The Fundamental Theorem of Calculus Solutions
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1 The Fundamenal Theorem of Calculus Soluions We have inenionally included more maerial han can be covered in mos Suden Sudy Sessions o accoun for groups ha are able o answer he quesions a a faser rae. Use your own judgmen, based on he group of sudens, o deermine he order and selecion of quesions o work in he session. Be sure o include a variey of ypes of quesions (muliple choice, free response, calculaor, and non-calculaor) in he ime alloed. Muliple Choice Quesions Soluions. A (985 AB) ( )( ) ( ). C (988 AB) k k k k 8 8 k k 8. C (988 AB6) The firs quadran inerval for y y avg avg y avg 7 9 y is [, ]. Copyrigh Naional Mah + Science Iniiaive, Dallas, TX. All righs reserved. Visi us online a
2 4. A (988 AB) Fundamenal Theorem of Calculus Using he Fundamenal Theorem of Calculus, f ( ) f ( b ) f ( a ), i follows direcly ha a c f ( ) f ( c ) f (). 5. D ( AB) f () f () f ( ) 5 8 Alernaively, he equaion for he derivaive shown is f ( ) 6 6. f ( ) ( 6 6) f ( ) 6 c Wih f () 5 implies 5 c and herefore b f () 6() E ( AB) Applying he Second Fundamenal Theorem, d 6 sin( ) d sin( ) ( ) sin( ) d g( ) a f ( ) d f ( g( )) g( ) 7. C (99 BC4 appropriae for AB) d ( ) f ( ) e d e ( ) f( ) only when. f for, and f for, hus f( ) has a minimum a. 8. D (988 AB4) cos d sin Le u sin u u du u ( ) u 9. C ( BC8 appropriae for AB) d g( ) f ( ) d f ( ) g() f () f (6) ( ). E (99 BC appropriae for AB) Q( ) p( ), so he degree of Q is n. Copyrigh Naional Mah + Science Iniiaive, Dallas, TX. All righs reserved. Visi us online a
3 Fundamenal Theorem of Calculus. E (97 BC45 appropriae for AB) F( ) g( ) wih and g( ), so F( ) ; herefore, F is decreasing (no increasing).. E ( AB9) d ln d ( )ln ( ) 6 ( )ln. B (988 AB9) u du u 5 and du 5 u (ln ln5) ln 5 ln, so u 4. D ( AB9) g ( ) is decreasing when g( ). d g( ) sin( ) d g( ) sin( ) Using a graphing calculaor, deermine where g( ) B (997 BC8 appropriae for AB) Since ( ) d d,. Using he calculaor, he greaes -value on he inerval [, 4] ha saisfies his inequaliy is found o occur a E (997 AB88) f ( ) h( ), so f( a) ; herefore, only choices (A) and (E) are possible. Bu a f ( ) h( ), so f is differeniable a b. This is rue for he graph in opion (E), bu no for he graph in opion (A), where here appears o be a sharp urn a b. Also, since h is increasing a firs, he graph of f mus sar ou concave up. This is also rue in (E) bu no (A). Copyrigh Naional Mah + Science Iniiaive, Dallas, TX. All righs reserved. Visi us online a
4 Fundamenal Theorem of Calculus 7. A (8 BC88 appropriae for AB) g( ) f ( ) d and f and f ; g( ) f ( ), so g ( ) f ( ). g() f ( ) d. Since f, g, so g is increasing, so f() g(), so g(). f, so g, so g g() g() g() g() is decreasing; herefore,, so he answer is choice A. 8. A ( AB8) The funcion r () is he rae of change in he aliude, so he aliude is decreasing when r ( ). The zeros of r () are.57 and.54, so he change in aliude when he aliude is decreasing can be found using r d. 9. A ( AB84) 5.4 T(5) T() ( e ) d F. A ( BC8 appropriae for AB). 4e T d ons 7 e Copyrigh Naional Mah + Science Iniiaive, Dallas, TX. All righs reserved. Visi us online a
5 Free Response Quesions. AB/BCacd Fundamenal Theorem of Calculus (a) 6 f d 4.74 or 4.75 cubic fee (c) h For 6, h h g s ds ds For 6 7, h h g s ds ds For 7 9, h h g s ds ds 7 7 h for 6 : : : h for 6 7 h for 7 9 Thus, ; 6 h 5( 6); ; (d) Amoun of snow is f d h cubic fee.. 9 AB/BCacd or 6.5 h 9 : (a) R d 98 people (c) w w w d R d 87.5 The oal wai ime for hose who ener he audiorium afer ime is 87.5 hours. R d On average, a person wais.775 or.776 hour. (d) w Copyrigh Naional Mah + Science Iniiaive, Dallas, TX. All righs reserved. Visi us online a
6 . 9 AB5/BC5b Fundamenal Theorem of Calculus (b) 5 f 5 f f f 5 8 : uses Fundamenal Theorem of Calculus 4. AB5a g 5 g 5 4 (a) g 5 g 5 : uses g 5 : g : g 5. 8B AB4 f 4 (a) sin g f cos 4 sin cos (c) For, g( ) only a. g g g d 4 The maimum value of g on, is 4 d. 4 : f : g g : ses : jusifies maimum a epression for g Copyrigh Naional Mah + Science Iniiaive, Dallas, TX. All righs reserved. Visi us online a
7 6. 9B ABac R 6 R() 6.6 or 6.6 (a) 6 sin Fundamenal Theorem of Calculus : epression for R : R (c) A d A A 4. or 4. From ime o years, he cross-secional area grows by 4. square cenimeers. : uses Fundamenal Theorem of Calculus : value of A d : meaning of A d 7. B AB6 f g cos 4 6 sin 4 4 (a) (c) h g 6 h g : aniderivaive : h Copyrigh Naional Mah + Science Iniiaive, Dallas, TX. All righs reserved. Visi us online a
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