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Unit 5 Progress Check: MCQ Part A1-Let / be the function given byf(z )c*'In Lr1 )3- The derivative offis given by/ ’ i r )=■5 cos f yi sin ( y ) 4- - - •-. What 'LTaiue ofrsatisfies the conclusion of the Mean Value Theoremapplied to/on the interval1, L ?(A)2.132 because / ( 2. 132} —1 1 , 1/'(B)2.749 because/'(2.749 1— - 3-(C)3.042 because / ' ( 3.012)0(D)3.252 because/'(3.252)'.The derivative of the function / is given by /r( r irJ2- 3.Tj . On which of the following intervalsit’-4 , 3 ]i s /decreasing?(A)1,3.441], [1.806,11.(if>11Land 1 . 5 0 9 . 3 ](B)4 ,2.805] andl.227,IM>37|(C)3.414,1.806 and0.660,1.509](D):2.805,1.227] and 10.637, 3]3.The temperature inside a vehicle is modeled by the function / . where / ( / } is measured in degrees Fahrenheitand - is measured in minutes. The first derivative of / is given by / ' ( f }t~3 t -b cos t - At what timest .for0. t45does the temperature attain a local minimum?(A)0.354 only(B)1.962(C)3.299 only(D)0.354 and 3.2994,Let / be the function given byon the closed interval7. 7] - Of the following intervals, onwhich can the Mean Value Theorem be applied to / ?I.1 , 3 ] because / is continuous on1 , 3 ] and differentiable on i1 . 3 .II.•>.7becauseis continuous on 5 .7and differentiable on; j , “ |-III.1 . 5because■is continuous onI . 5and differentiable on ; I5 | .(A)None(B)Ionly(C)Iand II only(D)LIL and IH5.Let / be a differentiable function with f (0)4 and /I■l1 I. Which of the following must be true forsomerin the interval 0.1(1?

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Unit 5 Progress Check: MCQ Part A(A)f ' ( c ) —IK since the Extreme Value Theorem applies.(B)/ ' ( c ) =—■— — , since the Mean Value Theorem applies.(C)f ' i ci —3 i] |?D*i3since the Mean Value Theorem applies.(D)/ ' ( <.■i — 1.5, since the Intermediate Value Theorem applies.Let j be the function given by / ] . <——r—*1*—jy ™ the closed interval—5,5]. On which of the followingclosed intervals is the function / guaranteed by the Extreme Value Theorem to have an absolute maximum and anabsolute minimum?6.(A)[ - 5 , 5 ](E)[-3,1](C)[ - 2 , 0(D)[0,5]Let jf be the function defined by / ( z ) =z sin z with domain 0, oc )■The functionfhas no absolute minimumand no absolute maximum on its domain. Why does this not contradict the Extreme Value Theorem?(A)The domain ofjis not an open interval.(B)The domain of / is not a closed and bounded interval.(C)The functionfis not continuous on its domain.(D)The function f is not differentiable on its domain.z2345/(*)1142'131Selected values of a continuous function / are given ui the table above. Which of the following statements could befalse?. , ,By the Intermediate Value Theorem applied tofon the interval2 5■there is a value r such that1J/(<0 = 1 0 _______________________________________________(B)By the Mean Value Theorem applied to / on the interval 2 , 5 , there is a valuet:such thatf‘ -cI — 1(1.By the Extreme Value Theorem applied to / on the interval2 , 5 . , there is a valueesuch that1J/(c) '1f(?)forallxin[2t5 | .,By the Extreme Value Theorem applied to J on the interval2 , 5 , there is a value r such that(f(c) >f ( x )for all x in]2,5|-9.Let■' be the function defined by / ( z )z3ftz’- 9 z■4 for O 'z3. Which of the following statementsis true?

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Unit 5 Progress Check: MCQ Part A(A)fisdecreasing on the interval ((), 1 I because y'j ,r- ) < (J on the interval ((J, | . .(B)y isincreasing o n the interval i0TIbecause_r ) ..lion the interval ((), |.(C)/ isdecreasing on the interval i0. 21 because,r )■(jo n the interval0 . 2 ) .(D)fisdecreasing on the interval i ] .3 1because ’ 'i r i0on the interval i 1 . 3 : .10.Letfbe the function defined by / ( J?)J"In Jfor,r■(). On what open interval is / decreasing?(A)0 < £ <|only(B)0 <j< 1(C)» > 1(D)There is no such internal.11.Let■' be a function with first derivative given byf( r ) -j— 5 ) ‘ ( j: -I- 1 J. At what values of j? does / have arelativemaximum?(A)1only(B)0 only(C)1and 5 only(D)] , 0, and 5 onlyyG r a p ho ff 'The graph off.the derivative of the function f. is shown above for (I ... j? < <J. XMiich of the followingstatements is true for [) <j? < 9 ?Unit 5 Progress Check: MCQ Part A(A)fhas one relative minimum and two relative maxima.(B)fhas two relative minima and one relative maximum.(C)fhas two relative minima and two relative maxima.(D)fhas three relative minima and two relative maxima.

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Unit 5 Progress Check: MCQ Part BThe second derivative of the function / is given byf *' (> )x3cos'' ,i . At what values of x in theinter._al■1. 3 > does the graph offhave a point of inflection?(A)2.229 only(B)0 and 2.229(C)—2.357 and 0.987(D)3.259,0, and 1.603The second derivative of the function ,< is given byf( x |■s i n ()2 cos ;r- The function / has manycritical points, two of which are at j(J and j-6.949. Which of the following statements is true?(A)( has a local minimum at j- .(I and at j*6-919-(B)fhas a local minimum atr0 and a local maximum at r6.9-19.(C)/ has a local maximum at r — 0 and a local minimum at j—6,949-(D)fhas a local maximum at jQ and at j=6,949.v3.Letfbe the function given by / 1xJ2_r■’ -I 3x- 1■What is the absolute maximum value of / on the closedinterval - 3 , 1| ?(A)1(B)2(C)<5(D)264.Letfbe the function defined bvf(x '■= sin xcos ,r. What is the absolute minimum value of / on the interval(A)2(B)-v2(C)- 1(D)05.Letbe the function defined by y]'z)| r2— X + 1 ' cr■What is the absolute maximum value ofyon theinterval-1,1?(A)1(B)e(C)|(P)£

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Unit 5 Progress Check: MCQ Part BThe graph off',the derivative of the functionf,is shown above. On which of the following open intervals is thegraph offconcave down?(A)7 - 5 ,- 3 )and ( h 6 )(B)7 - 3 .t | and6 , 8 )(C)( - 1 . 4 )(B)(4,8)

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Unit 5 Progress Check: MCQ Part BLetfbe the function defined byf; j:x '1()j . The graph offf.the derivative of f3is shown above. Onwhich of the following intervals is the graph offconcave up?(A)j<v 3 and 0■j* <y'3(B)v'''3x■(Jandxv3(c)JC <- v6™d a; >G(D)VO < xV'63,The Second DerivativeTestcannot be usedtoconclude that2 is the location of a relative minimum or relativemaximum for which of the following functions?(A)f ( j ] ~2 ):where / ' ( > } -sin fj'2 )(B)/(jr) _ J-f:, wherec74(C)/(X) -P4 x2:Where/'(j)4CD)/ ( x ) -xa-& r2+12x -Lwhere /{ z)-L2x4-12

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