( ) Same as above but m = f x = f x - symmetric to y-axis. find where f ( x) Relative: Find where f ( x) x a + lim exists ( lim f exists.

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1 AP Clculus Finl Review Sheet solutions When you see the words This is wht you think of doing Find the zeros Set function =, fctor or use qudrtic eqution if qudrtic, grph to find zeros on clcultor Find eqution of the line tngent to f(x Tke derivtive - f = m nd use t y y (, = m x x 3 Find eqution of the line norml to f(x t (, 4 Show tht ( x Sme s ove ut m = f f x = f x - symmetric to y-xis f is even Show tht 5 Show tht f ( x is odd Show tht f ( x = f ( x 6 Find the intervl where ( x f is incresing 7 Find intervl where the slope of f x is incresing 8 Find the minimum vlue of function 9 Find the minimum slope of function = - symmetric to origin = Find where f ( x = or DNE Mke sign chrt of ( x f nd determine where it is positive Find the derivtive of f ( x = f ( x = or DNE Mke sign chrt of f '' x nd determine where it is positive Reltive: Find where f ( x = or DNE Mke sign chrt of ( x find where f ( x chnges from + to f, Asolute: Find ll criticl numers nd endpoints Plug those vlues ck into f ( x nd choose the smllest Sme s ove, ut with Find criticl vlues Find where f ( x = or DNE Find inflection points Mke sign chrt of f ( x to find where ( x Show tht or to + lim f x exists Show tht lim f ( x= lim x x 3 Show tht f ( x is continuous Show tht f ( x x 4 Find verticl symptotes of ( x x + f ( x lim exists ( lim f exists lim f x = 3 f x f Fctor/cncel ( x x f chnges sign (+ to f ( x= lim x + f nd set denomintor = Find x= such tht lim ± ± 5 Find horizontl symptotes of f ( x Find y= lim f ( x nd y= lim f ( x x f ( x 6 Find the verge rte of chnge of x, f on [ ] Find f f

2 7 Find instntneous rte of chnge f x t of 8 Find the verge vlue of f ( x on [, ] 9 Find the solute mximum of x, f on [ ] Show tht piecewise function is differentile t the point where the function rule splits Given ( t v ( t Given v ( t trvels on [, ] s (position function, find, find how fr prticle 3 Find the verge velocity of Find f - Find f ( x Find where f ( x ck into ( x = or DNE nd endpoints nd Plug those vlues f nd choose the lrgest First, e sure tht the function is continuous t x = Tke the lim f x = lim f x derivtive of ech piece nd show tht Find v ( t = s ( t Find v ( t prticle on [, ] Find v( t 4 Given v ( t, determine if prticle is speeding up t t = k 5 Given v(t[velocity] nd s(, find s(t[position] 6 Show tht Men Vlue Theorem, holds on [ ] 7 Find f ( x y definition 8 Find derivtive of inverse to f ( x t x = 9 y is incresing proportionlly to y 3 Find the line x = c tht divides the re under f ( x on [, ] to two equl res d x 3 f ( t = d g ( x 3 f(t 33 The line y = mx + is tngent to f x t, s = Find v( k nd ( k s x x + If oth hve the sme sign, the prticle is speeding up, if different signs, then the prticle is slowing down s t + C Plug in t = to find C (= v( t Show tht f is continuous nd differentile on the intervl Then find some c such tht f ( c= f ( f f ( x + h f ( x f ( x= lim or h h f ( x f f ( x= lim x x = or = = ky (differentil eqution trnslting to y = Ce kt (solution c ( x f ( x f = c nd FTC: Answer is f ( x nd FTC: Answer is f ( g( x g'( x The two functions shre the sme slope ( m = f ( x 34 Find re using left Riemnn sums A = [ f x x + f ( x x + + f ( x n x] left ( Drw oxes string on the

3 35 Find re using right Riemnn sums [ f x x + f ( x x + + f ( x x] A = ( n Drw oxes strting on the right 36 Find re using midpoint rectngles A = [ f x x + f ( x x + + f ( x n x] 37 Find re using trpezoids ( 3 Typiclly done with tle of vlues Be sure to use only vlues tht re given If you re given 6 sets of points, you cn only do 3 midpoint rectngles A = f + f ( x f ( x + f ( x f ( xn x + x + + ( x + f ( xn x Drw the trpezoids nd find the re of ech 38 Solve the differentil eqution Seprte the vriles ll x s on one side, ll y s on the other The nd must e multiplied not dded x The ccumultion function ccumulted re under the function 39 Mening of f ( t f x strting t some constnt nd ending t x 4 Given se, cross sections perpendiculr to the x-xis re squres 4 Find where the tngent line to f ( x is horizontl 4 Find where the tngent line to f ( x is verticl 43 Find the minimum ccelertion v t given 44 Approximte the vlue of f ( y using the tngent line to f t x = 45 Given the vlue of F nd the fct tht the nti-derivtive of f is F, F find 46 Find the derivtive of f ( g( x f ( x, nd f(xg(x g( x The re etween the curves typiclly is the se of your squre So the volume is ( s where s is the length of side Find where x= If ( x to zero Find where x DNE If ( x equl to zero f is frction Set the numertor equl f s frction Set the denomintor First find the ccelertion ( t v ( t ( t Find the eqution of the tngent line to f using y y = m( x x where m = f ( nd the point is (, ( line eing sure to use n pproximte ( sign! = Then find the minimum using f Then plug in into this Usully, this prolem contins n ntiderivtive you cnnot tke F x is the ntiderivtive of f, then Utilize the fct tht if F F f ( x + = Use the clcultor = = = h h = + 47 Given f ( x, find [ f ( x k] + [ f ( x k] = f ( x + k = f ( x + k( +

4 48 Given picture of f ( x where f ( x is incresing 49 Given v ( t nd s (, find, find the gretest distnce from the origin of prticle on [, ] 5 Given wter tnk with g gllons F t gllons/min nd emptied t the rte of E ( t gllons/min on [ t,t ], find the mount of wter in the tnk t m minutes 5 the rte the wter mount is initilly eing filled t the rte of Mke sign chrt of f ( x nd determine where ( x (ove the x-xis f is positive Find where v(t= or DNE Plug criticl numers nd endpoints into s(t The lrgest is the mx m ( F( t E( t g + chnging t m ( F( t E( t = F( m E( m 5 c the time when the wter is t minimum d m t ( m E( m F =, testing the endpoints s well Plug ck into m ( F( t E( t g + t 53 Given chrt of x nd ( x f on selected vlues etween nd, estimte f ( c where c is etween nd 54 Given, drw slope field Strddle c, using vlue k greter thn c nd vlue h less thn c so f ( k f ( h f c k h Use the given points nd plug them into, drwing little lines with 55 Find the re etween curves x g x the indicted slopes t the points f, on [, ] A f ( x g( x 56 Find the volume if the re etween x g x = [ ], ssuming tht the f curve is ove the g curve = Or A f ( x g( x f, is rotted out the x-xis A = ( f ( x π ( g( x π [ ] ssuming tht the f curve is ove the g [ ] curve Or A = π ( f ( x π ( g( x 57 Given se, cross sections perpendiculr to the x-xis re semicircles, find the volume 58 Given s(t (position of prticle, find when the prticle hs incresing speed The re etween the curves typiclly is the se of your semicircles Typiclly the you cn find the dimeter s So the volume is π s where s is the length of side Find = = Mke sign chrts to determine where hve the sme sign

5 59 Find the volume if the re etween f(x, g(x is rotted out the line y = A = ( f ( x π ( g( x 6 Find lim f ( x lim if g ( x g(x = f (x = lim x π [ ] ssuming tht the f curve is [ ] ove the g curve Or A = π ( f ( x π ( g( x Use L Hopitl s Rule f ( x f '( x lim = lim g( x g' ( x

6 BC Prolems Find f (x lim f ( x dp = k M P(M P or dp = kp P M Signls logistic growth dp lim = M = P mx growth t M/ t 3 Find where x + x + x + x + fctors 4 The position vector of prticle moving in the plne is r(t = x(t, y(t Find the velocity 5 Find the ccelertion (t = x (t, y (t Fctor denomintor nd use prtil frction technique v(t = x (t, y (t 6 c Find the speed v(t = x (t [ ] + [ y (t ] 7 Given the velocity vector v(t = x(t, y(t nd position t time, find the position vector ( y( t + C to find C, rememering it is vector s(t = x t + Use s 8 When does the prticle stop? v(t = x t (= AND y t (= 9 c Find the slope of the tngent line to This is the ccelertion vector t t the vector t t Find the re inside the polr curve r = f (θ A = θ [ f ( θ ] dθ Find the slope of the tngent line to the polr curve r = f (θ θ x = r cosθ, y = r sinθ = dθ dθ Use Euler s method to pproximte f ( given, ( x, y = (,, nd x = 3 Is the Euler s pproximtion n underestimte or n overestimte? = ( x, ynew = yold + = + Look t sign of nd d y in the intervl This gives you incresing/decresing/concvity Drw picture to scertin nswer 4 Find x n e x where, n re integers Integrtion y prts, u dv = uv v du + C

7 5 Write series for x n cosx where n is n integer 6 Write series for ln(+ x centered t x = cos x = x! + x 4 4! x6 6! + Multiply ech term y x n Find Mclurin polynomil: P n ( x= f ( + f ( x + f! x + f 3! x f ( n n! x n 7 converges if p > n p n= 8 If f (x = + 6x +8x + 54 x 3 +, find f Plug in nd fctor This will e geometric series: r n = r n= 9 Find the intervl of convergence of series Let S 4 e the sum of the first 4 terms of n lternting series for f (x Approximte f (x S 4 (n + n! Suppose f (n (x = Write the n first four terms nd the generl term of series for f (x centered t x = c Use test (usully the rtio to find the intervl nd then test convergence t the endpoints This is the error for the 4 th term of n lternting series which is simply the 5 th term It will e positive since you re looking for n solute vlue You re eing given formul for the derivtive of f (x f ( x= f ( c+ f ( c ( x c+ f ( c ( x c + + f ( n ( c x c! n! n Given Tylor series, find the Lgrnge form of the reminder for the n th term where n is n integer t x = c 3 f ( x= + x + x! + x3 3! + f x You need to determine the lrgest vlue of the 5 th derivtive of f t some vlue of z Usully you re told this Then: R n ( x= f ( n+ ( z ( x c n+ ( n +! = e x 4 f ( x= x x3 3! + x5 5! 5 f ( x= x! + x 4 4! x6 6! n x n+! n n x n ( n! + m n 6 Find (sin x (cos x where m nd n re integers f ( x= sin x f ( x= cos x Integrtion y prts, u dv = uv v du + C or u-sustitution

8 7 Given x = f t (, y = g t (, find 8 Given x = f ( t, y = g( t, find d y 9 x = f (, t y = g(, t find rc length on t,t [ ] 3 Find horizontl tngents to polr curve r = f θ 3 Find verticl tngents to polr curve r = f θ 3 Given f ( x, find rc length on [,] 33 Given = find the rc length on [,] = d d y = d = t L = + t x = r cosθ, y = r sinθ Find where r sinθ = where r cosθ x = r cosθ, y = r sinθ Find where r cosθ = where r sinθ [ ] L = + f x β dr L = dθ α ( r + d

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