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1 Studet s Prited Name: Istructor: XID: C Sectio: No questios will be aswered durig this eam. If you cosider a questio to be ambiguous, state your assumptios i the margi ad do the best you ca to provide the correct aswer. Istructios: You are ot permitted to use a calculator o ay portio of this test. You are ot allowed to use a tetbook, otes, cell phoe, computer, or ay other techology o ay portio of this test. All devices must be tured off ad stored away while you are i the testig room. Durig this test, ay kid of commuicatio with ay perso other tha the istructor or a desigated proctor is uderstood to be a violatio of academic itegrity. No part of this test may be removed from the eamiatio room. Read each questio carefully. To receive full credit for the free respose portio of the test, you must:. Show legible, logical, ad relevat justificatio which supports your fial aswer.. Use complete ad correct mathematical otatio.. Iclude proper uits wherever appropriate. 4. Give aswers as eact values wheever possible. You have 9 miutes to complete the etire test. Do ot write below this lie. Free Respose Problem Possible Eared Free Respose Problem Possible Eared.a b. 5.a...b (Scatro) 4.a. 5 Free Respose 58 4.b. 5 Multiple Choice 4 Test Total Versio A KEY Page of 5

2 Multiple Choice: There are 4 multiple choice questios. Each questio is worth poits ad has oe correct aswer. The multiple choice problems will be 4% of the total grade. Circle your choice o your test paper ad bubble the correspodig aswer o your Scatro. Ay questios ivolvig iverse trigoometric fuctios should be aswered based o the domai restrictios for trigoometric fuctios used i Sectio.4.. Determie the locatio ad value of the absolute etreme values of the fuctio f o the give iterval. f ( ) = o [, 5] A) f has absolute maimum of at = ad absolute miimum of at = 4. B) f has absolute maimum of 4 at = ad absolute miimum of 9 at =. C) f has absolute maimum of at = ad absolute miimum of 5 at = 5. D) f has absolute maimum of at = ad absolute miimum of at = 4.. Determie if Rolle s Theorem applies to the fuctio f o the give iterval. If ot, state why. If so, fid all values c guarateed to eist by Rolle s Theorem. f = + ( ) 5 4 o [, 5] A) Rolle's Theorem does ot apply because f is ot differetiable o (, 5). B) Rolle's Theorem does ot apply because f () f (5). C) Rolle's Theorem does ot apply because f is ot cotiuous o [, 5]. D) Rolle's Theorem applies; c=,,, 4. Versio A KEY Page of 5

3 . Use a midpoit Riema sum to approimate the area A of the regio bouded by the graph of f ad the -ais betwee = ad =. Divide the iterval [, ] ito = subitervals. y = f () A) A 4 B) A C) A D) A 4. What is the smallest value of b > such that b cos d=? A) B) b= b= 4 C) b = D) b = Versio A KEY Page of 5

4 5. Use the followig limits to determie the asymptotes of the fuctio f. lim f ( ) =, lim f ( ) = lim f ( ) =, lim f ( ) = + lim f ( ) =, lim f ( ) = + lim f ( ) =, lim f ( ) = A) Horizotal: y= ; Vertical: =, =, = 5; Slat: oe B) Horizotal: y =, y =, y = ; Vertical: =, = 5; Slat: y = + C) Horizotal: y=, y= ; Vertical: =, =, = 5; Slat: oe D) Horizotal: y =, y = ; Vertical: =, = 5; Slat: oe. The elevatio h (i feet above the groud) of a stoe dropped from a height of ft is modeled by the equatio h( t) = t, where t is measured i secods ad air resistace is eglected. Use differetials to approimate the chage i elevatio over the iterval t. secods. A) h 85. ft B) h 9. ft C) h.8 ft D) h 84. ft Versio A KEY Page 4 of 5

5 7. Use the give graph ad geometry to evaluate the defiite itegral. (4 ) d A) B) C) D) (4 ) d = (4 ) d = 5 (4 ) d = (4 ) d = 8. Fid the liear approimatio to f ( ) = si( ) + at a =. A) L( ) = + B) L( ) = + C) L( ) = + D) L( ) = + Versio A KEY Page 5 of 5

6 9. Fid all atiderivatives of f ( ) =. A) F( ) = + C B) F( ) = + C 4 C) ( ) l ( ) F = + C 8 + C D) F( ) = 4. Evaluate d d p dp. A) B) C) 4 D) 4 Versio A KEY Page of 5

7 . Determie the followig idefiite itegral. sec v sec v dv sec v A) B) cosv ta v+ C 4 sec v sec v 4 sec v v + C C) ta v+ C D) sec v + C. Use sigma otatio to write the followig Riema sum: the right Riema sum for o [, 5] with = 5. f = + ( ) A) B) C) D) 5 k= 5 k= 5 k= 5 k= k k + k k + Versio A KEY Page 7 of 5

8 . The Mea Value Theorem applies to the fuctio f o the give iterval. Fid all values c guarateed to eist by the Mea Value Theorem. f ( ) = l o [, e] (HINT: l = l + l ) A) c = e B) c = C) c = l e l D) c= 4. Suppose f ( ) d=, Evaluate ( ( ) + ( )) f g d. A) ( ) f ( ) + g( ) d = B) ( ) f ( ) + g( ) d= C) ( ) f ( ) + g( ) d = 7 D) ( ) f ( ) + g( ) d= 5 f ( ) d = 5, g( ) d =, ad g( ) d=. Versio A KEY Page 8 of 5

9 Free Respose: The Free Respose questios will be 58% of the total grade. Read each questio carefully. To receive full credit, you must show legible, logical, ad relevat justificatio which supports your fial aswer. Give aswers as eact values. Questios ivolvig iverse trigoometric fuctios should be aswered based o the domai restrictios i Sectio.4.. ( pts.) Evaluate the followig itegrals. a. (5 pts.) 9 d = 9 / / 9 = 4 = 4 9 d = = 4 = 8 Fids a atiderivative poits Substitutes the upper ad lower limits ito the result ad subtracts poits Evaluates the result poit Notes: Subtract poit maimum for otatio errors such as missig or icorrect itegral otatio, missig or icorrect use of groupig symbols, etc. b. (5 pts.) ( θ ) si dθ [ θ cosθ] cos ( ) cos( ) ( ) [ () ] = + = + + = + = 4 Fids a atiderivative ( poit per term) poits Substitutes the upper ad lower limits ito the result ad subtracts poits Evaluates the result poit Notes: Subtract poit maimum for otatio errors such as missig or icorrect itegral otatio, missig or icorrect use of groupig symbols, etc. Subtract ½ poit maimum for ot evaluatig the trigoometric fuctio or evaluatig icorrectly b Subtract poits for ( ) a f ( ) g( ) d = f ( ) g( ) d b a Versio A KEY Page 9 of 5

10 . ( pts.) Use the graphs of f ad f to complete the followig steps. Parts (a) ad (d): poits each * Subtract poit for per missig ad/or additioal value Parts (b), (c), (e): poits each poit per blak Part (f): poits * May award poit partial credit if choice follows icorrect work i most parts (a) (e) a. ( pts.) The critical poits of f are =.,, (Separate values with a comma.) b. ( pts.) f is icreasig o (, ), (, ) ad decreasig o. (, ), (, ) (Separate itervals with a comma.) (Separate itervals with a comma.) c. ( pts.) f has local maimum at = ad local miimum at =., (Separate values with a comma.) (Separate values with a comma.) d. ( pts.) f has iflectio poits at =., (Separate values with a comma.) e. ( pts.) f is cocave up o (, ), (, ) ad cocave dow o. (, ) (Separate itervals with a comma.) (Separate itervals with a comma.) f. ( pts.) Select a possible graph of f. I. II. III. Versio A KEY Page of 5

11 . (9 pts.) A right triagle has legs of legth h ad r ad a hypoteuse of legth 4 i. (See figure.) It is revolved about the leg of legth h to sweep out a right circular coe (radius r ad height h). What values of h ad r maimize the volume of the coe? (Volume of a coe = r h.) I your work, you should: State the fuctio to be optimized i terms of h. Use V for the volume of the coe. State the domai of the volume fuctio. Show all work eeded to fid the value of h that maimizes the volume. Use the st or d derivative test to verify the locatio of the absolute maimum. Give the dimesios of the coe havig maimum volume, icludig uits. Maimize Volume: V = r h 4 By the Pythagorea Theorem: r + h = 4 r = h V ( h) = h h = h h, Domai:, 4 V ( h) = ( h ) V = whe ( h ) = V always eists 4 4 h = or h = Secod Derivative Test Solvig for : (ot i domai) V ( h) = ( h) = h < for all h i the domai of V V = = < 4 4 V has absolute maimum at h = = r r ( ) ( ) ( ) = = = r = = = 4 4 Radius of the coe should be i. ad the height should be i. States the Pythagorea Theorem i terms of the give variables poits States the volume i terms of oe variable poits States the domai of the volume poit Takes the derivative of the volume poit Determies where the derivative is zero (Ratioalizig deomiators is NOT required.) poit Verifies the locatio of the absolute maimum by the st or d Derivative Test poit States results with appropriate uits (Setece is ot required.) poit Notes: Subtract ½ poit for missig upper boud o domai Subtract ½ poit for missig or icorrect uits Subtract ½ poit for each otatio error such as icorrect use of equals sigs, missig or icorrect use of paretheses, missig or icorrect derivative otatio, with a maimum deductio of poit Versio A KEY Page of 5

12 4. ( pts.) Evaluate the limits. Use of L Hôpital s Rule must be idicated each time it is used, either symbolically or i words. No credit will be awarded without supportig work. Gradig Notes for both parts (a) ad (b): Subtract ½ poit for failig to idicate use of L Hopital s Rule Subtract ½ poit for otatio errors such as missig or icorrect limit otatio, iappropriately usig limit otatio after direct substitutio, with a maimum of poit deductio for all otatio errors (ecludig errors idicatig use of L Hopital s Rule) Subtract ½ poit for the icorrect statemet aythig = a idetermiate form Subtract ½ poit for idicatig the wrog idetermiate form a. (5 pts.) e 4 lim e ( ) L e e e L e = lim lim = = lim = ( i. f.) () e e + e ( ) ( i. f.) + Applies L Hopital s Rule correctly ( pt for umerator, pt for deomiator) poits Applies L Hopital s Rule correctly a d time ( pt for umerator, pt for deomiator) poits Uses direct substitutio to fid fial aswer poit Notes: May recogize from comparig growth rates that result is ifiity, but must show work for at least the st applicatio of L Hopital s Rule. b. (5 pts.) lim 5/ 5/ i. f. 5 lim 5 5l lim l lim L l = lim e = e = e = e = e = ( i. f.) Rewrites usig e ad atural log fuctio poit Uses log property to simplify ad epress as oe fractio poit Applies L Hopital s Rule correctly ( pt for umerator, pt for deomiator) poits Uses direct substitutio to fid fial aswer poit Notes: Award full credit for applyig other correct techiques o Defie y as a fuctio ad take atural log of both sides o Defie y as the limit i the epoet, determie the value of the limit, the raise e to that result ( ) Versio A KEY Page of 5

13 5. ( pts.) A car startig at rest accelerates at ft/s for secods o a straight road. How far does it travel durig this time? a( t) = ft/s v( t) = a( t) dt = dt = t + C { = t + v } Give: v() = ft/s C = v( t) = t s t = v t dt = t dt = t + D = t + s ( ) ( ) 8 { 8 } Implied: s() = ft D = s( t) = 8t Distace traveled i secods: s () = 8() = 8(9) = 7 ft Fids velocity from acceleratio poits (Determiatio of particular costat of itegratio does ot eed to be eplicit.) Fids positio from velocity poits (Determiatio of particular costat of itegratio does ot eed to be eplicit.) Substitutes time ito positio fuctio to fid distace traveled poits Notes: Subtract ½ poit for otatio errors with a maimum of poits deductio for all otatio errors Subtract poit for missig or icorrect uits o fial aswer Versio A KEY Page of 5

14 . ( pts.) Idicated below is the regio bouded by the curve a. ( pts.) State ad evaluate the defiite itegral to calculate the area of the give regio. ( + ) 4 4 = + d y = + ad the -ais o [, ]. y = + () 4 4 = + = + = 4 States the defiite itegral to calculate the area ( pt for itegrad, ½ pt per limit of itegratio) Fids a atiderivative Substitutes the upper ad lower limits ito the result ad subtracts Evaluates the result poits poits poit poit b. (4 pts.) Below is the limit of a Riema Sum which also calculates the area of the give regio. Usig the summatio formulas as eeded, evaluate the limit. (You must show all work to receive full credit.) ( + ) ( + )(+ ) ( + ) c= c, i=, i =, i = 4 i= i= i= i= k lim + k = k 8 = lim lim k + k = = + k= k = 8 = lim ( + ) 4 ( ) = lim + = lim = lim lim lim = = + + = + + = Uses summatio formulas to get a epressio i terms of oly Evaluates the limit (No work required to resolve / i.f.) poits poit Versio A KEY Page 4 of 5

15 Scatro ( pt.) My Scatro: Check to make sure your Scatro form meets the followig criteria. If ay of the items are NOT satisfied whe your Scatro is haded i ad/or whe your Scatro is processed oe poit will be subtracted from your test total. is bubbled with firm marks so that the form ca be machie read; is ot damaged ad has o stray marks (the form ca be machie read); has 4 bubbled i aswers; has MATH ad my sectio umber writte at the top; has my istructor s last ame writte at the top; has Test No. writte at the top; has the correct test versio writte at the top ad bubbled i below my XID; shows my correct XID both writte ad bubbled i; Bubble a zero for the leadig C i your XID. Please read ad sig the hoor pledge below. O my hoor, I have either give or received iappropriate or uauthorized iformatio at ay time before or durig this test. Studet s Sigature: Versio A KEY Page 5 of 5

CALCULUS AB SECTION I, Part A Time 60 minutes Number of questions 30 A CALCULATOR MAY NOT BE USED ON THIS PART OF THE EXAM.

AP Calculus AB Portfolio Project Multiple Choice Practice Name: CALCULUS AB SECTION I, Part A Time 60 miutes Number of questios 30 A CALCULATOR MAY NOT BE USED ON THIS PART OF THE EXAM. Directios: Solve