Differentiation 9G. 1 a. 2t 1 2t 1 dx 2 dy 2 t(2t 1) 2t. t 3t. d 2 2. dx 2(1 t ) 4t 2(1 t ) y t t t t t d 2 (1 ) 2 (1 ) 4.

Transcription

1 Differeniaion 9G a c d e x, y, x, y 6, 6 d 6 6 x, y 6, d 6 5 x, y, 5 d 5 5, x y, 6 d 6 f x, y ( ), d ( ) d ( ) ( ) y d ( ) d g x, y ( ) ( ) d ( ) ( ) d ( ) ( ) y d ( ) ( ) ( ) h x e, y e e, d e e ( )e i x sin, y cos cos, 9sin 9sin an cos j x sin, y cos cos, sin d sin an cos Pearson Educaion Ld 07. Copying permied for purchasing insiuion only. This maerial is no copyrigh free.

2 k x sec, y an secan, sec d sec sec cosec sec an an l x sin, y cos cos, sin d sin sin cos cos sin cos co ( sin ) m x e 5, y ln e, d e n o x ln, y 6, d x e, y e e, e d e e e e a x sin, y cos cos, sin cos sin cos an cos A poin P, where =, x, y and d x Equaion of angen is y ( ) ( x ) y x x 9, y 6, 6 6 A poin P, where =, 5 x 5, y6 and d x Equaion of angen is 5 y6 ( x 5) y 5 5x y5x 57 a x e, y e e e, e e e e e e A poin P, where = 0, 0 Gradien of curve is 0 normal is parallel o he y-axis. When 0, x and y Equaion of he normal is x =. Pearson Educaion Ld 07. Copying permied for purchasing insiuion only. This maerial is no copyrigh free.

3 x cos, y sin sin, cos cos co sin A poin P, where an, 6 gradien of he normal is When, x cos 6 and y sin Equaion of he normal is x, y Using he quoien rule, d x ( ) ( ) d ( ) ( ) d y ( ) ( ) d ( ) ( ) d x ( ) ( ) ( ) 0 when 0 or When 0, x 0 and y 0 When, x and y y x y x y x 5 a (0, 0) and (, ) are he poins of zero gradien on he curve. x e, y e e, e e e e When ln, x, y and d x Equaion of angen is y ( x ) y x A saionary poins d y 0 0 e 0 e This has no soluions, so he curve has no saionary poins. Pearson Educaion Ld 07. Copying permied for purchasing insiuion only. This maerial is no copyrigh free.

4 6 x, y x, l is parallel o yx 5 so gradien of l is so (ecause 0) When, x and y Equaion of l is y ( x ) y x7 7 a x sin, y co sin cos, cosec cosec sec cosec sin cos When, x, y and Equaion of angen is 8 y x y 6 8x 8x y0 0 8 a x sin, y cosec 9 a x sin sin, y cosec, which is he y-coordinae of poin A. So a poin A. cos, cosec co cosec co cos co cosec cos co cosec When, cos gradien of normal is Equaion of normal, l, is y ( x ) y6 9( x ) 9xy 0 x, y 0 5, 0 0 When gradien is, A P, x ( 6) 6 0 and y ( 6) 0( 6) 5 0 Coordinaes of P are (0, 0). Pearson Educaion Ld 07. Copying permied for purchasing insiuion only. This maerial is no copyrigh free.

5 9 Equaion of angen a P is y0 ( x 0) yx c Susiuing for y and x in he angen equaion: 0 5 ( ) 6 0 Discriminan 6 0 Therefore he curve and he line only inersec once, so he angen a P does no inersec he curve again. 0 a x sin, y cos cos, sin sin sin cos cos cos sin When : x, y and 6 Equaion of normal a A is y ( x ) 6 6y x x 6y 0 a c Susiuing for y and x in he normal equaion: sin 6 cos 0 sin cos 0 sin ( sin ) 0 sin sin 0 (sin )( sin ) 0 sin or sin (nd opion no possile since sin ) sin or When : x sin, y cos, which is he same as poin A, so l does no inersec C oher han a poin A. x cos, y sin sin, cos cos sin When : x, y 6 and Equaion of angen a A is y x i.e. y x Pearson Educaion Ld 07. Copying permied for purchasing insiuion only. This maerial is no copyrigh free. 5

6 c l and l oh have gradien values of a poins where he angens cu he curve will e soluions o cos sin sin sin sin sin 0 (sin )(sin ) 0 sin or 5, or 6 6 So lines l and l ouch he curve when 5 and. 6 5 x, y 6 Equaion of l is y x i.e. y x x 0, y 0 Equaion of l is y0 ( x 0) i.e. y x Pearson Educaion Ld 07. Copying permied for purchasing insiuion only. This maerial is no copyrigh free. 6