus, F Applictions of Integrl Clculus, G Prticle Motion nd Rtes This is wht you think of doing 1 (B) Show tht f ( x) is continuous. 2 Find the eqution of the tngent line to f t ( x 1, y 1 ). 3 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. 4 Find points of reltive extrem of f ( x). 5 Find rnge of f ( x) on [, ] 6 Given the position function s( t), find the verge velocity on [ t 1 7 (E) Approximte " f ( x) using left Riemnn sums with n rectngles. 8 Given grph of f "( x), determine intervls where f ( x) is incresing/decresing. 9 Given piecewise function, show it is differentile t x = where the function rule splits. 10 Find criticl vlues of f ( x). 11 Find the zeros of f ( x) Stu Schwrtz

2 Legend: A Preclculus, B Limits, C Differentil Clculus, D Applictions of Differentil Clculus, E Integrl Clculus, F Applictions of Integrl Clculus, G Prticle Motion nd Rtes This is wht you think of doing 12 (B) Find lim f x x " ( ). Find x-vlues of horizontl tngents to f. Show tht Rolle s Theorem holds for f x ( ) on [, ]. Find inflection points of f ( x). 16 (F) Find the re etween f ( x) nd g( x). 17 (F) Given dy, drw slope field. 18 determine the difference of position of prticle on [ t 1 19 (F) Find the volume when the re under f ( x) is rotted out the x- xis on the intervl [, ]. 20 Find the verge rte of chnge of f on [, ] (B) Find the derivtive of f ( g( x) ). Find the derivtive of function using the derivtive definition. Show tht f ( x) is odd. Find lim f ( x) nd lim f ( x). x "# x "$# Stu Schwrtz

3 Legend: A Preclculus, B Limits, C Differentil Clculus, D Applictions of Differentil Clculus, E Integrl Clculus, F Applictions of Integrl Clculus, G Prticle Motion nd Rtes Given the position function s( t), find the instntneous velocity t t = k. Given the velocity function v( t) nd s( 0), find s( t). This is wht you think of doing 27 (F) Find the verge vlue of f ( x) on [, ]. 28 (E) 29 (E) Mening of Find d x x dt. dt. Find x-vlues of horizontl tngents to f. Find domin of f ( x). 32 Given chrt of x nd f ( x) nd selected vlues of x etween nd, pproximte f "( c) where c is vlue etween nd. 33 Given the velocity function v( t) nd s( 0), find the gretest distnce of the prticle from the strting position on [ 0,t 1 ]. 34 (F) y is incresing proportionlly to y. 35 (E) Given " f ( x), find " [ f ( x) + k]. Find the eqution of the norml line to f t x 1, y 1 ( ). Show tht the Men Vlue Theorem holds for f x ( ) on [, ] Stu Schwrtz

4 Legend: A Preclculus, B Limits, C Differentil Clculus, D Applictions of Differentil Clculus, E Integrl Clculus, F Applictions of Integrl Clculus, G Prticle Motion nd Rtes Find rnge of f ( x) on ("#,#). Find the instntneous rte of chnge of f t x =. This is wht you think of doing 40 (F) Find the line x = c tht divides the re under f x equl res. 41 (E) Approximte " f ( x) using ( ) on [, ] into two trpezoidl summtion. 42 (F) Solve the differentil eqution dy = f ( x )g( y). 43 (F) 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. 44 (E) Approximte " f ( x) using 45 midpoint Riemnn sums. Find the intervl(s) where f x incresing/decresing. ( ) is 46 (B) Find horizontl symptotes of f x ( ). The line y = mx + is tngent to the grph of f ( x) t ( x 1,y 1 ). Find verticl symptotes of f ( x). Find x-vlues of verticl tngents to f Stu Schwrtz

5 Legend: A Preclculus, B Limits, C Differentil Clculus, D Applictions of Differentil Clculus, E Integrl Clculus, F Applictions of Integrl Clculus, G Prticle Motion nd Rtes This is wht you think of doing ( ) where the 50 (E) Given the vlue of F ntiderivtive of f is F, find F( ). 51 (F) Find the re under the curve f ( x) on the intervl [, ]. 52 Find the minimum slope of f ( x) on [, ]. 53 (E) Find " f ( x) where <. 54 Given the position function s( t) of prticle moving long stright line, find the velocity nd ccelertion. 55 (F) Find the volume when the re etween f ( x) nd g x out the x-xis. ( ) is rotted determine the distnce prticle trvels on [ t 1 determine if prticle is speeding up or slowing down t t = k. Find intervls where the slope of f x ( ) is incresing. 59 (E) Approximte " f ( x) using right Riemnn sums with n rectngles Stu Schwrtz

6 Legend: A Preclculus, B Limits, C Differentil Clculus, D Applictions of Differentil Clculus, E Integrl Clculus, F Applictions of Integrl Clculus, G Prticle Motion nd Rtes 60 Approximte the vlue of f x 1 + if you know the function goes through point ( x 1, y 1 ). t 61 2 Clculte " v t t 1 ( ) dt without clcultor. Given the velocity function v t [ t 1,t 2 ], find the minimum ccelertion of prticle. Find the solute mximum or minimum of f x ( ) on [, ]. ( ) ( ) on This is wht you think of doing 64 Show tht f ( x) is even. 65 (F) Find the verge rte of chnge of F " x [ ]. ( ) on t 1,t 2 66 The mening of # R "( t) dt. 67 find the verge velocity on [ t 1 68 Find the intersection of f ( x) nd g( x) (E) The volume of solid is chnging t the rte of Given the ccelertion function ( t) of prticle t rest nd s( 0), find s( t). Find d g( x) dt Stu Schwrtz

7 Legend: A Preclculus, B Limits, C Differentil Clculus, D Applictions of Differentil Clculus, E Integrl Clculus, F Applictions of Integrl Clculus, G Prticle Motion nd Rtes (B) Determine whether the liner pproximtion for f ( x 1 + ) overestimtes or under-estimtes f ( x 1 + ). Find lim ( ) where f ( x) is f x x " piecewise function. This is wht you think of doing 73 Find the derivtive of the inverse to f x ( ) t x =. 74 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 [ t 1,t 2 ] ) 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 Stu Schwrtz

8 AB Clculus Review Sheet Solutions Legend: A Preclculus, B Limits, C Differentil Clculus, D Applictions of Differentil Clculus, E Integrl Clculus, F Applictions of Integrl Clculus, G Prticle Motion nd Rtes This is wht you think of doing 1 (B) Show tht f x 2 Find the eqution of the tngent line to f t x 1, y 1 ( ). ( ) is continuous. Show tht 1) lim f x x " ( ) exists 2) f ( ) exists 3) lim ( ) = f ( ) ( ) hs 3 If continuous function f x f ( ) < k nd f ( ) > k, explin why there must e vlue c such tht < c < nd f ( c) = k. 4 Find points of reltive extrem of f ( x). 5 Find rnge of f x f x x " Find slope m = f "( x i ). Then use point slope eqution: y " y 1 = m( x " x 1 ) This is the Intermedite Vlue Theorem. Mke sign chrt of f "( x). At x = c where the derivtive 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 is reltive minimum t x = c. If f " c mximum t x = c. (2 nd Derivtive test). ( ) > 0, there ( ) < 0, there is reltive ( ) on [, ] Use reltive extrem techniques to find reltive mx/mins. Evlute f t these vlues. Then exmine f ( ) nd f exmine f ( ) nd f ( ). ( ), ( ) " s( t 1 ) ( ). Then 6 Given the position function s t Avg. vel. = s t 2 find the verge velocity on [ t 1 t 2 " t 1 7 (E) # Approximte " f ( x) using left A = " & % ([ f ( x $ n ' 0 ) + f ( x 1 ) + f ( x 2 ) f ( x n"1 )] Riemnn sums with n rectngles. 8 Given grph of f "( x), determine Mke sign chrt of f "( x) nd determine the intervls where intervls where f ( x) is f "( x) is positive nd negtive. incresing/decresing. 9 Given piecewise function, show it First, e sure tht f ( x) is continuous t x =. Then tke the is differentile t x = where the derivtive of ech piece nd show tht lim f $ ( x) = lim f $ ( x). function rule splits. x " # x " + 10 Find criticl vlues of f ( x). Find nd express f "( x) s frction. Set oth numertor nd denomintor equl to zero nd solve. 11 Find the zeros of f ( x). Set function equl to 0. Fctor or use qudrtic eqution if qudrtic. Grph to find zeros on clcultor Stu Schwrtz

9 Legend: A Preclculus, B Limits, C Differentil Clculus, D Applictions of Differentil Clculus, E Integrl Clculus, F Applictions of Integrl Clculus, G Prticle Motion nd Rtes This is wht you think of doing 12 (B) Find lim f x x " ( ). Step 1: Find f ( ). If you get zero in the denomintor, 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 x " # for equlity. ( ) nd lim 13 Find x-vlues of horizontl tngents to f. Write f " x f x 14 Show tht Rolle s Theorem holds Show tht f is continuous nd differentile on [, ]. If for f ( x) on [, ]. f ( ) = f ( ), then find some c on [, ] such tht f " c 15 Find inflection points of f ( x). Find nd express f " x denomintor equl to zero nd solve. Mke sign chrt of f " ( x). Inflection points occur when f " ( x) witches from positive to negtive or negtive to positive. 16 (F) Find the re etween Find the intersections, nd of f ( x) nd g( x). If f ( x) nd g( x). f ( x) " g( x) on [,], then re A = # [ f ( x) " g( x) ]. 17 (F) 18 Given dy, drw slope field. determine the difference of position of prticle on [ t 1 x " + ( ) s frction. Set numertor of "( ) = 0. f ( x) ( ) = 0. ( ) s frction. Set oth numertor nd dy Use the given points nd plug them into, drwing little lines with the clculted slopes t the point. Displcement = t 2 " t 1 v( t) dt 19 (F) Find the volume when the re under f ( x) is rotted out the x- Disks: Rdius = f ( x): V = " # [ f ( x) ] 2 xis on the intervl [, ]. 20 Find the verge rte of chnge of f on [, ]. Find f ( ) " f ( ) " 21 Find the derivtive of f ( g( x) ). This is the chin rule. You re finding f "( g( x) )# g "( x). 22 Find the derivtive of function f ( x + h) # f ( x) f ( x) # f ( ) using the derivtive definition. Find lim or lim h " 0 h x " x # 23 Show tht f ( x) is odd. Show tht f ("x) = " f ( x). This shows tht the grph of f is symmetric to the origin. 24 Find lim f ( x) nd lim f ( x). Express f x x "# x "$# (B) ( ) s frction. Determine loction of the highest power: Denomintor: lim f ( x) = lim f ( x) = 0 x "# x "$# Both Num nd Denom: rtio of the highest power coefficients Numertor: lim x "# f x ( ) = ±# (plug in lrge numer) Stu Schwrtz

10 Legend: A Preclculus, B Limits, C Differentil Clculus, D Applictions of Differentil Clculus, E Integrl Clculus, F Applictions of Integrl Clculus, G Prticle Motion nd Rtes 25 Given the position function s( t), find the instntneous velocity t t = k. 26 Given the velocity function v( t) nd s( 0), find s( t). 27 (F) Find the verge vlue of f x [, ]. 28 (E) Mening of x dt. ( ) on This is wht you think of doing Inst. vel. = s "( k). ( ) = v( t) dt s t " + C. Plug in s 0 ( ) to find C. " f ( x) F vg = # The ccumultion function ccumulted re under function f strting t some constnt nd ending t some vrile x. 29 (E) Find d x x d dt. dt = f ( x). The 2nd Fundmentl Theorem. 30 Find x-vlues of horizontl tngents Write f "( x) s frction. Set numertor of f "( x) = 0. to f. 31 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. 32 Given chrt of x nd f ( x) nd Strddle c, using vlue of k c nd vlue of selected vlues of x etween nd h c. f "( c) # f k), pproximte f "( c) where c is k $ h vlue etween nd. 33 Given the velocity function v( t) nd s( 0), find the gretest distnce Generte sign chrt of v( t) to find turning points. s( t) = " v( t) dt + C. Plug in s( 0) to find C. of the prticle from the strting Evlute s t position on [ 0,t 1 ]. mximum distnce from s( 0). 34 (F) y is incresing proportionlly to y. dy = ky which trnsltes to y = Cekt dt 35 (E) Given " f ( x), find " [ f ( x) + k] = " f ( x) + " k " [ f ( x) + k]. Find the eqution of the norml line to f t x 1, y 1 ( ). Show tht the Men Vlue Theorem holds for f x ( ) on [, ]. ( ) t ll turning points nd find which one gives the Find slope m"= #1 f $ x i ( ) ( ). Then use point slope eqution: y " y 1 = m x " x 1 Show tht f is continuous nd differentile on [, ]. If f ( ) = f ( ), then find some c on [, ] such tht ( ) = f ( ) # f ( ) f " c # Stu Schwrtz

11 Legend: A Preclculus, B Limits, C Differentil Clculus, D Applictions of Differentil Clculus, E Integrl Clculus, F Applictions of Integrl Clculus, G Prticle Motion nd Rtes This is wht you think of doing 38 Find rnge of f ( x) on ("#,#). Use reltive extrem techniques to find reltive mx/mins. Evlute f t these vlues. Then exmine f ( ) nd f ( ). Then exmine lim f x x "# ( ) nd lim f x x "$# ( ). 39 Find the instntneous rte of chnge of f t x =. Find f "( ) c c 40 (F) Find the line x = c tht divides the re under f ( x) on [, ] into two " f ( x) = " f ( x) or " f ( x) = 2 " f ( x) c equl res. 41 (E) # Approximte " f ( x) using A = " & % ([ f ( x $ 2n ' 0 ) + 2 f ( x 1 ) + 2 f ( x 2 ) f ( x n"1 ) + f ( x n )] trpezoidl summtion. This formul only works when the se of ech trpezoid is the sme. If not, clculte the res of individul trpezoids. 42 (F) Solve the differentil eqution Seprte the vriles: x on one side, y on the other with the dy = f ( x nd dy in the numertors. Then integrte oth sides, )g( y). rememering the +C, usully on the x-side. 43 (F) Given se ounded y Bse = f ( x) " g( x). Are = se 2 = [ f ( x) " g( x) ] 2. f ( x) nd g( x) on [, ] the cross sections of the solid perpendiculr Volume = # [ f ( x) " g( x) ] 2 to the x-xis re squres. Find the volume. 44 (E) Typiclly done with tle of points. Be sure to use only Approximte " f ( x) using vlues tht re given. If you re given 7 points, you cn only clculte 3 midpoint rectngles. midpoint Riemnn sums. 45 Find the intervl(s) where f ( x) is Find criticl vlues of f "( x). Mke sign chrt to find sign of incresing/decresing. f "( x) in the intervls ounded y criticl vlues. Positive mens incresing, negtive mens decresing. 46 Find horizontl symptotes of lim (B) f ( ( x ) nd lim x "# x "$# x). 47 The line y = mx + is tngent to Two reltionships re true: the grph of f ( x) t ( x 1,y 1 ). 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 Find verticl symptotes of f ( x). Express f ( x) s frction, express numertor nd denomintor in fctored form, nd do ny cncelltions. Set denomintor equl to Find x-vlues of verticl tngents to f. Write f "( x) s frction. Set denomintor of f "( x) = Stu Schwrtz

12 Legend: A Preclculus, B Limits, C Differentil Clculus, D Applictions of Differentil Clculus, E Integrl Clculus, F Applictions of Integrl Clculus, G Prticle Motion nd Rtes This is wht you think of doing ( ) where the 50 (E) Given the vlue of F ntiderivtive of f is F, find F( ). 51 (F) Find the re under the curve f ( x) on the intervl [, ]. Find the minimum slope of f ( x) (E) on [, ]. Find Use the fct tht F( ) = F( ) + f ( x) integrl. " f ( x) " f ( x) where <. " f ( x) = # " f ( x) ( ) of 54 Given the position function s t prticle moving long stright line, find the velocity nd ccelertion. 55 (F) Find the volume when the re etween f ( x) nd g( x) is rotted out the x-xis (E) determine the distnce prticle trvels on [ t 1 determine if prticle is speeding up or slowing down t t = k. Find intervls where the slope of f x ( ) is incresing. Approximte " f ( x) using right Riemnn sums with n rectngles. " f ( x) = F ( ) # F( ) so ". Use the clcultor to find the definite 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. v t ( ) = s "( t) ( t) = v "( t) = s " ( t) Wshers: Outside rdius = f ( x). Inside rdius = g( x). Estlish the intervl where f x nd, where f ( x) = g x Distnce = t 2 " t 1 v( t) dt ( ) " g( x) nd the vlues of $ ([ ( )] 2 # [ g ( x ) ] 2 ) ( ). V = " f x 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. Find the derivtive of f " x vlues of f " x positive intervls. # A = " & % ( f ( x $ n ' 1 ) + f x 2 ( ) which is f " ( x). Find criticl ( ) nd mke sign chrt of f " ( x) looking for [ ( ) + f ( x 3 ) f ( x n )] Stu Schwrtz

13 Legend: A Preclculus, B Limits, C Differentil Clculus, D Applictions of Differentil Clculus, E Integrl Clculus, F Applictions of Integrl Clculus, G Prticle Motion nd Rtes 60 Approximte the vlue of f x 1 + if you know the function goes through point ( x 1, y 1 ). t 61 2 Clculte " v t clcultor. t 1 ( ) dt without ( ) This is wht you think of doing 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 + ). Set v( t) = 0 nd mke sign chrge of v t [ ]. On intervls [, ] where v t On intervls [, ] where v t ( ) > 0, v( t) dt ( ) = 0 on t 1,t 2 " = " v t ( ) < 0, v( t) dt ( ) dt " = " v t ( ) dt Given the velocity function v t [ t 1,t 2 ], find the minimum ccelertion of prticle. Find the solute mximum or minimum of f x ( ) on [, ]. ( ) on ( ) is even. Show tht f "x 64 Show tht f x 65 (F) Find the verge rte of chnge of F " x [ ] (E) ( ) on t 1,t 2 The mening of # R "( t) dt. find the verge velocity on [ t 1 Find the intersection of f ( x) nd g( x). The volume of solid is chnging t the rte of Given the ccelertion function ( t) of prticle t rest nd s( 0), find s( t). Find d g( x) dt. ( ) nd set "( ) = 0. Set up sign chrt nd find criticl Find t t vlues. Evlute the ccelertion t criticl vlues nd lso t 1 nd t 2 to find the minimum. 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. ( ) = f ( x). This shows tht the grph of f is symmetric to the y-xis. t d 2 # F "( x) dt ( ) $ "( ) t 1 = F " t 2 F t 1 t 2 $ t 1 t 2 $ t 1 This gives the ccumulted chnge of R t # R "( t ) dt = R t 2 " ( ) on [, ]. ( ) $ R( ) or R( ) = R( ) + R "( t) v( t) dt # dt t Avg. vel. = 1 t 2 # t 1 Set the two functions equl to ech other. Find intersection on clcultor. dv dt =... v( t) = " ( t) dt + C 1. Plug in v( 0) = 0 to find C 1. s( t) = " v( t) dt + C 2. Plug in s( 0) to find C 2. d g( x) dt = f g x ( ( )) # $ g ( x). The 2nd Fundmentl Theorem Stu Schwrtz

14 Legend: A Preclculus, B Limits, C Differentil Clculus, D Applictions of Differentil Clculus, E Integrl Clculus, F Applictions of Integrl Clculus, G Prticle Motion nd Rtes (B) Determine whether the liner pproximtion for f x 1 + ( ) overestimtes or under-estimtes f x 1 + ( ). Find lim f x x " piecewise function. ( ) where f ( x) is Find the derivtive of the inverse to f x ( ) t x =. 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 [ t 1,t 2 ] ) 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 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 + ). Determine if lim f ( x) = lim f ( x) y plugging in to x " # x " + f ( x),x < nd f ( x),x > for equlity. If they re not equl, the limit doesn t exist. 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. 4) Plug the y-vlue you found in 2) to dy m ) g + # [ F( t) " E( t) ] dt ) d dt 0 m # [ F( t) " E( t) ] dt = F m 0 c) set F m m ( ) " E( m) ( ) " E( m) = 0, solve for m, nd evlute g + # [ F( t) " E( t) ] dt t vlues of m nd lso the endpoints Stu Schwrtz

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