TMJC Term 4 Revision Chapter 7 and 8 Differentiation Questions

Term 4 Revision Topical Quick Check: Differentiation and its Applications TMJC 2024 Page 1 of 3 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] 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] 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] x y (2,3) O

Term 4 Revision Topical Quick Check: Differentiation and its Applications TMJC 2024 Page 2 of 3 4 RVHS Promo 9758/2022/Q11(i)-(iii) To ease food security concerns, the government is building tents at the community garden plots to allow urban farming to take place more efficiently among residents. The organizing committee is building a tent consisting of two rectangular pieces on the ro of and two rectangular vertical sides as shown in the diagram below. The base of the tent has sides x m by y m, and covers a total floor area of 40 m2. The vertical sides of the tent are 4 m tall, and the roof adds another 0.01y2 m to the overall height of the tent. (Diagram not drawn to scale) (i) The total external surface area of the tent is denoted by A m2. Show that A is given by 2 168 40 1 25Ax x= + + . [3] (ii) Suppose that A has a stationary value at some x, show that x satisfies the equation 6425 16 256 0xx+ − = . [3] (iii) The design team decides that the material for the tent costs $3.10 per m 2, estimate the minimum total cost of the material for the whole tent. [3] 5 MJC Prelim 9740/2008/P1/Q13 A piece of wire of length d units is cut into two pieces. One piece is bent to form a circle of radius r units, and the other piece is bent to form a regular hexagon. Prove that, as r varies, the sum of the areas enclosed by the two shapes is a minimum when the radius of the circle is approximately 0.076d units. [7] 4 m x m y m 0.01y2 m

Term 4 Revision Topical Quick Check: Differentiation and its Applications TMJC 2024 Page 3 of 3 6 JPJC Promo 9758/2022/Q5 The diagram shows a V -shaped tank with dimensions = 4 mL , = 0.6 mW and = 0.7 mH . The tank is initially empty. Water is pumped into the tank at a rate of 0.0025 m3/s. At any instant from the start of water flowing into the tank, the water in the tank has a depth of y m and a surface width of x m. (i) Find the rate of change of the water depth when = 0.4 my , leaving your answer to 4 decimal places. [4] (ii) Find, to the nearest second, the time taken to completely fill up the tank from the instant when = 0.4 my . [2] Answer Key No. Year JC Answers 1 2022 SAJC (i) 2 1 1 x xx− + (ii) 1 2 2sec3 3tan 3 sin 2 14 x x x x − + − 2 2022 DHS 3 2022 MI (iii) 13 24yx=− + (iv) 9 16 4 2022 RVHS (iii) $177.53 6 2022 JPJC (i) d 0.0018 m/s (4 d.p)d y t = (ii) 226 seconds L = 4 m y H = 0.7 m W = 0.6 m x