( ) where f ( x ) is a. AB/BC Calculus Exam Review Sheet. A. Precalculus Type problems. Find the zeros of f ( x).


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1 AB/ Clculus Exm Review Sheet A. Preclculus Type prolems A1 Find the zeros of f ( x). This is wht you think of doing A2 Find the intersection of f ( x) nd g( x). A3 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) Stu Schwrtz
2 B6 B7 B8 B9 f ( x) lim Find x 0 g( x) if lim f ( x) = 0 nd limg x x 0 x 0 Find lim f x x 0 # g( x) = 0 ( ±!) Find lim f x x 0 # g( x) = $ # $ Find lim f ( x) g( x) =1 # or 0 0 or # 0 x 0 = 0 This is wht you think of doing C. Derivtives, differentiility, nd tngent lines C1 C2 C3 C4 C5 C6 C7 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 xvlues of horizontl tngents to f. This is wht you think of doing C8 Find xvlues of verticl tngents to f. C9 Approximte the vlue of f x 1 + if you know the function goes through point x 1, y Stu Schwrtz
3 C10 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. 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 Stu Schwrtz
4 D11 D12 D13 Determine whether the liner pproximtion for f ( x 1 + ) overestimtes or underestimtes f ( x 1 + ). Find intervls where the slope of f ( x) is incresing. Find the minimum slope of f ( x) on [, ]. This is wht you think of doing E. Integrl Clculus E1 E2 E3 E4 E5 E6 E7 E8 E9 E10 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 f ( t) dt. f ( x), find [ f ( x) + k]. Given the vlue of F ntiderivtive of f is F, find F. Find d Find d x f ( t) dt. g( x) f ( t) dt. where the This is wht you think of doing Stu Schwrtz
5 E11 E12 Find Find # f ( x). 0 f ( x) # g( x) 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 F4 F5 F6 F7 F8 F9 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 xxis on the intervl [, ]. Find the volume when the re etween f ( x) nd g( x) is rotted out the xxis. Given se ounded y f ( x) nd g( x) on [, ] the cross sections of the solid perpendiculr to the xxis 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 t 1, F10 y is incresing proportionlly to y. F11 Given dy, drw slope field. F12 Find x 2 + x + c Stu Schwrtz
6 F13 F14 F15 F16 F17 F18 Use Euler s method to pproximte f ( 1.2) given formul for dy, ( x, y 0 0) nd x = 0.1 Is the Euler s pproximtion n overor underpproximtion? A popultion P is incresing logisticlly. Find the crrying cpcity of popultion growing logisticlly. Find the vlue of P when popultion growing logisticlly is growing the fstest. Given continuous f ( x), find the rc length on [, ] This is wht you think of doing G. Prticle Motion nd Rtes of Chnge G1 G2 G3 G4 G5 G6 G7 G8 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 Given the velocity function v( t), determine if prticle is speeding up or slowing down t t = k. Given the position function s( t), find the verge velocity on [ t 1, ]. Given the position function s( t), find the instntneous velocity t t = k. on Given the velocity function v t [ t 1, ], find the minimum ccelertion of prticle. Given the velocity function v( t), find the verge velocity on [ t 1, ]. This is wht you think of doing Stu Schwrtz
7 G9 G10 G11 G12 G13 G14 G15 Given the velocity function v( t), determine the difference of position of prticle on [ t 1, ]. Given the velocity function v( t), determine the distnce prticle trvels on [ t 1, ]. Clculte t 1 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,t 1 ]. 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 [ t 1, ] ) 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 H. Prmetric nd Polr Equtions  H1 H2 H3 H4 H5 Given x = f ( t),y = g( t), find dy. Given x = f ( t),y = g( t), find d 2 y. 2 Given x = f ( t),y = g( t), find rc length on [ t 1, ]. Express polr eqution in the form of r = f in prmetric form. Find the slope of the tngent line to r = f. This is wht you think of doing Stu Schwrtz
8 H6 H7 H8 H9 Find horizontl tngents to polr curve r = f (). Find verticl tngents to polr curve r = f. Find the re ounded y the polr curve r = f [ ]. on 1, 2 Find the rc length of the polr curve r = f [ ]. on 1, 2 This is wht you think of doing I. Vectors nd Vectorvlued functions  This is wht you think of doing I1 Find the mgnitude of vector v v 1,v 2 I2 Find the dot product: u 1,u 2 v 1,v 2 I3 I4 I5 I6 I7 The position vector of prticle moving in the plne is r( t) = x( t),y t vector nd ) the ccelertion vector. The position vector of prticle moving in the plne is r( t) = x( t),y t prticle t time t. Given the velocity vector v( t) = x( t), y t = 0, find the position vector. Given the velocity vector v( t) = x( t), y( t), when does the prticle stop? The position vector of prticle moving in the plne is r( t) = x( t),y t prticle trvels from t 1 to.. Find ) the velocity. Find the speed of the nd position t time t. Find the distnce the Stu Schwrtz
9 J. Tylor Polynomil Approximtions  J1 J2 J3 J4 J5 J6 J7 Find the nth degree Mclurin polynomil to f ( x). Find the nth degree Tylor polynomil to f ( x) centered t x = c. Use the firstdegree Tylor polynomil to f ( x) centered t x = c to pproximte f ( k) nd determine whether the pproximtion is greter thn or less thn f ( k). Given n nth degree Tylor polynomil for f out x = c, find f ( c), f ( c), f ( c),..., f ( n) ( c). Given Tylor polynomil centered t c, determine if there is enough informtion to determine if there is reltive mximum or minimum t x = c. Given n nth degree Tylor polynomil for f out x = c, find the Lgrnge error ound (reminder). Given n nth degree Mclurin polynomil P for f, find the f k P( k). This is wht you think of doing K. Infinite Series  K1 K2 K3 Given n, determine whether the sequence n converges. Given n, determine whether the series n could converge. Determine whether series converges. This is wht you think of doing K4 Find the sum of geometric series. K5 Find the intervl of convergence of series Stu Schwrtz
10 K n This is wht you think of doing K7 K8 K9 K10 f ( x) =1+ x + x x 3 3! x n n! +... f ( x) = x x 3 3! + x 5 5! x 7 7! +... f ( x) =1 x 2 2! + x 4 4! x 6 6! +... f ( x) =1+ x + x 2 + x 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) centered t x = c. K12 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 Stu Schwrtz
11 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 yxis. 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 nonnegtive 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 tht 1) 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 $# Stu Schwrtz
12 B6 B7 B8 B9 f ( x) lim Find x 0 g( x) if lim f x x 0 = 0 nd lim x 0 Find lim f x x 0 # g( x) = 0 ( ±!) Find lim f x x 0 # g x Find lim f ( x) g x x 0 g( x) = 0 This is wht you think of doing Use L Hopitl s Rule: lim x 0 = lim x 0 f x g x Express g x = $ # $ Express f x f # x g # x = 1 nd pply L Hopitl s rule. 1 g( x) g( x) with common denomintor nd use L Hopitl s rule. =1 # or 0 0 or # 0 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 C2 C3 C4 C5 C6 C7 C8 C9 C10 C11 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 xvlues of horizontl tngents to f. Find xvlues 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 underpproximtion 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 yvlue t x 1. 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 ) Stu Schwrtz
13 C12 C13 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. This is wht you think of doing Follow this procedure: 1) Interchnge x nd y in f ( x). 2) Plug the xvlue 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 yvlue 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 + $. D. Applictions of Derivtives D1 D2 D3 D4 D5 D6 D7 D8 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 f is continuous nd differentile on [, ]. If f = f, then find some c on [, ] such tht f c = Stu Schwrtz
14 D9 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 underestimtes f ( x 1 + ). D12 D13 Find intervls where the slope of f ( x) is incresing. Find the minimum slope of f ( x) on [, ]. 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. 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 + ). 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 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) [ f ( x n1 )] [ + 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 ) f ( x n1 ) + f ( x n )] This formul only works when the se of ech trpezoid is the sme. If not, clculte the res of individul trpezoids Stu Schwrtz
15 E6 E7 E8 Mening of Given x f ( t) dt. f ( x), find [ f ( x) + k]. where the Given the vlue of F ntiderivtive of f is F, find F. This is wht you think of doing 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. E9 Find d x x d f ( t) dt. f ( t) dt = f ( x). The 2nd Fundmentl Theorem. E10 Find d g( x) g( x) d f ( t) dt. f ( t) dt = f ( g( x) ) # g $ ( x). The 2nd Fundmentl Theorem. h E11 Find # f ( x). # f ( x) = lim # f ( x) = lim F ( h ) % F( 0). h $ h $ E12 Find f ( x) # g( x) If usustitution doesn t work, try integrtion y prts: # u dv = uv $ # v du 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 xxis on the intervl [, ]. Find the volume when the re etween f ( x) nd g x the xxis. 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 Stu Schwrtz
16 This is wht you think of doing F6 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 to Volume = # [ f ( x) g( x) ] 2 the xxis 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 xside. 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 [ t 1, ]. # F ( x) dt t 1 = ( ) $ F ( t 1 ) $ t 1 $ t 1 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. F12 Fctor x Find + x + c into nonrepeting fctors to get x 2 + x + c ( mx + n) ( px + q) prtil frctions nd integrte ech frction. F13 Use Euler s method to pproximte f ( 1.2) given formul for dy = dy ( x ), y new = y old + dy dy, ( x, y 0 0) nd x = 0.1 F14 Is the Euler s pproximtion n over or underpproximtion? Look t sign of dy nd d 2 y in the intervl. This gives 2 incresing/decresing nd concvity informtion. Drw picture to scertin the nswer. ). F15 A popultion P is incresing logisticlly. dp dt = kp ( C P F16 Find the crrying cpcity of dp popultion growing logisticlly. = 0 # C = P. F17 F18 Find the vlue of P when popultion growing logisticlly is growing the fstest. Given continuous f ( x), find the rc length on [, ] dt = kp C P dp dt = kp( C P) # Set d 2 P dt = 0 2 # [ ] 2 L = 1+ f x Stu Schwrtz
17 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). Given the velocity function v( t), determine if prticle is speeding up or slowing down t t = k. Given the position function s( t), find the verge velocity on [ t 1, ]. Given the position function s( t), find the instntneous velocity t t = k. Given the velocity function v( t) on [ t 1, ], find the minimum ccelertion of prticle. Given the velocity function v( t), find the verge velocity on [ t 1, ]. Given the velocity function v( t), determine the difference of position of prticle on [ t 1, ]. Given the velocity function v( t), determine the distnce prticle trvels on [ t 1, ]. Clculte clcultor. t 1 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,t 1 ]. 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( t 1 ) Avg. vel. = s t 1 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 t 1 nd to find the minimum. Avg. vel. = t 1 Displcement = Distnce = t 1 v( t) dt # t 1 t 1 v( t) dt v( t) dt Set v( t) = 0 nd mke sign chrge of v( t) = 0 on t 1, 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 Stu Schwrtz
18 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 [ t 1, ] ) 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 g + m E( m) # dt E( m) = 0, solve for m, nd evlute # [ F( t) E( t) ] dt t vlues of m nd lso the endpoints. 0 H. Prmetric nd Polr Equtions  H1 H2 H3 H4 H5 H6 H7 H8 H9 Given x = f ( t),y = g( t), find dy. dy dy = dt dt This is wht you think of doing Given x = f ( t),y = g( t), find d 2 y. d dy % 2 x = f ( t),y = g( t), find d 2 y = d $ ' dy % dt # & $ ' = 2 # & dt Given x = f ( t),y = g( t), find rc 2 % length on [ t 1, ]. L = $ ' + dy 2 % ( $ ' dt # dt & # dt & t 1 Express polr eqution in the form x = rcos = f ()cos y = rsin = f ()sin of r = f () in prmetric form. Find the slope of the tngent line to r = f (). x = rcos y = rsin # dy dy = d d Find horizontl tngents to polr x = rcos y = rsin curve r = f (). Find where rsin = 0 when rcos # 0 Find verticl tngents to polr x = rcos y = rsin curve r = f (). Find where rcos = 0 when rsin # 0 Find the re ounded y the polr curve r = f () on [ 1, 2 ]. A = # r 2 d = # [ f ()] 2 d 2 2 Find the rc length of the polr 2 curve r = f () on [ 1, 2 ]. s = $ [ f ()] 2 + [ f #( )] 2 2 % d = r 2 + ' dr 2 ( $ * d & d ) Stu Schwrtz
19 I. Vectors nd Vectorvlued functions  I1 Find the mgnitude of vector v v 1,v 2. This is wht you think of doing v = v v 2 2 I2 Find the dot product: u 1,u 2 v 1,v 2 u 1,u 2 v 1,v 2 = u 1 v 1 + u 2 v 2 I3 The position vector of prticle moving in the plne is ) v( t) = x ( t), y ( t) r( t) = x( t),y( t). Find ) the ) ( t) = x ( t), y ( t) velocity vector nd ) the ccelertion vector. I4 The position vector of prticle moving in the plne is r( t) = x( t),y( t). Find the speed of the prticle t time t. Speed = v( t) = [ x ( t) ] 2 + y t I5 Given the velocity vector s( t) = x( t) dt + y( t) dt + C v( t) = x( t), y( t) nd position t Use s 0 time t = 0, find the position vector. I6 Given the velocity vector v( t) = x( t), y( t), when does the prticle stop? I7 The position vector of prticle moving in the plne is Distnce = # [ x ( t) ] 2 + y ( t) r( t) = x( t),y( t). Find the distnce t 1 the prticle trvels from t 1 to. J. Tylor Polynomil Approximtions  [ ] 2  sclr to find C, rememering tht it is vector. v( t) = 0 x( t) = 0 AND!y ( t) = 0 [ ] 2 dt J1 J2 J3 Find the nth degree Mclurin polynomil to f ( x). Find the nth degree Tylor polynomil to f x x = c. centered t Use the firstdegree Tylor polynomil to f ( x) centered t x = c 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 ( 0) + f ( 0)x + f ( 0) x 2 + 2! f ( 0) 3! ( x) = f c x f n 0 n! x n + f ( c) ( x # c) + f ( c) x # c ( x # c) 3 2 f c f ( n) c 3! n! Write the firstdegree TP nd find f ( k). Use the signs of f c f c 2! ( x # c) n nd to determine incresing/decresing nd concvity nd drw your line (1 st degree TP) to determine whether the line is under the curve (underpproximtion) or over the curve (overpproximtion) Stu Schwrtz
20 J4 J5 J6 J7 Given n nth degree Tylor polynomil for f out x = c, find f ( c), f ( c), f ( c),..., f ( n) ( c) Given Tylor polynomil centered t c, determine if there is enough informtion to determine if there is reltive mximum or minimum t x = c. Given n nth degree Tylor polynomil for f out x = c, find the Lgrnge error ound (reminder). Given n nth degree Mclurin polynomil P for f, find the f k P( k). f c f c f c This is wht you think of doing will e the constnt term in your Tylor polynomil (TP) will e the coefficient of the x term in the TP. 2! f ( n) c will e the coefficient of the x 2 term in the TP. will e the coefficient of the x n term in the TP. n! If there is no firstdegree xterm in the TP, then the vlue of c out which the function is centered is criticl vlue. Thus the coefficient of the x 2 term is the second derivtive divided y 2! Using the second derivtive test, we cn tell whether there is reltive mximum, minimum, or neither t x = c. R n ( x) = f ( n +1) ( z) ( n +1)! x c n +1. The vlue of z is some numer z represents the ( n +1) st derivtive of etween x nd c. f n +1 z. This usully is given to you. This is looking for the Lgrnge error the difference etween the vlue of the function t x = k nd the vlue of the TP t x = k. K. Infinite Series  K1 K2 K3 K4 K5 Given n, determine whether the sequence n converges. Given n, determine whether the series n could converge. Determine whether series converges. Find the sum of geometric series. Find the intervl of convergence of series. This is wht you think of doing n converges if lim n exists. n # If lim n = 0, the series could converge. If lim n $ 0, the n # n # series cnnot converge. (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 convergence re: Geometric: # r n  converges if r <1 n= 0 1 pseries: #  converges if p > 1 n p n=1 # Alternting: $ (1) n n  converges if 0 < n +1 < n Rtio: n=1 # n  converges if lim n +1 n # n= 0 n <1 # r n = n= 0 1$ r If not given, you will hve to generte the nth term formul. Use test (usully the rtio test) to find the intervl of convergence nd then check out the endpoints Stu Schwrtz
21 K6 K7 K8 K9 K n f ( x) =1+ x + x x 3 3! x n n! +... f ( x) = e x f ( x) = x x 3 3! + x 5 5! x 7 7! +... f x f ( x) =1 x 2 f x 2! + x 4 4! x 6 6! +... f x =1+ x + x 2 + x 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) centered t x = c. K12 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 2. This is wht you think of doing The hrmonic series divergent. = sin x = cos x f ( x) = 1 1 x Convergent : (1,1 ) f c f ( x) = f ( c) + f ( c) ( x # c) + ( x # c) 3 f c 3! f ( n) c x # c 2 2! ( x # c) n n! This is the error for the 4 th term of n lternting series which is simply the 5 th tern. It will e positive since you re looking for n solute vlue. Rther thn go through generting Tylor polynomil, use the fct tht if f x = e x, then f x 2 = e x 2. So f ( x) = e x =1+ x + x x 3 3! + x 4 4! x n +... nd n! f ( x 2 ) = e x 2 =1+ x 2 + x x 6 3! + x 8 2n x ! n! Stu Schwrtz
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