2022 J1 RVHS End of Year Revision Package Solutions Ch5-7

H2 Mathematics (9758) JC 1 River Valley High School, Mathematics Department, 2021 End-of-Year Revision Package 91 5 Differentiation & Applications Basic Skills 1. Prove these trigonometric identities. (a) 1 cos sin 2 sin 1 cos sin x x x x x    (b) tan cotsin cos sec cosec x xx x x x    (c)   2 2 1 cot1 sec 22 cot 1 xx x   (d) tan cotsec 2 cot tan x xx x x   (e) sin sin 2 sin 3 tan 2cos cos 2 cos3 x x x xx x x     (a)       2 2 2 2 1 cos sin sin 1 cos 1 cos sin sin 1 cos 1 2cos cos sin sin 1 cos 2 1 cos sin 1 cos 2 sin x x x x x x x x x x x x x x x x x              (b)    2 2 tan cot sec cosec sin cos cos sin1 1 cos sin sin cos sin cossin cos sin cos sin cos sin cos sin cos sin cos x x x x x x x x x x x x x xx x x x x x x x x x x x              (c)   2 2 2 2 2 2 2 2 2 cot cot 1 cos sin cos 1sin cos cos sin cos 2 1 2 cos 2 1 cos 2 1 2 cos 2 1 1 sec 22 x x x x x x x x x x x x x x               (d) 2 2 2 2 tan cot cot tan sin cos sin cos cos sin sin cos 1 cos 2 sec 2 x x x x x x x x x x x x x x       

H2 Mathematics (9758) JC 1 River Valley High School, Mathematics Department, 2021 End-of-Year Revision Package 92 (e)    sin sin 2 sin 3 cos cos 2 cos3 2sin 2 cos sin 2 2 cos 2 cos cos 2 sin 2 2cos 1 cos 2 2 cos 1 tan 2 x x x x x x x x x x x x x x x x x          2. Differentiate the following expressions with respect to x: (a) 3 24x x (b) 4(3 1) , 33 x xx   (c) cos(sin x) (d) sin1(1 – x) (e) 3ln 3 x x   (f) e e e x x x +  (a) ଵ ଷ(4𝑥−𝑥ଶ)ିమ య(4−2𝑥) (b) (௫ିଷ)(ସ)(ଷ௫ାଵ)య(ଷ)ି(ଷ௫ାଵ)ర(ଵ) (௫ିଷ)మ =(9𝑥−37)(2𝑥+1)ଷ (𝑥−3)ଶ (c) −sin(sin𝑥)∙cos𝑥 (d) ଵ ඥଵି(ଵି௫)మ(−1)= ିଵ √ଶ௫ି௫మ (e) ୢ ୢ௫ቀଵ ଶቁ(ln(𝑥+3)−ln(𝑥−3)) =1 2൬ 1 𝑥+ 3− 1 𝑥− 3൰ = −3 𝑥ଶ−9 (f) ୢ ୢ௫ቀ ୣమೣ ୣమೣାଵቁ =(eଶ௫+1)(2eଶ௫)−eଶ௫(2eଶ௫) (eଶ௫+ 1)ଶ = 2eଶ௫ (eଶ௫+1)ଶ Tutorial Review Tutorial 5A Questions 1, 4, 7 and 10. Tutorial 5B Question 1 and 2. Tutorial 5C Question 1 and 2. Tutorial 5D Question 2 and 5. Revision Questions 1. 2020/Promo/SAJC/Q3 (a) Differentiate 2 1e cos (3 )x x with respect to x, where 1 1 3 3x   . [3]   2 1 2 2 1 2 2 1 2 d 3e cos (3 ) e 2e cos (3 )d 1 3 3 e 2cos (3 )1 9 x x x x x xx x x x                     (b) Given that  x y a , where 0, 0x y  and a is a positive constant , show that 2 2 d d2 1 0. dd y yx xx       [4]

H2 Mathematics (9758) JC 1 River Valley High School, Mathematics Department, 2021 End-of-Year Revision Package 93 2 2 2 Differentiating with respect to , 1 1 d 0d2 2 d 0d d d (Note that 0)d d d d Differentiating once more with respect to , d d d2 d dd x y a x y xx y yy x x yy yx yx x x yx yx x y y yx x xx                                    2 2 2 2 2 2 2 d d d d d2 1 0d d d d d d0