AP Exam Practice Questions for Chapter 9
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1 AP Eam Practice Questios for Chater 9 AP Eam Practice Questios for Chater 9. Evaluate each series. I: Because ad 0, the series diverges. e II: si Because r e si.99 >, the series diverges. 5 III:! ( ) + ( + ) 5! lim lim ( +! ) By the Ratio Test, the series coverges. So, the aswer is C.. ( ) a ( + ) + + lim 0 By the Alteratig Series Test, the series coverges. ( ) ( ) ( ) lim lim lim By the Root Test, the series coverges absolutely. So, the aswer is B.. Let l.. ( l ) ( l ) l + l e e!!!! So, the aswer is C. 5( + ) ( ) Use the Limit Comariso test to comare with, which coverges for. > > 5( + ) ( ) ( ) lim lim 5 Because lim is fiite ad ositive, 5( + ) ( ) So, the aswer is C. 5. cos ( ) ( ) 0! ( ) ( ) coverges whe >. ( ) ( ) ( ) ( ) cos 0 ( )!!! 6! 8!!! 6! 8! cos !! 6! 8!!! 6! 8! So, the aswer is C. 08 Cegage Learig. All Rights Reserved. May ot be scaed, coied or dulicated, or osted to a ublicly accessible website, i whole or i art.
![(a) P ( ) g( ) + g ( )( ) 50 + ( ) So, ( ) ( ) Because g ( ) is icreasig o [ ] g > So, g( ) is cocave uward at, ad the taget lie at will lie below the grah of g. So, the aroimatio is less g.](/docs-images/90/104086804/images/2-1.jpg ". tha the actual value of ( ) (b) () () ()( ) () g () ( ) ( ) g P g + g + +!! 60 50 + ( ) + ( ) + ( )!! 60 7 50 + ( ) + ( ) + ( ) 8 60 7 So, g(.) 50 + (. ) + (. ) + (. ) 8 55.5. (c) g(.) P(.) R(.")
2 AP Eam Practice Questios for Chater 9 6. ( ) + a + + ( ) a ( ) ( ) ( ) + > + > So, the aswer is B ( ) Grah f( ) y y 5 ad y. + f() + 5 The grahs itersect at So, the aswer is C. 60 g ,, () (a) P ( ) g( ) + g ( )( ) 50 + ( ) So, ( ) ( ) Because g ( ) is icreasig o [ ] g > So, g( ) is cocave uward at, ad the taget lie at will lie below the grah of g. So, the aroimatio is less g.. tha the actual value of ( ) (b) () () ()( ) () g () ( ) ( ) g P g + g + +!! ( ) + ( ) + ( )!! ( ) + ( ) + ( ) So, g(.) 50 + (. ) + (. ) + (. ) (c) g(.) P(.) R(.) ( ) ( z) ( ) (. ) g! 8! Notes: Roud each aswer to at least three decimal laces to receive credit for the aroimatio. You do ot eed to simlify the coefficiets i these Taylor olyomials. I these aroimatios, be sure to write rather tha Because this is a aroimatio, a oit may be deducted if a equal sig is used. 08 Cegage Learig. All Rights Reserved. May ot be scaed, coied or dulicated, or osted to a ublicly accessible website, i whole or i art.
3 AP Eam Practice Questios for Chater 9 9. (a) e !!! ( ) ( ) e + ( ) !! ( ) !!! ( ) ( ) e !!! !!! So, the first four ozero terms are +!!. (b) lim ( ) + + f e lim 0 0 t: aswer with justificatio [usig results from art (a)] t g te dt 0 (c) ( ) ( ) !!! + t t t t t dt ( ) ! + t t t t t dt ( ) ( + ) t t t t t 8 0! ( ) ( + ) 8 0! Notes: This aroimatio does ot eed to be simlified. Write rather tha Because this is a aroimatio, a oit may be deducted if a equal sig is used. So, g (d) a ( ) ( + )( + ) 6 ( ) ( )( ) 90,000! 90, ! 90,000,50,000 90,000 Note: This error boud does ot eed to be simlified. 08 Cegage Learig. All Rights Reserved. May ot be scaed, coied or dulicated, or osted to a ublicly accessible website, i whole or i art.
4 AP Eam Practice Questios for Chater 9 0. (a) ( ) ( ) cos + + +!!! cos ( ) ( ) ( ) ( ) ( ) !! 6!! ( ) ( ) ( ) ! Note: You do ot eed to simlify the coefficiets i these Taylor olyomials. (b) + ( + ) ( ) ( ) ( ) ( ) ( + )( + )! lim lim 0 +! Because the series coverges for all, R. (c) 8 + cos () () 8 cos() + a + 6! () 6! Notes: Write rather tha Because this is a aroimatio, a oit may be deducted if a equal sig is used. This error boud does ot eed to be simlified. 08 Cegage Learig. All Rights Reserved. May ot be scaed, coied or dulicated, or osted to a ublicly accessible website, i whole or i art.
5 AP Eam Practice Questios for Chater 9 5. (a) f( )!!! 5! f ( ) f ( ) f 0 0 ad f 0. So, ( ) ( ) Because f ( 0) 0, f has a critical value at 0. Because f ( 0) 0, f is cocave dowward at 0. So, by the Secod Derivative Test, f has a relative maimum at 0. Note: Elicitly idetify each fuctio by ame. Referrig to it, the fuctio, or the grah will ot receive credit o the eam. (b) f( ) f 6 + () ()!! () () 6 +!!!! Use the Alteratig Series Test to fid the error. () 8 8 a +!! 0 So, f () with a error of less tha.!! 0 y f!!! 5! y (c) ( ) y y !!! 5! Cegage Learig. All Rights Reserved. May ot be scaed, coied or dulicated, or osted to a ublicly accessible website, i whole or i art.
![Whe : ( ) ( ) ( ) + + The series coverges by the Alteratig Series Test. So, the iterval of covergece is ( 0, ].](/docs-images/90/104086804/images/6-1.jpg "(b) f( ) ( ) ( ) ( ) ( ) + + ( ) ( ) ( 0) ( ) ( ) ( ) g f + + ( ) ( ) ( ) ( ) ( ) + + + (c) g ( ) ( ) + ( ) ( ) + + ( ) ( ) t: aswer (d) h ( ) f( + ) h (")
6 6 AP Eam Practice Questios for Chater 9. (a) f( ) lim ( ) ( ) ( ) + a + lim a + ( ) lim + 0 ( ) ( ) ( ) ( ) + + Whe 0: The series diverges (diverget -series). Whe : ( ) ( ) ( ) + + The series coverges by the Alteratig Series Test. So, the iterval of covergece is ( 0, ]. (b) f( ) ( ) ( ) ( ) ( ) + + ( ) ( ) ( 0) ( ) ( ) ( ) g f + + ( ) ( ) ( ) ( ) ( ) (c) g ( ) ( ) + ( ) ( ) + + ( ) ( ) t: aswer (d) h ( ) f( + ) h ( ) f ( + ) ( ) g ( + )( ) + + ts: fids [usig g from art (c)] 08 Cegage Learig. All Rights Reserved. May ot be scaed, coied or dulicated, or osted to a ublicly accessible website, i whole or i art.
7 AP Eam Practice Questios for Chater 9 7. (a) Because g ( ), the equatio of the taget lie at (, ) is y ( ) y 5. So, g(.) (.) 5.6. Because g ( ) 0, g ( ) > is cocave uward at, ad the taget lie at will lie below the grah of g. So, the aroimatio is less tha g (. ).. (b) g ( ) d 0.g (.) g (.) 0.( 5 + 8) +.6. (.) ( ) + ( ) g g g d (c) g(.) g( ) + 0.g ( ) + 0.( ).8 g(.) g(.) 0.g (.) ( 7) 5. + (d) P ( ) g( ) g ( )( ) ( ) ( ) ( )( ) g + +! 0 + ( ) + ( ) So, ( ) ( ) ( ) Error: g ( z) ( ) g 6 8.! Notes: These aroimatios do ot eed to be simlified. Write rather tha Because these are aroimatios, a oit may be deducted if a equal sig is used. 08 Cegage Learig. All Rights Reserved. May ot be scaed, coied or dulicated, or osted to a ublicly accessible website, i whole or i art.
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