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) dx using left Riemnn sums with n rectngles. Approximte " f ( x) dx using right Riemnn sums with n rectngles. Approximte Riemnn sums. Approximte " f ( x) dx using midpoint " f ( x) dx using trpezoidl summtion. Find " f ( x) dx where <. Mening of Given x " f ( t) dt. " f ( x) dx, find " [ f ( x) + k] dx. Given the vlue of F ntiderivtive of f is F, find F( ). Find d dx Find d dx x " f ( t) dt. g( x) " f ( t) 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 dx = f ( x )g( y). Find the verge vlue of f ( x) on [, ]. Find the verge rte of chnge of F " x [ ]. ( ) on t 1,t 2 This is wht you think of doing F10 y is incresing proportionlly to y. F11 Given dy, drw slope field. dx 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 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. 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 [ t 1,t 2 ], 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 t 2 " 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,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 www.mstermthmentor.com - 6 - Stu Schwrtz