1. Find the zeros Find roots. Set function = 0, factor or use quadratic equation if quadratic, graph to find zeros on calculator


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1 AP Clculus Finl Review Sheet When you see the words. This is wht you think of doing. Find the zeros Find roots. Set function =, fctor or use qudrtic eqution if qudrtic, grph to find zeros on clcultor. Show tht f ( ) is even Show tht f ( ) = f ( ) symmetric to yis 3. Show tht f ( ) is odd Show tht f ( ) = f ( ) OR f ( ) = f ( ) 4. Show tht f ( ) 5. Find ( ) 6. Find ( ) symmetric round the origin lim eists Show tht lim f( )= lim f( ); eists nd re equl + lim f, clcultor llowed Use TABLE [ASK], find y vlues for vlues close to from left nd right lim f, no clcultor Sustitute = ) limit is vlue if c, incl. ; c c = ) DNE for 3) DO MORE WORK! 7. Find lim ( ) f 8. Find lim ( ) f ) rtionlize rdicls ) simplify comple frctions c) fctor/reduce d) known trig limits sin. lim = cos. lim = e) piecewise fcn: check if RH = LH t rek, clcultor llowed Use TABLE [ASK], find y vlues for lrge vlues of, i.e , no clcultor Rtios of rtes of chnges ) fst DNE slow = 9. Find horizontl symptotes of ( ) slow ) fst = 3) sme rtio sme = of coefficients f Find lim f ( ) nd lim f ( ). Find verticl symptotes of f ( ) Find where lim f ( ) = ± ± ) Fctor/reduce ( ) f nd set denomintor = ) ln hs VA t =
2 . Find domin of f ( ) Assume domin is (, ). Restrictle domins: denomintors, squre roots of only nonnegtive numers, log or ln of only positive numers, relworld constrints. Show tht f ( ) is continuous Show tht ) f ( ) ) f ( ) eists 3) lim f ( ) = f ( ) 3. Find the slope of the tngent line to ( ) =. f t 4. Find eqution of the line tngent to ( ) (, ) 5. Find eqution of the line norml, f t (perpendiculr) to f ( ) t ( ) 6. Find the verge rte of chnge of f ( ) on [, ] 7. Show tht there eists c in [ ] f ( c) = n 8. Find the intervl where ( ), such tht 9. Find intervl where the slope of ( ) incresing f is incresing Find ( ) f is. Find instntneous rte of chnge of ( ). Given () t f t lim eists ( lim Find derivtive f ( ) = m f ( ) = m nd use y = m( ) sometimes need to find = f ( ) f( )= lim + f( )) Sme s ove ut m = f ( ) f ( ) f ( ) Find Intermedite Vlue Theorem (IVT) Confirm tht f ( ) is continuous on[, ], then show tht f ( ) n f( ). f, set oth numertor nd denomintor to zero to find criticl points, mke sign chrt of f ( ) nd determine where f ( ) is positive. Find the derivtive of f ( ) = f ( ), set oth numertor nd denomintor to zero to find criticl points, mke sign chrt of f ( ) nd determine where f ( ) is positive. f Find ( ) s (position function), find v ( t) Find v ( t) = s ( t). Find f ( ) y the limit definition f ( + h) f ( ) f ( ) = Frequently sked ckwrds 3. Find the verge velocity of prticle on, [ ] 4. Given v () t, determine if prticle is speeding up t t = k 5. Given grph of f ( ), find where ( ) incresing f is lim or h h f ( ) f ( ) f ( ) = lim s s Find v() t dt OR s t know v( t ) or ( ) Find v( k) nd ( ) ( ) ( ) depending on if you k. If signs mtch, the prticle is speeding up; if different signs, then the prticle is slowing down. f is positive (ove the is.) Determine where ( )
3 6. Given tle of nd f ( ) on selected vlues etween nd, estimte f ( c) where c is etween nd. reltive mimum. 7. Given grph of f ( ), find where ( ) f hs 8. Given grph of f ( ), find where ( ) concve down. f is 9. Given grph of f ( ), find where ( ) point(s) of inflection. f hs Strddle c, using vlue, k, greter thn c nd vlue, h, f less thn c. so () ( k) f ( h) f c k h Identify where f ( ) to elow OR where ( ) = crosses the is from ove f is discontinuous nd jumps from ove to elow the is. f is decresing. Identify where ( ) Identify where ( ) f chnges from incresing to decresing or vice vers. 3. Show tht piecewise function is differentile t the point where the function rule splits 3. Given grph of f ( ) nd h( ) f ( ) ' find h ( ) 3. Given the eqution for f ( ) nd ' h( ) = f ( ), find h ( ) 33. Given the eqution for f ( ) 34. Given reltion of nd y, find dy lgericlly. First, e sure tht the function is continuous t = y evluting ech function t =. Then tke the derivtive of ech piece nd show tht lim f = lim f ( ) ( ) + Find the point where is the yvlue on f ( ) tngent line nd estimte f '( ) =,, sketch t the point, then h' ( ) = f '( ) Understnd tht the point (, ) is on h( ) so the point (, ) is on f ( ). So find where f ( ) = h' ( ) = f '( ), find its ) know product/quotient/chin rules derivtive lgericlly. ) know derivtives of sic functions. Power Rule: polynomils, rdicls, rtionls. e ; c. ln ;log d. sin ;cos ; tn e. rcsin ;rccos ;rctn ;sin ; etc Implicit Differentition Find the derivtive of ech term, using product/quotient/chin ppropritely, especilly, chin rule: every derivtive of y is multiplied y dy ; then group ll dy dy terms on one side; fctor out nd solve. f ( g ) Chin Rule f g g 35. Find the derivtive of ( ) ( ( )) ( )
4 36. Find the minimum vlue of function on, [ ] 37. Find the minimum slope of function on, [ ] = or DNE, mke sign chrt, find sign chnge from negtive to positive for reltive minimums nd evlute those cndidtes long with endpoints ck f nd choose the smllest. NOTE: e creful to Solve f ( ) 38. Find criticl vlues Epress ( ) 39. Find the solute mimum of ( ) 4. Show tht there eists c in [, ] such tht f '( c ) = 4. Show tht there eists c in [, ] such tht f '( c) = m 4. Find rnge of f ( ) on [ ] 43. Find rnge of ( ) into ( ) confirm tht f ( ) eists for ny vlues tht mke f '( ) DNE. Solve f "( ) = or DNE, mke sign chrt, find sign chnge from negtive to positive for reltive minimums nd evlute those cndidtes long with endpoints ck f ' nd choose the smllest. NOTE: e creful to into ( ) confirm tht f ( ) eists for ny vlues tht mke f "( ) DNE. f s frction nd solve for numertor nd denomintor ech equl to zero. f = or DNE, mke sign chrt, find sign chnge from positive to negtive for reltive mimums nd evlute those cndidtes into f ( ), lso find f lim f ; choose the lrgest. f Solve ( ) lim ( ) nd ( ) Rolle s Theorem Confirm tht f is continuous nd differentile on the intervl. Find k nd j in [, ] such tht f ( k) = f ( j), then there is some c in [ k, j ] such tht f ( c)=. Men Vlue Theorem Confirm tht f is continuous nd differentile on the, such tht intervl. Find k nd j in [ ] f ( k) f ( j). m = k j tht f ( c) = m., then there is some c in [ k, j ] such, Use m/min techniques to find vlues t reltive f (endpoints) f on ( ) 44. Find the loctions of reltive etrem of ' f ". f ( ) given oth f ( ) nd ( ) Prticulrly useful for reltions of nd y where finding chnge in sign would e difficult. m/mins. Also compre f ( ) nd ( ), Use m/min techniques to find vlues t reltive m/mins. Also compre lim f(). ± Second Derivtive Test Find where '( ) OR DNE f "( ) there. If f "( ) is positive, ( ) minimum. If f "( ) is negtive, ( ) mimum. f = then check the vlue of f hs reltive f hs reltive
5 45. Find inflection points of f ( ) lgericlly. Epress f ( ) denomintor equl to zero. Mke sign chrt of f ( ) 46. Show tht the line y = m + is tngent to f ( ) t (, y ) 47. Find ny horizontl tngent line(s) to f ( ) or reltion of nd y. 48. Find ny verticl tngent line(s) to ( ) reltion of nd y. 49. Approimte the vlue of (.) f or f y using the tngent line to f t = 5. Find rtes of chnge for volume prolems. 5. Find rtes of chnge for Pythgoren Theorem prolems. 5. Find the verge vlue of f ( ) on [, ] 53. Find the verge rte of chnge of f ( ) on [, ] 54. Given v () t, find the totl distnce prticle trvels on [, ] 55. Given v () t, find the chnge in position prticle trvels on [, ] s frction nd set oth numertor nd to f chnges sign. (+ to or to +) find where ( ) NOTE: e creful to confirm tht f ( ) eists for ny  vlues tht mke f "( ) DNE. Two reltionships re required: sme slope nd point of m f ', y is intersection. Check tht = ( ) nd tht ( ) on oth f ( ) nd the tngent line. Write dy s frction. Set the numertor equl to zero. NOTE: e creful to confirm tht ny vlues re on the curve. Eqution of tngent line is y =. My hve to find. Write dy s frction. Set the denomintor equl to zero. NOTE: e creful to confirm tht ny vlues re on the curve. Eqution of tngent line is =. My hve to find. Find the eqution of the tngent line to f using y y = m( ) where m = f ( ) nd the point is (, f ( ) ). Then plug in. into this line; e sure to use n pproimte ( ) sign. Alterntive lineriztion formul: y = f + f '( )( ) ( ) Write the volume formul. Find dv. Creful out dt product/ chin rules. Wtch positive (incresing mesure)/negtive (decresing mesure) signs for rtes. + y = z dy dz + y = z ; cn reduce s dt dt dt Wtch positive (incresing distnce)/negtive (decresing distnce) signs for rtes. f Find ( ) ( ) f ( ) f Find v () t dt Find vtdt ()
6 56. Given v () t nd initil position of prticle, find the position t t = d d f g( ) () t dt = Find vtdt () + s( ) Red crefully: strts t rest t the origin mens v = s ( ) = nd ( ) f ( ) f () t dt ( ) f g ( ) g'( ) 59. Find re using left Riemnn sums A se[ ] =... n Note: sketch numer line to visulize A = se Find re using right Riemnn sums [ ] Note: sketch numer line to visulize 6. Find re using midpoint rectngles 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. Note: sketch numer line to visulize 6. Find re using trpezoids se A = [ n + n ] This formul only works when the se (width) is the sme. Also trpezoid re is the verge of LH nd RH. If different widths, you hve to do individul trpezoids, A = h+ 63. Descrie how you cn tell if rectngle or trpezoid pproimtions over or underestimte re. 64. Given f ( ), find [ f ( ) + k] ( ) 3 Overestimte re: LH for decresing; RH for incresing; nd trpezoids for concve up Underestimte re: LH for incresing; RH for decresing nd trpezoids for concve down DRAW A PICTURE with shpes. n ( ) + = ( ) + = ( ) + ( ) f k f k f k 65. dy Given, drw slope field dy Use the given points nd plug them into, drwing little lines with the indicted slopes t the points. 66. y is incresing proportionlly to y ky y = Ae kt dt 67. Solve the differentil eqution Seprte the vriles on one side, y on the other. The nd dy must ll e upstirs. Integrte ech side, dd C. Find C efore solving for y,[unless ln y, then solve for y first nd find A]. When solving for y, choose + or (not oth), solution will e continuous function pssing through the initil vlue. 68. Find the volume given se ounded y The distnce etween the curves is the se of your f ( ) nd g( ) with f ( ) > g( ) nd squre. So the volume is ( ( ) ( )) cross sections perpendiculr to the is re f g squres
7 69. Given the vlue of F ( ) nd F '( ) f ( ) find F ( ) 7. Mening of f () t dt 7. Given () t distnce v nd s ( ), find the gretest =, from the origin of prticle on [, ] 7. Given wter tnk with g gllons initilly eing filled t the rte of F () t gllons/min nd emptied t the rte of E () t gllons/min on [, ], find ) the mount of wter in the tnk t m minutes 73. ) the rte the wter mount is chnging cnnot do. Utilize the fct tht if ( ) Usully, this prolem contins n ntiderivtive you F is the ntiderivtive of f, then f ( ) = F( ) F( ) for ( ). So solve F using the clcultor to find the definite integrl, ( ) = ( ) + ( ) F f F The ccumultion function: net (totl if ( ) mount of yunits for the function ( ) = nd ending t =. OR DNE Solve v( t ) =. Then integrte ( t) s ( ) to find s ( t) f is positive) f eginning t v dding. Finlly, compre s(ech cndidte) nd s(ech endpoint). Choose gretest distnce (it might e negtive!) m ( () ()) g+ F t E t dt t m () () 74. c) the time when the wter is t minimum 75. Find the re etween f ( ) nd ( ) f ( ) > g( ) on [, ] 76. Find the volume of the re etween f ( ) nd g( ) with f ( ) g( ) the is. 77. Given () t v nd ( ) g with >, rotted out s, find s () t 78. Find the line = c tht divides the re d dt m ( F t E t ) dt = F( m) E( m) Solve ( ) ( ) F t E t = to find cndidtes, evlute cndidtes nd endpoints s = in ( () ()) g+ F t E t dt, choose the minimum vlue ( ) ( ) A= f g π ( ( )) ( ) ( ) V = f g t () = ( ) + ( ) s t v s under f ( ) on [, ] to two equl res ( ) = ( ) c f f Note: this pproch is usully esier to solve thn c ( ) = ( ) f f c
8 79. Find the volume given se ounded y f ( ) nd g( ) with f ( ) > g( ) nd cross sections perpendiculr to the is re semicircles The distnce etween the curves is the dimeter of your circle. So the volume is ( ) ( ) f g π
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