EXERCISE 97 Page 9 CHAPTER MACLAURIN S SERIES. Determine the irst our terms o the power series or sin using Maclaurin s series. Let () sin () sin () cos () cos () sin () sin () 8 cos () 8 cos 8 i( ) 6 sin i() 6 sin ( ) cos () cos ( ) 6 sin () 6 sin i ( ) 8 cos i () 8 cos 8 Maclaurin s series states: () () + () +! () +! () + i.e. () 5 6 7 + () + () + ( 8) + () + () + () + ( 8)!!! 5! 6! 7! 8 5 87 + 6 5 8 i.e. sin + 5 7 5 5. Use Maclaurin s series to produce a power series or cosh as ar as the term in 6 Let () cosh () cosh () sinh () sinh () 9 cosh () 9 cosh 9 () 7 sinh () 7 sinh i( ) 8 cosh i() 8 cosh 8 59, John Bird
( ) sinh () sinh ( ) 79 cosh () 79 cosh 79 Maclaurin s series states: () () + () +! () +! () + 9 8 79 i.e. () + + + 7 5 6 + () + (9) + () + (8) + () + (79) +...!!! 5! 6! 6 9 7 8 i.e. cosh + + + 6 8 8. Use Maclaurin's theorem to determine the irst three terms o the power series or ln( + e ). Let () ln ( + e ) () ln ( e ) + ln () e + e () e + e () ( + e ) e e ( e ) ( + e ) + e e e e () ( ) ( ) ( + e ) Maclaurin s series states: () () + () + () +! i.e. ln ( e ) + ln + + +...! ln + + 8! () +. Determine the power series or cos t as ar as the term in t 6 Let (t) cos t () cos (t) sin t () sin (t) 6 cos t () 6 cos 6 (t) 6 sin t () 6 sin 6, John Bird
i () t 56 cos t i() 56 cos 56 () t sin t () sin () t 96 cos t () 96 cos 96 Maclaurin s series states: (t) () + t () + t! () + t! () + 6t 56t 96t i.e. () + 7 t t t t5 t6 + t() + ( 6) + () + (56) + () + ( 96) +...!!! 5! 6! 6 56 i.e. cos t 8t + t t6 5 5. Epand e (/) in a power series as ar as the term in Let () e (/) () e () e () e () 9 e 9 () e 9 () 7 e 8 7 () e 8 7 8 Maclaurin s series states: () () + () +! () +! () + 9 7 + + + +...!! 8 9 7 i.e. () + + + 8 8 9 9 i.e. e (/) + + + 8 6 6. Deelop, as ar as the term in, the power series or sec. Let () sec () sec 6, John Bird
() sec tan () () ( sec )( sec ) + ( tan )( sec tan ) sec [ sec + tan ] sec [ sec + sec ] [ ] sec sec 8sec sec () 8 () ( ) sec sec tan 8sec tan 8sec tan 8sec tan () i( ) ( 8sec )( sec ) + ( tan )( sec (sec tan ) ) ( 8sec )( sec ) ( tan )( 6sec tan ) + i() 96 + 6 8 Maclaurin s series states: () () + () + () +! + () + () + () + (8)!!! 8 + + () +! i.e. sec + + as ar as the term in 7. Epand e cos as ar as the term in using Maclaurin s series. Let () e cos () e cos () ( e )( sin ) + ( cos)( e ) e ( cos sin ) () e ( cos sin ) () ( e )( 6sin 9cos ) ( cos sin)( e ) + () 9 + 5 Maclaurin s series states: () () + () +! + () + ( 5)! () +! () + 6, John Bird
5 i.e. e cos + as ar as the term in 8. Determine the irst three terms o the series or sin by applying Maclaurin s theorem. Let () sin () sin () sin cos () sin cos () ( sin )( sin ) + (cos )( cos ) ( ) sin + cos cos sin cos () cos () sin () sin i( ) 8 cos i() 8 cos 8 ( ) 6 sin () 6 sin ( ) cos () cos Maclaurin s series states: () () + () + () +! i.e. 6 sin () +! 5 6 + () + () + () + ( 8) + () + () +...!!! 5! 6! + to three terms 5 9. Use Maclaurin s series to determine the epansion o ( + t) Let (t) ( + t) () 8 (t) ( + t) () 8( + t) () 8() 6 (t) ( + t) () 8( + t) () 8() (t) 96( + t)() 9( + t) () 9() 576 i( ) t 9() 8 i() 8 t Maclaurin s series states: (t) () + t () + () +! t! () + t t t 8 + t(6) + () + (576) + (8) +...!!! 6, John Bird
i.e. (t) ( + t) 8+ 6t+ 6t + 96t + 6t 6, John Bird
EXERCISE 98 Page. Ealuate.6 e sin d., correct to decimal places, using Maclaurin s series. Let () esin () e sin () ( esin )( cos ) () sin ( )( ) () ( e sin )( sin sin ) + ( cos)( e cos) e cos esin ( cos sin) () esin ( cos sin ) () ( esin)( cos sin cos sin ) + ( cos sin)( e cos) () ( esin )( cos sin cos sin ) ( cos sin )( e cos ) + ()( ) + ()() Maclaurin s series states: () () + () +! + + + () +! () + Hence,.6 e sin d..6.6 d. + + + +. ( ) ( ) ( ).6. (.6) +.6 + (.) + (.) + (.8 +.5 +.8) (.6 +.6 +.).8.66.78, correct to decimal places. Use Maclaurin s theorem to epand cos and hence ealuate, correct to decimal places, cos d Let () cos () cos () sin () sin () cos () cos () 8 sin () 8 sin 65, John Bird
i( ) 6 cos i() 6 cos 6 ( ) sin () sin ( ) 6 cos () 6 cos 6 Maclaurin s series states: () () + () +! () +! () + 5 6 + () + ( ) + () + (6) + () + ( 6) +...!!! 5! 6! i.e. 6 6 6 () + +... 7 i.e. cos + 6 +... 5 Hence, Hence, 6... 6 cos + 5 + 5 + +... cos 5 7 d d + 5 + 5 5 7 8 + 8 5 7 75.5.75 + ( ).88, correct to decimal places. Determine the alue o cosd, correct to signiicant igures, using Maclaurin s series. 6 From Problem, page 6, cos + +...!! 6! Since then 5 9 6 cos + +... + +...!! 6! 7 66, John Bird
Thus, cosd + +... d 7 5 9 7 + +... 7 () () (7) 7 68 + ( ).5, correct to signiicant places. Use Maclaurin s theorem to epand ln( + ) as a power series. Hence ealuate, correct to decimal places,.5 ln( + ) d. From page 7, ln( + ) + +... Hence, 5 ln( )d + + + +... d 5 5 7 9.5... + + 5 6.5.5 5 7 9 5 + + +... 5 7 9 5 5 6.5 5 7 7 65 5 7 9 (.5) (.5) + (.5 ) (.5) + (.5) +... [ ].77.6 +.7. +..67.6, correct to decimal places 67, John Bird
EXERCISE 99 Page. Determine: + lim + 5 + + 5 lim 6 + 6 + 9. Determine: sin sin cos cos. Determine: ln( + ) ln( + ) +. Determine: lim sin + sin + cos lim + 5. Determine: sin cos lim ( )( ) cos sin cos sin lim + sin cos lim cos + sin ( sin ) + cos () + cos + 6 6 6 6 68, John Bird
6. Determine: ln t t t ln t lim lim t t t t t 7. Determine: sinh sin lim sinh sin lim cosh cos sinh + sin cosh + cos 6 6 + 6 6 8. Determine: sin ln sin π sin ln sin π cos π lim lim sin sin cos sin { } π π 9. Determine: sect t tsin t sect secttan t (sec t)(sec t) + (tan t)(sec ttan t) t tsin t t tcost sin t t + t( sin t) + cost+ cost + + + 69, John Bird