RVHS H2 Math Differentiation Rate of change & min max Qns

River Valley High School, Mathematics Department 9758 H2 Mathematics 2026 Differentiation: Rate of Change, Maxima & Minima Problems 1 Differentiation: Rate of Change, Maxima & Minima Problems 1. ACJC Prelim/2020/01/Q11 O D E F C The diagram above shows part of a roller coaster track, made up by a curve with parametric equations 23 10 1x t t= − − , 1316tan 4 16y t t −= − + , for 20 t−   , with points B and P on the curve. Rods are used to support the track. AB is a horizontal metal rod. Another straight metal rod, fixed from point B to C, is tangential to the track at point P. In order to further strengthen the support, vertical metal rods BD and PF, of length m metres and ( )45 π− metres respectively, are fixed. (i) Find the x-coordinate of point P. [2] Hence show that the equation of line BC is 5 4 16 π 140 0xy+ + − = . [3] (ii) Find m, giving your answer to 3 decimal places. [3] For regular maintenance, an extendable ladder, fixed at point E, is extended till it meets rod BC. If the ladder and rod BC meet at point P, then the length of the extended ladder, l, is at its shortest length. Find the exact value of l. [4] 2. PJC/I/6 The diagram shows a rectangle ABCD inscribed in a semi -circle with fixed radius cmr . Two vertices of the rectangle lie on the arc of the semi-circle. If AB = x cm, show that the perimeter P of the rectangle ABCD is 2242 xrx −+ . [2] Given that as x varies, the maximum value of P occurs when AB : BC = 1 : k, find k. [4] x D A C B

River Valley High School, Mathematics Department 9758 H2 Mathematics 2026 Differentiation: Rate of Change, Maxima & Minima Problems 2 3. YJC/I/Q5 (a) (i) Show that            + − 2 1 1 2cosd d x x x = 21 2 x+ − for 11 − x . Explain, in your working, how you use the condition 11 − x to conclude your answer. [3] (ii) Hence, determine the constants A and B satisfying the equation BxA x x =+     + −− 1 2 1 tan 1 2cos , for 11 − x . [3] (b) A right cone with base area A has a fixed height of 10 cm. Given that the base area is increasing at a rate of 2 cm2s−1, calculate the rate of change of the volume of the cone. [2] 4. DHS/2013/I/11 The point Q(x, y) lies on the curve 224 36,x xy y+ + = where 0,y as shown in the diagram above. The curve cuts the x-axis at the points P and R. (i) Show that A, the area of triangle PQR, is given by 3.Ay= [3] (ii) Find d d y x in terms of x and y. [2] (iii) Hence find the value of x for which A has a stationary value. Using the second derivative test, determine the nature of this stationary value. [6] (iv) If x increases at a constant rate of 8 units/s, find the rate of change of A when x = 0. [3] x O P R Q y

River Valley High School, Mathematics Department 9758 H2 Mathematics 2026 Differentiation: Rate of Change, Maxima & Minima Problems 3 5. PJC/2010/I/Q4 An athlete at point C is running towards the finishing line AB, at a constant speed of 10 1ms− in a direction perpendicular to AB, as shown in the diagram above. (i) Given that ∡𝐴𝐶𝐵 is  and x is the distance of the athlete from AB, show that 1132tan tanxx −−=+ . [2] (ii) Find the exact rate of change of  when the athlete is 10 m from the finishing line. [3] 6. ACJC/2011/I/7 An isosceles triangle has fixed base of length b cm. The other 2 equal sides of the triangle are each decreasing at the constant rate of 3 cm per second. How fast is the area changing when the triangle is equilateral? Leave your answer in terms of b. 7. AJC/2011/I/7 The above 48 cm by 18 cm plastic sheet is cut out to form a shape represented by the shaded region. The shape is folded into a box with width x, length y and height z as shown in the diagram. (i) Express the volume of the box, V, in terms of y. [2] (ii) Using differentiation, find the maximum possible volume of the box. [3] A B C 10 3 m 2 m x m Base Side Side Side Side Top cover

River Valley High School, Mathematics Department 9758 H2 Mathematics 2026 Differentiation: Rate of Change, Maxima & Minima Problems 4 Two small robots of negligible dimensions, A and B, are initially placed on the edge of the base of the box where m = 0 cm and n = 0 cm respectively as shown in the diagram. Robot A starts to travel along the length of the base with speed 2 cm/s. One second later, Robot B starts to travel in the opposite direction along the length of the base with speed 1 cm/s. (iii) Given that the base of the box has length 20 cm and breadth 10 cm, find the rate of change of the distance between the two robots when n = 4 cm. [4] 8. RVHS/2014/I/6 A right circular cone has a base radius of r cm and height h cm. It is given that the cone has a fixed volume of 10 cm3 and a curved surface area of A cm2. (i) Show that 2 2 4 2 900πAr r=+ . [3] (ii) Using differentiation, find the value of r such that the curved surface area of the cone is a minimum, giving your answer correct to 3 decimal places. [4] 9. CJC2012/I/8 Candy is being stored in a closed container in the form of a regular hexagonal prism, with sides x cm and height h cm (as shown in the figure below). Given that the container has a volume of 972 cm3, (i) show that its base area given by 2 33 2x cm2. [1] (ii) Using differentiation, find the minimum area of the material, A cm2, that is used to make the container, leaving your answer to 2 decimal places. [6] (iii) Given that the cost of the material for the packaging is $ 0.05 per 100 cm2, find the minimum cost required for the container. [1] A B Base of the box 2cm/s 1cm/s m n

River Valley High School, Mathematics Department 9758 H2 Mathematics 2026 Differentiation: Rate of Change, Maxima & Minima Problems 5 10. TJC/2013/II/1 The diagram below shows the points P and Q on the circumference of a circle with centre O, and radius 2a cm, where POQ = . Points P and Q are moving on the circumference so that  is increasing at a constant rate. Find the acute angle  at the instant when the rate of change of the area of the shaded segment is 2 a times the rate of change of the length of the minor arc PQ. [5] 11. ACJC/2014/I/2 P is a variable point on the circumference of a circle with diameter AB, and Q is the point on AB such that AQ AP= . Given that angle PAQ = radians and AB=  , show that the area S of triangle PAQ is given by ( ) 231 sin sin2S =−  . [3] Use differentiation to find, in surd form and in terms of  , the maximum value of S, proving that it is a maximum. [4] 12. IJC/2017/I/11 [It is given that the volume of a circular cone with base radius r and height h is 1 3 𝜋𝑟2ℎ and the volume and surface area of a sphere of radius r are 4 3 𝜋𝑟3 and 24 r respectively.] In a distant Northern kingdom of Drivenbell, Elsanna builds a spherical snowball with radius 3 m. The snowball is inscribed in a right conical container of base radius r m and height h m. The container is specially designed to allow the snowball to remain intact with fixed radius 3 m (see diagram). P Q ⚫ ⚫ ⚫ O

River Valley High School, Mathematics Department 9758 H2 Mathematics 2026 Differentiation: Rate of Change, Maxima & Minima Problems 6 (i) By considering the slant height of the cone, show that ( ) 2 3 6h hr h = − . [3] (ii) Use differentiation to find the values of h and r that give a minimum volume for the container. Find the value of the minimum volume. [6] The snowball is being removed from the container and it starts to melt under room temperatur