# AB Calculus Review Sheet

Size: px
Start display at page:

Transcription

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

### ( ) as a fraction. Determine location of the highest

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

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

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

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

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

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

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

### ( ) 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.

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

### ( ) as a fraction. Determine location of the highest

AB/ Clulus Exm Review Sheet Solutions A Prelulus Type prolems A1 A A3 A4 A5 A6 A7 This is wht you think of doing Find the zeros of f( x) Set funtion equl to Ftor or use qudrti eqution if qudrti Grph to

### critical number where f '(x) = 0 or f '(x) is undef (where denom. of f '(x) = 0)

Decoding AB Clculus Voculry solute mx/min x f(x) (sometimes do sign digrm line lso) Edpts C.N. ccelertion rte of chnge in velocity or x''(t) = v'(t) = (t) AROC Slope of secnt line, f () f () verge vlue

### Topics Covered AP Calculus AB

Topics Covered AP Clculus AB ) Elementry Functions ) Properties of Functions i) A function f is defined s set of ll ordered pirs (, y), such tht for ech element, there corresponds ectly one element y.

### Definition of Continuity: The function f(x) is continuous at x = a if f(a) exists and lim

Mth 9 Course Summry/Study Guide Fll, 2005 [1] Limits Definition of Limit: We sy tht L is the limit of f(x) s x pproches if f(x) gets closer nd closer to L s x gets closer nd closer to. We write lim f(x)

### MATH 144: Business Calculus Final Review

MATH 144: Business Clculus Finl Review 1 Skills 1. Clculte severl limits. 2. Find verticl nd horizontl symptotes for given rtionl function. 3. Clculte derivtive by definition. 4. Clculte severl derivtives

### 5.1 How do we Measure Distance Traveled given Velocity? Student Notes

. How do we Mesure Distnce Trveled given Velocity? Student Notes EX ) The tle contins velocities of moving cr in ft/sec for time t in seconds: time (sec) 3 velocity (ft/sec) 3 A) Lel the x-xis & y-xis

### AB Calculus Path to a Five Problems

AB Clculus Pth to Five Problems # Topic Completed Definition of Limit One-Sided Limits 3 Horizontl Asymptotes & Limits t Infinity 4 Verticl Asymptotes & Infinite Limits 5 The Weird Limits 6 Continuity

### lim f(x) does not exist, such that reducing a common factor between p(x) and q(x) results in the agreeable function k(x), then

AP Clculus AB/BC Formul nd Concept Chet Sheet Limit of Continuous Function If f(x) is continuous function for ll rel numers, then lim f(x) = f(c) Limits of Rtionl Functions A. If f(x) is rtionl function

### 2 b. , a. area is S= 2π xds. Again, understand where these formulas came from (pages ).

AP Clculus BC Review Chpter 8 Prt nd Chpter 9 Things to Know nd Be Ale to Do Know everything from the first prt of Chpter 8 Given n integrnd figure out how to ntidifferentite it using ny of the following

### Math 1431 Section M TH 4:00 PM 6:00 PM Susan Wheeler Office Hours: Wed 6:00 7:00 PM Online ***NOTE LABS ARE MON AND WED

Mth 43 Section 4839 M TH 4: PM 6: PM Susn Wheeler swheeler@mth.uh.edu Office Hours: Wed 6: 7: PM Online ***NOTE LABS ARE MON AND WED t :3 PM to 3: pm ONLINE Approimting the re under curve given the type

### Instantaneous Rate of Change of at a :

AP Clculus AB Formuls & Justiictions Averge Rte o Chnge o on [, ]:.r.c. = ( ) ( ) (lger slope o Deinition o the Derivtive: y ) (slope o secnt line) ( h) ( ) ( ) ( ) '( ) lim lim h0 h 0 3 ( ) ( ) '( ) lim

### A. Limits - L Hopital s Rule ( ) How to find it: Try and find limits by traditional methods (plugging in). If you get 0 0 or!!, apply C.! 1 6 C.

A. Limits - L Hopitl s Rule Wht you re finding: L Hopitl s Rule is used to find limits of the form f ( x) lim where lim f x x! c g x ( ) = or lim f ( x) = limg( x) = ". ( ) x! c limg( x) = 0 x! c x! c

### The Fundamental Theorem of Calculus, Particle Motion, and Average Value

The Fundmentl Theorem of Clculus, Prticle Motion, nd Averge Vlue b Three Things to Alwys Keep In Mind: (1) v( dt p( b) p( ), where v( represents the velocity nd p( represents the position. b (2) v ( dt

### Chapter 5 1. = on [ 1, 2] 1. Let gx ( ) e x. . The derivative of g is g ( x) e 1

Chpter 5. Let g ( e. on [, ]. The derivtive of g is g ( e ( Write the slope intercept form of the eqution of the tngent line to the grph of g t. (b Determine the -coordinte of ech criticl vlue of g. Show

### AP Calculus AB Unit 5 (Ch. 6): The Definite Integral: Day 12 Chapter 6 Review

AP Clculus AB Unit 5 (Ch. 6): The Definite Integrl: Dy Nme o Are Approximtions Riemnn Sums: LRAM, MRAM, RRAM Chpter 6 Review Trpezoidl Rule: T = h ( y + y + y +!+ y + y 0 n n) **Know how to find rectngle

### Practice Final. Name: Problem 1. Show all of your work, label your answers clearly, and do not use a calculator.

Nme: MATH 2250 Clculus Eric Perkerson Dte: December 11, 2015 Prctice Finl Show ll of your work, lbel your nswers clerly, nd do not use clcultor. Problem 1 Compute the following limits, showing pproprite

### Math 1431 Section 6.1. f x dx, find f. Question 22: If. a. 5 b. π c. π-5 d. 0 e. -5. Question 33: Choose the correct statement given that

Mth 43 Section 6 Question : If f d nd f d, find f 4 d π c π- d e - Question 33: Choose the correct sttement given tht 7 f d 8 nd 7 f d3 7 c d f d3 f d f d f d e None of these Mth 43 Section 6 Are Under

### A. Limits - L Hopital s Rule. x c. x c. f x. g x. x c 0 6 = 1 6. D. -1 E. nonexistent. ln ( x 1 ) 1 x 2 1. ( x 2 1) 2. 2x x 1.

A. Limits - L Hopitl s Rule Wht you re finding: L Hopitl s Rule is used to find limits of the form f ( ) lim where lim f or lim f limg. c g = c limg( ) = c = c = c How to find it: Try nd find limits by

### MATH SS124 Sec 39 Concepts summary with examples

This note is mde for students in MTH124 Section 39 to review most(not ll) topics I think we covered in this semester, nd there s exmples fter these concepts, go over this note nd try to solve those exmples

### Calculus AB Bible. (2nd most important book in the world) (Written and compiled by Doug Graham)

PG. Clculus AB Bile (nd most importnt ook in the world) (Written nd compiled y Doug Grhm) Topic Limits Continuity 6 Derivtive y Definition 7 8 Derivtive Formuls Relted Rtes Properties of Derivtives Applictions

### III. AB Review. The material in this section is a review of AB concepts Illegal to post on Internet

III. AB Review The mteril in this section is review of AB concepts. www.mstermthmentor.com - 181 - Illegl to post on Internet R1: Bsic Differentition The derivtive of function is formul for the slope of

### Calculus AB. For a function f(x), the derivative would be f '(

lculus AB Derivtive Formuls Derivtive Nottion: For function f(), the derivtive would e f '( ) Leiniz's Nottion: For the derivtive of y in terms of, we write d For the second derivtive using Leiniz's Nottion:

### First midterm topics Second midterm topics End of quarter topics. Math 3B Review. Steve. 18 March 2009

Mth 3B Review Steve 18 Mrch 2009 About the finl Fridy Mrch 20, 3pm-6pm, Lkretz 110 No notes, no book, no clcultor Ten questions Five review questions (Chpters 6,7,8) Five new questions (Chpters 9,10) No

### Time in Seconds Speed in ft/sec (a) Sketch a possible graph for this function.

4. Are under Curve A cr is trveling so tht its speed is never decresing during 1-second intervl. The speed t vrious moments in time is listed in the tle elow. Time in Seconds 3 6 9 1 Speed in t/sec 3 37

### Section 6: Area, Volume, and Average Value

Chpter The Integrl Applied Clculus Section 6: Are, Volume, nd Averge Vlue Are We hve lredy used integrls to find the re etween the grph of function nd the horizontl xis. Integrls cn lso e used to find

### First Semester Review Calculus BC

First Semester Review lculus. Wht is the coordinte of the point of inflection on the grph of Multiple hoice: No lcultor y 3 3 5 4? 5 0 0 3 5 0. The grph of piecewise-liner function f, for 4, is shown below.

### 1. Find the derivative of the following functions. a) f(x) = 2 + 3x b) f(x) = (5 2x) 8 c) f(x) = e2x

I. Dierentition. ) Rules. *product rule, quotient rule, chin rule MATH 34B FINAL REVIEW. Find the derivtive of the following functions. ) f(x) = 2 + 3x x 3 b) f(x) = (5 2x) 8 c) f(x) = e2x 4x 7 +x+2 d)

### A REVIEW OF CALCULUS CONCEPTS FOR JDEP 384H. Thomas Shores Department of Mathematics University of Nebraska Spring 2007

A REVIEW OF CALCULUS CONCEPTS FOR JDEP 384H Thoms Shores Deprtment of Mthemtics University of Nebrsk Spring 2007 Contents Rtes of Chnge nd Derivtives 1 Dierentils 4 Are nd Integrls 5 Multivrite Clculus

### Ch AP Problems

Ch. 7.-7. AP Prolems. Willy nd his friends decided to rce ech other one fternoon. Willy volunteered to rce first. His position is descried y the function f(t). Joe, his friend from school, rced ginst him,

### Review of basic calculus

Review of bsic clculus This brief review reclls some of the most importnt concepts, definitions, nd theorems from bsic clculus. It is not intended to tech bsic clculus from scrtch. If ny of the items below

### Unit #10 De+inite Integration & The Fundamental Theorem Of Calculus

Unit # De+inite Integrtion & The Fundmentl Theorem Of Clculus. Find the re of the shded region ove nd explin the mening of your nswer. (squres re y units) ) The grph to the right is f(x) = -x + 8x )Use

### Unit Six AP Calculus Unit 6 Review Definite Integrals. Name Period Date NON-CALCULATOR SECTION

Unit Six AP Clculus Unit 6 Review Definite Integrls Nme Period Dte NON-CALCULATOR SECTION Voculry: Directions Define ech word nd give n exmple. 1. Definite Integrl. Men Vlue Theorem (for definite integrls)

### 7.1 Integral as Net Change Calculus. What is the total distance traveled? What is the total displacement?

7.1 Integrl s Net Chnge Clculus 7.1 INTEGRAL AS NET CHANGE Distnce versus Displcement We hve lredy seen how the position of n oject cn e found y finding the integrl of the velocity function. The chnge

### B Veitch. Calculus I Study Guide

Clculus I Stuy Guie This stuy guie is in no wy exhustive. As stte in clss, ny type of question from clss, quizzes, exms, n homeworks re fir gme. There s no informtion here bout the wor problems. 1. Some

### Section Areas and Distances. Example 1: Suppose a car travels at a constant 50 miles per hour for 2 hours. What is the total distance traveled?

Section 5. - Ares nd Distnces Exmple : Suppose cr trvels t constnt 5 miles per hour for 2 hours. Wht is the totl distnce trveled? Exmple 2: Suppose cr trvels 75 miles per hour for the first hour, 7 miles

### Shorter questions on this concept appear in the multiple-choice sections. As always, look over as many questions of this kind from past exams.

22 TYPE PROBLEMS The AP clculus exms contin fresh crefully thought out often clever questions. This is especilly true for the free-response questions. The topics nd style of the questions re similr from

### x 2 1 dx x 3 dx = ln(x) + 2e u du = 2e u + C = 2e x + C 2x dx = arcsin x + 1 x 1 x du = 2 u + C (t + 2) 50 dt x 2 4 dx

. Compute the following indefinite integrls: ) sin(5 + )d b) c) d e d d) + d Solutions: ) After substituting u 5 +, we get: sin(5 + )d sin(u)du cos(u) + C cos(5 + ) + C b) We hve: d d ln() + + C c) Substitute

### Interpreting Integrals and the Fundamental Theorem

Interpreting Integrls nd the Fundmentl Theorem Tody, we go further in interpreting the mening of the definite integrl. Using Units to Aid Interprettion We lredy know tht if f(t) is the rte of chnge of

### Distance And Velocity

Unit #8 - The Integrl Some problems nd solutions selected or dpted from Hughes-Hllett Clculus. Distnce And Velocity. The grph below shows the velocity, v, of n object (in meters/sec). Estimte the totl

### MORE FUNCTION GRAPHING; OPTIMIZATION. (Last edited October 28, 2013 at 11:09pm.)

MORE FUNCTION GRAPHING; OPTIMIZATION FRI, OCT 25, 203 (Lst edited October 28, 203 t :09pm.) Exercise. Let n be n rbitrry positive integer. Give n exmple of function with exctly n verticl symptotes. Give

### We divide the interval [a, b] into subintervals of equal length x = b a n

Arc Length Given curve C defined by function f(x), we wnt to find the length of this curve between nd b. We do this by using process similr to wht we did in defining the Riemnn Sum of definite integrl:

### Summary Information and Formulae MTH109 College Algebra

Generl Formuls Summry Informtion nd Formule MTH109 College Algebr Temperture: F = 9 5 C + 32 nd C = 5 ( 9 F 32 ) F = degrees Fhrenheit C = degrees Celsius Simple Interest: I = Pr t I = Interest erned (chrged)

### Topics for final

Topics for 161.01 finl.1 The tngent nd velocity problems. Estimting limits from tbles. Instntneous velocity is limit of verge velocity. Slope of tngent line is limit of slope of secnt lines.. The limit

### DERIVATIVES NOTES HARRIS MATH CAMP Introduction

f DERIVATIVES NOTES HARRIS MATH CAMP 208. Introduction Reding: Section 2. The derivtive of function t point is the slope of the tngent line to the function t tht point. Wht does this men, nd how do we

### Definite Integrals. The area under a curve can be approximated by adding up the areas of rectangles = 1 1 +

Definite Integrls --5 The re under curve cn e pproximted y dding up the res of rectngles. Exmple. Approximte the re under y = from x = to x = using equl suintervls nd + x evluting the function t the left-hnd

### Mat 210 Updated on April 28, 2013

Mt Brief Clculus Mt Updted on April 8, Alger: m n / / m n m n / mn n m n m n n ( ) ( )( ) n terms n n n n n n ( )( ) Common denomintor: ( ) ( )( ) ( )( ) ( )( ) ( )( ) Prctice prolems: Simplify using common

### x = b a n x 2 e x dx. cdx = c(b a), where c is any constant. a b

CHAPTER 5. INTEGRALS 61 where nd x = b n x i = 1 (x i 1 + x i ) = midpoint of [x i 1, x i ]. Problem 168 (Exercise 1, pge 377). Use the Midpoint Rule with the n = 4 to pproximte 5 1 x e x dx. Some quick

### Main topics for the Second Midterm

Min topics for the Second Midterm The Midterm will cover Sections 5.4-5.9, Sections 6.1-6.3, nd Sections 7.1-7.7 (essentilly ll of the mteril covered in clss from the First Midterm). Be sure to know the

### Chapters 4 & 5 Integrals & Applications

Contents Chpters 4 & 5 Integrls & Applictions Motivtion to Chpters 4 & 5 2 Chpter 4 3 Ares nd Distnces 3. VIDEO - Ares Under Functions............................................ 3.2 VIDEO - Applictions

### Properties of Integrals, Indefinite Integrals. Goals: Definition of the Definite Integral Integral Calculations using Antiderivatives

Block #6: Properties of Integrls, Indefinite Integrls Gols: Definition of the Definite Integrl Integrl Clcultions using Antiderivtives Properties of Integrls The Indefinite Integrl 1 Riemnn Sums - 1 Riemnn

### Review Exercises for Chapter 4

_R.qd // : PM Pge CHAPTER Integrtion Review Eercises for Chpter In Eercises nd, use the grph of to sketch grph of f. To print n enlrged cop of the grph, go to the wesite www.mthgrphs.com... In Eercises

### 0.1 Chapters 1: Limits and continuity

1 REVIEW SHEET FOR CALCULUS 140 Some of the topics hve smple problems from previous finls indicted next to the hedings. 0.1 Chpters 1: Limits nd continuity Theorem 0.1.1 Sndwich Theorem(F 96 # 20, F 97

### 4.4 Areas, Integrals and Antiderivatives

. res, integrls nd ntiderivtives 333. Ares, Integrls nd Antiderivtives This section explores properties of functions defined s res nd exmines some connections mong res, integrls nd ntiderivtives. In order

### 6.5 Numerical Approximations of Definite Integrals

Arknss Tech University MATH 94: Clculus II Dr. Mrcel B. Finn 6.5 Numericl Approximtions of Definite Integrls Sometimes the integrl of function cnnot be expressed with elementry functions, i.e., polynomil,

### Math 1102: Calculus I (Math/Sci majors) MWF 3pm, Fulton Hall 230 Homework 2 solutions

Mth 1102: Clculus I (Mth/Sci mjors) MWF 3pm, Fulton Hll 230 Homework 2 solutions Plese write netly, nd show ll work. Cution: An nswer with no work is wrong! Do the following problems from Chpter III: 6,

### Keys to Success. 1. MC Calculator Usually only 5 out of 17 questions actually require calculators.

Keys to Success Aout the Test:. MC Clcultor Usully only 5 out of 7 questions ctully require clcultors.. Free-Response Tips. You get ooklets write ll work in the nswer ooklet (it is white on the insie)

### Calculus AB Section I Part A A CALCULATOR MAY NOT BE USED ON THIS PART OF THE EXAMINATION

lculus Section I Prt LULTOR MY NOT US ON THIS PRT OF TH XMINTION In this test: Unless otherwise specified, the domin of function f is ssumed to e the set of ll rel numers for which f () is rel numer..

### Review of Calculus, cont d

Jim Lmbers MAT 460 Fll Semester 2009-10 Lecture 3 Notes These notes correspond to Section 1.1 in the text. Review of Clculus, cont d Riemnn Sums nd the Definite Integrl There re mny cses in which some

### Thomas Whitham Sixth Form

Thoms Whithm Sith Form Pure Mthemtics Unit C Alger Trigonometry Geometry Clculus Vectors Trigonometry Compound ngle formule sin sin cos cos Pge A B sin Acos B cos Asin B A B sin Acos B cos Asin B A B cos

### Math 116 Calculus II

Mth 6 Clculus II Contents 5 Exponentil nd Logrithmic functions 5. Review........................................... 5.. Exponentil functions............................... 5.. Logrithmic functions...............................

### Big idea in Calculus: approximation

Big ide in Clculus: pproximtion Derivtive: f (x) = df dx f f(x +h) f(x) =, x h rte of chnge is pproximtely the rtio of chnges in the function vlue nd in the vrible in very short time Liner pproximtion:

### Precalculus Spring 2017

Preclculus Spring 2017 Exm 3 Summry (Section 4.1 through 5.2, nd 9.4) Section P.5 Find domins of lgebric expressions Simplify rtionl expressions Add, subtrct, multiply, & divide rtionl expressions Simplify

### 5: The Definite Integral

5: The Definite Integrl 5.: Estimting with Finite Sums Consider moving oject its velocity (meters per second) t ny time (seconds) is given y v t = t+. Cn we use this informtion to determine the distnce

### Chapter 6 Notes, Larson/Hostetler 3e

Contents 6. Antiderivtives nd the Rules of Integrtion.......................... 6. Are nd the Definite Integrl.................................. 6.. Are............................................ 6. Reimnn

### Chapter 9 Definite Integrals

Chpter 9 Definite Integrls In the previous chpter we found how to tke n ntiderivtive nd investigted the indefinite integrl. In this chpter the connection etween ntiderivtives nd definite integrls is estlished

### Sample Problems for the Final of Math 121, Fall, 2005

Smple Problems for the Finl of Mth, Fll, 5 The following is collection of vrious types of smple problems covering sections.8,.,.5, nd.8 6.5 of the text which constitute only prt of the common Mth Finl.

### Improper Integrals. Introduction. Type 1: Improper Integrals on Infinite Intervals. When we defined the definite integral.

Improper Integrls Introduction When we defined the definite integrl f d we ssumed tht f ws continuous on [, ] where [, ] ws finite, closed intervl There re t lest two wys this definition cn fil to e stisfied:

### 5 Accumulated Change: The Definite Integral

5 Accumulted Chnge: The Definite Integrl 5.1 Distnce nd Accumulted Chnge * How To Mesure Distnce Trveled nd Visulize Distnce on the Velocity Grph Distnce = Velocity Time Exmple 1 Suppose tht you trvel

### Final Exam Review. Exam 1 Material

Lessons 2-4: Limits Limit Solving Strtegy for Finl Exm Review Exm 1 Mteril For piecewise functions, you lwys nee to look t the left n right its! If f(x) is not piecewise function, plug c into f(x), i.e.,

### An Overview of Integration

An Overview of Integrtion S. F. Ellermeyer July 26, 2 The Definite Integrl of Function f Over n Intervl, Suppose tht f is continuous function defined on n intervl,. The definite integrl of f from to is

### The area under the graph of f and above the x-axis between a and b is denoted by. f(x) dx. π O

1 Section 5. The Definite Integrl Suppose tht function f is continuous nd positive over n intervl [, ]. y = f(x) x The re under the grph of f nd ove the x-xis etween nd is denoted y f(x) dx nd clled the

### Higher Checklist (Unit 3) Higher Checklist (Unit 3) Vectors

Vectors Skill Achieved? Know tht sclr is quntity tht hs only size (no direction) Identify rel-life exmples of sclrs such s, temperture, mss, distnce, time, speed, energy nd electric chrge Know tht vector

### Logarithmic Functions

Logrithmic Functions Definition: Let > 0,. Then log is the number to which you rise to get. Logrithms re in essence eponents. Their domins re powers of the bse nd their rnges re the eponents needed to

### Loudoun Valley High School Calculus Summertime Fun Packet

Loudoun Vlley High School Clculus Summertime Fun Pcket We HIGHLY recommend tht you go through this pcket nd mke sure tht you know how to do everything in it. Prctice the problems tht you do NOT remember!

### Fundamental Theorem of Calculus

Fundmentl Theorem of Clculus Recll tht if f is nonnegtive nd continuous on [, ], then the re under its grph etween nd is the definite integrl A= f() d Now, for in the intervl [, ], let A() e the re under

### Objectives. Materials

Techer Notes Activity 17 Fundmentl Theorem of Clculus Objectives Explore the connections between n ccumultion function, one defined by definite integrl, nd the integrnd Discover tht the derivtive of the

### Chapter 7 Notes, Stewart 8e. 7.1 Integration by Parts Trigonometric Integrals Evaluating sin m x cos n (x) dx...

Contents 7.1 Integrtion by Prts................................... 2 7.2 Trigonometric Integrls.................................. 8 7.2.1 Evluting sin m x cos n (x)......................... 8 7.2.2 Evluting

### Calculus Notes BC CH.7 Applications of Integration CH. 8 Integration Techniques CH. 9 Infinite Series

PG. Topic Clculus Notes BC PG. Topic 6 CH. Limits / Continuity - Find Limits Grphiclly nd Numericlly - Evluting Limits Anlyticlly - Continuity nd One-sided Limits -5 Infinite Limits/ -5 Limits tht Approch

### Polynomials and Division Theory

Higher Checklist (Unit ) Higher Checklist (Unit ) Polynomils nd Division Theory Skill Achieved? Know tht polynomil (expression) is of the form: n x + n x n + n x n + + n x + x + 0 where the i R re the

### MA Exam 2 Study Guide, Fall u n du (or the integral of linear combinations

LESSON 0 Chpter 7.2 Trigonometric Integrls. Bsic trig integrls you should know. sin = cos + C cos = sin + C sec 2 = tn + C sec tn = sec + C csc 2 = cot + C csc cot = csc + C MA 6200 Em 2 Study Guide, Fll

### Overview of Calculus

Overview of Clculus June 6, 2016 1 Limits Clculus begins with the notion of limit. In symbols, lim f(x) = L x c In wors, however close you emn tht the function f evlute t x, f(x), to be to the limit L

### Student Session Topic: Particle Motion

Student Session Topic: Prticle Motion Prticle motion nd similr problems re on the AP Clculus exms lmost every yer. The prticle my be prticle, person, cr, etc. The position, velocity or ccelertion my be

### MA123, Chapter 10: Formulas for integrals: integrals, antiderivatives, and the Fundamental Theorem of Calculus (pp.

MA123, Chpter 1: Formuls for integrls: integrls, ntiderivtives, nd the Fundmentl Theorem of Clculus (pp. 27-233, Gootmn) Chpter Gols: Assignments: Understnd the sttement of the Fundmentl Theorem of Clculus.

### Integrals - Motivation

Integrls - Motivtion When we looked t function s rte of chnge If f(x) is liner, the nswer is esy slope If f(x) is non-liner, we hd to work hrd limits derivtive A relted question is the re under f(x) (but

### Section 6.1 Definite Integral

Section 6.1 Definite Integrl Suppose we wnt to find the re of region tht is not so nicely shped. For exmple, consider the function shown elow. The re elow the curve nd ove the x xis cnnot e determined

### Anti-derivatives/Indefinite Integrals of Basic Functions

Anti-derivtives/Indefinite Integrls of Bsic Functions Power Rule: In prticulr, this mens tht x n+ x n n + + C, dx = ln x + C, if n if n = x 0 dx = dx = dx = x + C nd x (lthough you won t use the second

### Suppose we want to find the area under the parabola and above the x axis, between the lines x = 2 and x = -2.

Mth 43 Section 6. Section 6.: Definite Integrl Suppose we wnt to find the re of region tht is not so nicely shped. For exmple, consider the function shown elow. The re elow the curve nd ove the x xis cnnot

### MAT137 Calculus! Lecture 20

officil website http://uoft.me/mat137 MAT137 Clculus! Lecture 20 Tody: 4.6 Concvity 4.7 Asypmtotes Net: 4.8 Curve Sketching 4.5 More Optimiztion Problems MVT Applictions Emple 1 Let f () = 3 27 20. 1 Find

### INTRODUCTION TO INTEGRATION

INTRODUCTION TO INTEGRATION 5.1 Ares nd Distnces Assume f(x) 0 on the intervl [, b]. Let A be the re under the grph of f(x). b We will obtin n pproximtion of A in the following three steps. STEP 1: Divide

### 7.1 Integral as Net Change and 7.2 Areas in the Plane Calculus

7.1 Integrl s Net Chnge nd 7. Ares in the Plne Clculus 7.1 INTEGRAL AS NET CHANGE Notecrds from 7.1: Displcement vs Totl Distnce, Integrl s Net Chnge We hve lredy seen how the position of n oject cn e

### 1 The fundamental theorems of calculus.

The fundmentl theorems of clculus. The fundmentl theorems of clculus. Evluting definite integrls. The indefinite integrl- new nme for nti-derivtive. Differentiting integrls. Theorem Suppose f is continuous

### Math 190 Chapter 5 Lecture Notes. Professor Miguel Ornelas

Mth 19 Chpter 5 Lecture Notes Professor Miguel Ornels 1 M. Ornels Mth 19 Lecture Notes Section 5.1 Section 5.1 Ares nd Distnce Definition The re A of the region S tht lies under the grph of the continuous

### Table of Contents. 1. Limits The Formal Definition of a Limit The Squeeze Theorem Area of a Circle

Tble of Contents INTRODUCTION 5 CROSS REFERENCE TABLE 13 1. Limits 1 1.1 The Forml Definition of Limit 1. The Squeeze Theorem 34 1.3 Are of Circle 43. Derivtives 53.1 Eploring Tngent Lines 54. Men Vlue