2 f(x) dx = 1, 0. 2f(x 1) dx d) 1 4t t6 t. t 2 dt i)

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1 Mah PracTes Be sure o review Lab (ad all labs) There are los of good quesios o i a) Sae he Mea Value Theorem ad draw a graph ha illusraes b) Name a impora heorem where he Mea Value Theorem was used i he proof c) I your ow works, sae he defiiio of he defiie iegral of a coiuous fucio o a closed ierval [a, b] as a limi of some process d) Sae he Fudameal Theorem of Calculus (FTC I) e) Deermie d [ ] si () d Wha heorem(s) apply? d f) If g() d = si e, wha is g()? Eplai Draw ad he esimae Righ() for he graph of f o [, ] o he lef below Be careful of he scale I he graph of f o he righ above, assume ha f() d =, f() d =, ad f() d = Evaluae he followig a) f() d b) f() d c) f( ) d d) e) If f() is symmeric abou he origi (a odd fucio), wha is f() d? Calculae hese look-alike idefiie iegrals a) d b) d f) d g) + c) d h) 9 + d + d i) d) f() d d e) e e d j) + d + d a) If a aiderivaive of f() is si si +, wha was f()? b) O a rece es Judy said ha cos si d = si + c Elaie said ha cos si d = cos + c I gave hem boh full-credi How ca boh be correc if heir aswers are differe? Did I make a misake? a) Fill i he able for Righ() for ( ) d Be sure o simplify f( i ) f() [a, b] i f( i ) Righ() b) If you were o graph f() o [, ] i would be a icreasig fucio Is Righ() a over or uder esimae of ( ) d? Eplai c) Fid ( ) d usig a limi of Riema sums Check your aswer by evaluaig ( ) d usig he FTC d) Thik: O a es oce, I asked sudes o compue Righ() ad Righ() for f() = e o a ierval [a, b] A sude correcly compued boh wih his calculaor bu failed o label which is which Which of her wo values is Righ(): 788 or 7? Eplai clearly how you ca ell

2 7 Deermie hese idefiie ad defiie iegrals: a) e) 8 sec() a(), d b) e e e d f) + e e d c) si () d g) z a(z + ) dz d) sec( ) d h) si(si ) cos d sec ( ) d 8 The graph of f() is give below o he lef Aswer he followig quesios abou F () = a) O wha ierval(s) is F icreasig? Eplai b) A wha poi(s), if ay, does F have a local mi? Eplai c) O wha ierval(s) is F cocave dow? Eplai d) Does F have ay pois of iflecio? Eplai e) Fill i he he able below Your values should be reasoably accurae 7 8 F () = f() d f() d f) Graph F () o he aes o he righ usig your able values ad kowledge abou mas ad mis f() Thik ad Ierpre: The velociy of a objec v() i m/s is graphed above o he lef (same graph as i previous quesio) a) Whe is he objec movig backwards? Eplai b) Assume ha he iiial posiio of he objec is s() = Fill i he row i he able for he posiio s() for he objec The graph he posiio fucio o he aes above o he righ 7 8 Posiio s() Disace Traveled = b v() d c) Wha is v ave o he ierval [, 8]? (Hi: Use he v ave formula ad your work above) A wha ime(s) did v ave occur (ie, whe was v() acually equal o v ave? Eplai d) Speed is v() Graph he speed fucio for he objec o he aes o he lef Disace raveled is he iegral of he speed: b v() d Fill i he row i he able above for he disace raveled by he objec

3 Whe makig up his quesio, he prier jammed ad oly par of he graph of a differeiable fucio F () was pried ou, as show below Noeheless, he graph sill provides eough iformaio for you o precisely evaluae 8 F () d Wha is he value of his iegral (Look carefully a he iegrad) 7 8 Graph of F () a) Suppose ha he acceleraio of a objec is a() = e ad ha v() = ad s() =, where v ad s have heir usual meaig Fid s() b) Suppose isead ha a() = + ad ha s() = ad s() = 8 Fid s() Evaluae he followig epressio: d e d d O Mars he acceleraio due o graviy equals 9 ha of earh or f/sec/sec Suppose ha a Maria calculus sude hrows his calculus e upward a f/sec off he roof of a buildig a) If i akes secods o hi he groud, how high is he buildig? b) Wha was he velociy of he book whe i he groud? Le f() = o he ierval [, ] Suppose we wa o fid Righ() for his fucio a) Deermie he epressios for, k, ad f( k ) Be sure o simplify where possible b) Deermie ad simplify he epressio for Righ() c) Evaluae d as a limi of Riema sums d) Check your aswer by usig he Fudameal Theorem of Calculus Oe of he iegral properies saes ha if we swich he limis of iegraio, he sig of he iegral is swiched: a b f() d = b a f() d Give a brief eplaaio of why his should be so Hi: Thik abou Use he graph f below o draw ad he esimae he lef-had sum Lef() for he give fucio o [, ]

4 7 Three similar iegrals + a) d b) d c) + + d 8 My Hoda Accord acceleraes from o 88 f/sec ( mph) i secods Assume ha acceleraio is a cosa, a a) Fid he velociy fucio of he car (usig he velociies a he wo imes you ca elimiae ay cosas from his fucio) b) How far does i ravel i his secod period? 9 A balloo, risig verically wih a velociy of 8 f/s, releases a sadbag a he isa i is f above he groud a) How may secods afer is release will he bag srike he groud? Remember a() = f/s b) A wha velociy does i hi he groud? a) Fid he average value of f() = + o [, ] b) Fid he average value of f() = cos() o [ /, /] Why is his EZ? a) Page 79 #9 (Aswers i e) b) Page 79 # (Aswers i e) Area bewee curves: See Assigme from Day ad he associaed WeBWork problems (ad all of he eamples o lie i he Noes for Day ) Deermie a) cos () d b) si ( ) d Added From Moday s Class Se up he iegrals usig he fucios f(), g(), ad h() ad heir pois of iersecio ha would be used o fid he shaded areas i he hree regios below y = g() Shaded y = f() y = g() Shaded h() f() Skech he regios for each of he followig problems before fidig he areas a) Fid he area eclosed by he curves y = ad y = (Aswer: /) b) Fid he area eclosed by he curves y = + ad y = (Aswer: /) g() Sh f() Sh c) Fid he area bewee he curves f() = cos + si ad g() = cos si over [, ] (Aswer: 8)

5 Mah : Aswers o Praces a) Mea Value Theorem: See page 7 b) FTC II or Mea Value Theorem for Iegrals c) Pariio he ierval [a, b] io equal widh subiervals usig pois a = < < < < = b The defiie iegral of a coiuous fucio f o a closed ierval [a, b] is b a f() d = lim f(c i), where = b a ad ci is some poi i he i i-h subierval [ ] [ d d) si () d = d ] si () d = si ( ), where we used FTC II ad he chai rule d d [ e) FTC II implies g() = d ] g() d = d [ ] si = e cos e si So g() = e cos +e si = d d e e e i= = = So Righ() = ( 9) + ( 7) + ( ) + () + () = Use basic iegral properies a) b) d) f() d = f() d = f() d f() d f() d = ( ) = f() d = = c) f() d = + ( )[ ( )] = e) f( ) d = f() d = f() d = f() d = (horizoal shif) a) u =, du = d u / du = u/ + c = ( )/ + c b) Use a u du = arcsi u + c wih a a = so a =, u =, so u =, du = d arcsi( ) + c c) u =, du = 8 d, du = d d = 8 8 du = u / u/ + c = + c u du = arcsi( u ) + c = d) Mehod : u =, so u =, so u = so u du = d d = (u ) u u du = u 8u du = u 8 u + c = u 8 u + c = ( )/ 8 ( )/ + c d) Mehod : u =, so u = ad du = d ( u)u / du = u / u / du = 8 u/ + u/ + c = ( )/ 8 ( )/ + c e) u =, u =, du = d du = arca u + c = +u arca( ) + c f) u = +, du = d du = l u + c = l( + u ) + c g) a = 9 =, u =, u =, du = d du = arca( u ) + c = arca( ) + c +u 8 h) + d = + d = + + c (Jus divide firs by ) i) Use a du = arcsi u +c wih u a a = so a =, u = e, so u = e, du = e d e arcsi( ) + c u du = arcsi( u )+c =

6 j) u = u = du = d du = d So d = du = u arcsi u + c = arcsi + c k) + d = + d Use u = u = du = d du = d So + d = + u du = arca u + c = arca + c a) Sice f() d = si si +, he f() = d (si si + ) = cos si cos d d b) No Jus ake he derivaives of each: d (si + c) = si cos while d ( d cos + c) = cos ( si ) = si cos also The wo aidervaives differ by a cosa sice si = cos + a) f() [a, b] i f( i) Righ() ( ) ( ) ( ) [, ] + i i i + i ( + i ) i= b) Sice he fucio is icreasig (f () = > o [, ]), Righ() is a overesimae of d because f( i) will be greaer ha ay value of f() i [ i, i] 7 a) c) Righ() = So [ i i= + i = ] = i= i + i= [ ] [ ] i = ( + )( + ) + ( + ) [ ] = = + + ( ) d = lim Righ() = ad usig he FTC, d = = (7 9) ( ) = d) Sice e is a icreasig fucio Righ() is a overesimae Sice he esimaes improve (smaller overesimae) as ges larger, he Righ() > Righ() So Righ() = sec() + c = sec() + c (Meal adusme) b) u =, du = d e u du = e u + c = e + c c) u = z +, du = z dz, du = z dz So z a(z + ) dz = d) u = si, du = cos d si u du = cos(u) + c = cos(si ) + c a u du = l sec u + c = l sec(z + ) + c e) u = e + e, du = (e e )d Chage he limis: Whe =, u = e + e = ; whe =, u = e + e So f) si () d = Iegral propery g) sec( ) d = l sec( ) + a sec( ) + c Adjus! h) sec ( ) d = a( ) + c Adjus! e e e+e e + e d = e+e u du = l u = l e + e l 8 The graph of f() is give below o he lef Aswer he followig quesios abou F () = f() d a) F icresasig, so F = f > : [, ] ad [, 8] b) Local mi whe F = f chages from egaive o posiive: = c) b f() d is jus he e area uder he curve from o b b 7 8 F (b) = b f() d 8

7 f() F () = s() a) Whe is he objec movig backwards? v < so [,, ] b) Negaive acceleraio whe a = v < : so [, ] c) Posiio is he e area uder he curve (same able as above) 7 8 Posiio s() 8 Disace Traveled d) v ave = 8 v() d = [s(8) s()] = ( ) = vave occurred a c = ad 8 [Use he graph of v() = f()] From he graph we see ha F () = ad F (8) = So 8 F () d = F () 8 = F (8) F () = = a) Iegrae wice: v() = e d = e + c v() = = + c c = s() = e + d = e + + c s() = = + c c = So s() = e + + b) v() = + d = c So s() = c d = + + c + d Now s() = = d So s() = 8 = + + c + c = So s() = + + By FTC II ( d e d = d ) e d = ( ) e = 8 7 e d d a) Give s () =, s () = Fid s() The s () = d = + c Bu s () = = () + c meas c = ad s () = + Ne, s() = + d = ++d Now se s() = = () +()+d d = 7 f So s() = Ad s() = 7 b) Fid s () = () + = f/s Le f() = o he ierval [, ] Suppose we wa o fid Righ() for his fucio a) = possible b) So Righ() = c) Righ() =, i = + i ( i i= [ i i= i, ad f(i) = ( + ) ( + i ) = + i + i ( + i ) = i + i Be sure o simplify where + i + i ] = ) ( ) = 8 i + i= So d = lim Righ() = d) By FTC, d = = ( 9 9 ) ( i= [ ] [ ] i = 8 ( + )( + ) + ( + ) [ ] = = + + ) = 7

8 For a a b f() d, i he Riema sum = which is he egaive of he for he Riema sum for b f() d while he b a values of f( k ) remai he same Thus he sums will be he egaives of each oher Noe = = So L = (8) + (8) + () + ( 8) + ( 8) + () = 7 Similar iegrals doe differely ad oe era + a) d = + d = l + + c b) u = + du = d d = du = l u + c = l( + + u ) + c c) u = u = du = d d = du = arca u + c = + +u arca( ) + c 8 Give: a() = a cosa v = f/s s = Ad v() = 88 f/s a) For cosa acceleraio: v() = a + v = a Bu v() = a = 88 a = So v() = b) For cosa acceleraio: s() = a + v + s = Cosequely, s() = () = 7 f 9 Give: a() = f/s is cosa v = 8 f/s s = f So check ha v() = + 8 f/s ad s() = f a) The bag his he groud whe he posiio s() = +8+ = Usig he quadraic formula he soluio is = + (o ) b) The velociy whe i his he groud is v( + ) = ( + ) + 8 = 8 f/s a) f ave = d u = +, du = d Whe =, u = ; =, u = So + f ave = b) Use symmery, spli io odd ad eve erms: f ave = = / d + / / u du = l u / ( /) / cos() d = + / = l cos() d / cos() d = si / = ( ) = a) Noe he meal adjusme cos () d = + cos() d = + si() + c b) si ( ) d = cos( ) d = si() + c 8 Lef: f() d+ g() d Middle: Skech he curves! I have oly doe he iegrals here a) A = d = = ( ) = h() f() d+ g() f() d Righ: 7 b) A = ( +) ( ) d = + d = + + = ( + 8 ) ( + ) = A + A = + = c) A = (cos + si ) (cos si ) d = si d = cos si ) d = si d = cos = ( ) = Toal area is 8 g() f() d+ 7 f() g() d 7 = ( +) = A = ( ) ( +) d = = ( ) = A = (cos si ) (cos + 8