ISM_T11_C04_C - 212 Chapter 4 Applications of Derivatives #...

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212 Chapter 4 Applications of Derivatives 18. g(t) 1 t 3.1 g (t) 0 g(t) is increasing for t in ( 1 1); œ±± ² Ê œ ± ³ Ê ² ß "" " w ² t( t ) 21t È # È g( 0.99) 2.5 and g(0.99) 98.3 g(t) has exactly one zero in ( 1 1). ²œ ² œ Ê ² ß 19. r( ) sin 8 r ( ) 1 sin cos 1 sin 0 on ( ) r( ) is )) ) ) œ± ² Ê ³ ²_ ß_Ê #w " ˆ‰ ˆ ‰ ) ) 33 3 3 3 3 22 increasing on ( ); r(0) 8 and r(8) sin 0 r( ) has exactly one zero in ( ). ²_ß_ œ ² œ ³ Ê # 8 3 ) 20. r( ) 2 cos 2 r ( ) 2 2 sin cos 2 sin 2 0 on ( ) r( ) is increasing on ) ) ) ) ) ) œ ² ± Ê ³ È ( ); r( ) 4 cos ( ) 2 4 1 2 0 and r(2 ) 4 1 2 0 r( ) has ²# œ ² ² ²# ± œ ² ² ± ´ œ ² ± ³ Ê 11 1 1 1 1 ) ÈÈ È exactly one zero in ( ). 21. r( ) sec 5 r ( ) (sec )(tan ) 0 on r( ) is increasing on ; ) ) ) ) œ² ± Êœ ± ³ ! ß Ê ! ß " w ## ) ) $ % 3 r(0.1) 994 and r(1.57) 1260.5 r( ) has exactly one zero in . ¸² ¸ Ê ! ß ) 1 # 22. r( ) tan cot r ( ) sec csc 1 sec cot 0 on r( ) is increasing ) ) œ ² ² Ê œ ± ² œ ± ³ Ê w# # # # # 1 on 0 ; r 0 and r(1.57) 1254.2 r( ) has exactly one zero in . ˆ ßœ ² ´ ¸ Ê ! ß 1 1 # # 44 ) 23. By Corollary 1, f (x) 0 for all x f(x) C, where C is a constant. Since f( 1) 3 we have C 3 w œÊ œ ² œ œ f(x) 3 for all x. 24. g(x) 2x 5 g (x) 2 f (x) for all x. By Corollary 2, f(x) g(x) C for some constant C. Then œ±Ê œœ œ ± ww f(0) g(0) C 5 5 C C 0 f(x) g(x) 2x 5 for all x. Ê œ ± Ê œ Êœœ ± 25. g(x) x g (x) 2x f (x) for all x. By Corollary 2, f(x) g(x) C. œ ± w (a) f(0) 0 0 g(0) C 0 C C 0 f(x) f(2) 4 œ Ê œ ± Ê œ # (b) f(1) 0 0 g(1) C 1 C C 1 f(x) x 1 f(2) 3 œÊœ ±œ±Êœ ²Ê œ²Ê œ # (c) f( 2) 3 3 g( 2) C 3 4 C C 1 f(x) x 1 f(2) 3 ²œÊœ²±Êœ±Êœ # 26. g(x) mx g (x) m, a constant. If f (x) m, then by Corollary 2, f(x) g(x) b mx b where œ œ œ ± b is a constant. Therefore all functions whose derivatives are constant can be graphed as straight lines ym xb . 27. (a) y C (b) y C (c) y C xxx 34 # $ % # 28. (a) y x C (b) y x x C (c) y x x x C œ²± œ±²± ##$ # 29. (a) y x y C (b) y x C (c) y 5x C # " Ê œ ± œ ± ± œ ² ± xx x 30. (a) y x y x C y x C (b) y 2 x C w ±"Î# "Î# " # œ ± Ê œ ± œ ± (c) y 2x 2 x C ± # È 31. (a) y cos 2t C (b) y 2 sin C ± œ ± " # # t (c) y cos 2t 2 sin C ± ± " t 32. (a) y tan C (b) y y C (c) y tan C œ ± œ Ê ± ) ) ) w "Î# $Î# $Î# 33. f(x) x x C; 0 f(0) 0 0 C C 0 f(x) x x œ²± œ œ²±ÊœÊ œ² #
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Section 4.2 The Mean Value Theorem 213 34. g(x) x C; 1 g( 1) ( 1) C C 1 g(x) x 1 œ± ² ² œ ± œ± ² ± ² Ê œ± Ê ± "" " ## # ± x1 x 35. r( ) 8 cot C; 0 r 8 cot C 0 2 1 C C 2 1 )) ) 1 1 œ² ² œ œ ² ²Êœ²²Êœ ±± ˆ‰ 11 1 44 4 r( ) 8 cot 2 1 Êœ ²± ± ) 1 36. r(t) sec t t C; 0 r(0) sec (0) 0 C C 1 r(t) sec t t 1 œ± ²œ œ ± ² Ê œ ± Ê ± 37. v t s t t C; at s and t we have C s t t œ œ* Þ) ²&Ê œ% Þ* ²& ² œ"! œ! œ"!Ê œ% Þ* ²& ²"! ds dt # # 38. v t s t t C; at s and t we have C s t t œ œ $# ± # Ê œ "' ± # ² œ % œ œ " Ê œ ' ± # ² " ds dt " # 39. v sin t s cos t C; at s and t we have C s œ œ Ê œ± ² œ Ê œ ds dt cos t ab 1 1 40. v cos s sin C; at s and t we have C s sin œ œ ² œ " œ œ "Êœ ² " ds 2 t t t dt 1 1 # # 1 41. a v t C ; at v and t we have C v t s t t C ; at s and œ$#Ê œ$# ² œ#! œ#!Ê œ$# ²#!Ê œ"' ²#! ² œ& # # t we have C s t t œ&Ê œ"' ²#! ²& # # 42. a 9.8 v 9.8t C ; at v and t we have C v t s t t C ; at s and œ Ê œ ² œ±$ œ±$Ê œ* Þ) ±$Ê œ% Þ* ±$ ² # # t we have C s t t œ!Ê œ% Þ* ±$ # # 43. a sin t v cos t C ; at v
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This note was uploaded on 09/20/2010 for the course MATHEMATIC 09991051 taught by Professor Dr.maenshadeed during the Fall '10 term at Norwegian Univ. of Science & Technology.

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ISM_T11_C04_C - 212 Chapter 4 Applications of Derivatives #...

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