575 Answers Chapter 4

AnswersSelected answers are given here. For a full set of worked solutions, see Specialist Maths Dimensions Units 3 & 4 Solutions and Study Guide.8 i (a, 1)ii From f ′(x) =f ″(x) =iii–( x – a) 2---------------------2= e (– 1 + ( a – x) 2 )–( x – a) 2---------------------2= e (( x – a) 2 – 1)1⎛ –--2⎞and ⎜a– 1,e ⎟⎝ ⎠9 i Show that f ′(x) = 0 gives x 2 + 4x + 2a + 2a 2 = 0.Solutions exist if Δ > 0, i.e. 16 – 8a – 8a 2 > 0.∴ –2 < a < 1ii Asymptotes: y = x + a – 2 and x = –2.iii Sketch for 1 < a < 210 a Given f(x) = tan – 1 ⎛ x – x 2 – 1 ⎞ then1x 2 – 1 – x= ------------------------------------------------------------------ ⋅ ------------------------1 + x 2 – 2x x 2 – 1 + x 2 – 1 x 2 – 11– x–x 2 – 1⎝⎛ ⎠⎞= ------------------------------------------ ⋅ ------------------------------– 2x⎛x – x 2 – 1⎞x 2 – 1⎝ ⎠–1= ---------------------.2 x 2 – 1–( x – a) 2---------------------2e ( a – x),–( x – a) 2---------------------⎛ –( x – a) 2--------------------- ⎞22e × – 1 + ( a – x)⎜e( a – x)⎟⎜⎟⎝⎠1⎛ –--2⎞⎜a+ 1,e ⎟⎝ ⎠–2a 2 – aa – 2 ax = –2y y = x + a – 2–a 2⎝ ⎠1f ′(x) = ---------------------------------x× 1 – ---------------1 + x – x 2 – 1 x 2 – 1xb1 2 a 2 – 1= – 1 --------------------- + ---------------------2 a 2 1– 1– 11 [ + 4a 2 ( a 2 – 1)]= -------------------------------------------------2a a 2 – 1– 14a [ 4 – 4a 2 + 1]= ------------------------------------------2a a 2 – 1– 12a [ 2 – 1] 2= -----------------------------.2a a 2 – 111 Here f(x) = x 2 y 2 –1 yln( + ) + 2 tan -- and⎝ ⎛ x⎠⎞⎛1f ′(x) = ----------------x 2 + y 2 2x + 2y-----dyx----- dy – y⎞⎛ ⎞ 2 ⎝ dx ⎠+ ------------------- -----------------------⎝ dx⎠y1 + --⎝ ⎛ x⎠⎞2 x 2⎛2= ---------------- ⎛x 2 + y 2 x + y-----dy ⎞ 2x 2 x----- dy – y⎞⎝ dx ⎠+ ----------------⎝ dx⎠x 2 + y 2 -----------------------x 22= ---------------- ⎛x 2 + y 2 x + y-----dy x dy + ----- – y ⎞ .⎝ dx dx ⎠Since f(x) = x 2 y 2 –1 yln( + ) + 2 tan -- = 0 then⎝ ⎛ x⎠⎞f ′(x) = 0 implies x + y-----dy + x----- dy – y = 0.dx dxHence12 a a = 1, b = 0b13 y = 2x1f ′( a)+ -------------- –1 2 a 2 – 1= --------------------- + ---------------------f ′( a)2 a 2 –1– 1x +y-----dydx⎛ 1–1,-- ⎞⎝ 2⎠y – x= -----------.y + xyO y = 0x⎛1,–--1⎞⎝ 2⎠588Specialist Maths Dimensions Units 3 & 4