TMJC Term 4 Revision Chapter 7 and 8 Differentiation Solutions

Term 4 Revision Topical Quick Check: Differentiation and its Applications TMJC 2024 Page 1 of 11 JC1 H2 Mathematics (9758) Term 4 Revision Topical Quick Check Chapter 7 Differentiation Chapter 8 Applications of Differentiation 1 SAJC Promo 9758/2022/Q3b / JPJC Promo 9758/2022/Q2aiii Differentiate the following with respect to x. (i) 2 2ln 1 x x   + [2] (ii) 1sec3 sin 2xx − [3] Solutions (i) ( ) 2 2 2 2ln ln 2 ln ln 1 1 1ln 2 ln ln 1 2 x xx x xx  = + − + + = + − + ( ) 2 2 2 2 d 2 d 1ln ln 2 ln ln 1d d 2 1 11 (2 )2( 1) 1 1 x xxxx x xxx x xx   = + − +   + =− + =− + (ii) 1d sec3 sin 2d xxx − ( ) ( ) 1 2 1sec3 2 sin 2 3sec3 tan 3 12 x x x x x −=+ − 1 2 2sec3 3tan 3 sin 2 14 x x x x −=+  −

Term 4 Revision Topical Quick Check: Differentiation and its Applications TMJC 2024 Page 2 of 11 2 DHS Promo 9758/2022/Q1b The graph of f ( )yx= has a maximum turning point at (2,3) and passes through the origin. The lines 1x=− and 2y= are asymptotes to the graph, as shown in the diagram below. Sketch the graph of f '( ),yx= showing clearly the axial intercepts and the asymptotes. [3] Solutions x y (2,3) O ( )fyx =

Term 4 Revision Topical Quick Check: Differentiation and its Applications TMJC 2024 Page 3 of 11 Revision Guide Page 3 Let’s Try Now. 3 MI PU2 P1 Promo 9758/2022/Q7 A curve C has parametric equations 1cos 2 , sin , for 0 2xy    = =   . (i) Show that d1 d 4sin y x =− . [3] (ii) Sketch C, showing clearly the features of the curve at the points where 0 = and 1 2= . [2] (iii) The tangent to the curve C at the point where p = is parallel to the line 2 0.yx+= Find the equation of this tangent. [4] (iv) The tangent from part (iii) meets the x-axis at P and the y-axis at Q. Find the area of the triangle OPQ. [3] Tangents and Normals The equation of the tangent and normal at any point ( )00,xy on a curve ( )fyx= is given by: Tangent ( )00y y m x x− = − where d d ym x= is the gradient of tangent at the point ( )00,xy Normal ( )00 1y y x x m− =− − Note: • Tangent parallel to x-axis d 0d y x= • Tangent parallel to y-axis d d y x is undefined, i.e. DENOMINATOR of d d y x is ZERO.

Term 4 Revision Topical Quick Check: Differentiation and its Applications TMJC 2024 Page 4 of 11 Solutions (i) cos 2 sin dd 2sin 2 cosdd d d d dd d cos 2sin 2 cos 4sin cos 1 4sin xy xy y y xx          == =− = = − = − =− (ii) (iii) 120 2y x y x+ =  =− Tangent parallel to 20yx+= means d1 d2 y x =− when p = . 11 4sin 2 1sin 2 6 p p p  − =− = = 11When , cos , sin6 3 2 6 2p x y  = = = = = Equation of tangent: 1 1 1 2 2 2 13 24 yx yx − =− −  =− +

Term 4 Revision Topical Quick Check: Differentiation and its Applications TMJC