DHS Y4 Math 2 Supplementary Worksheet Applications of Differentiation

Differentiation & Applications_2023 KGH_DHS resources 1 Year 4 Mathematics 2 Supplementary Worksheet Differentiation & Applications of Differentiation Name : _________________________________ ( ) Class : _______ Date : __________ 1 (a) The force F newtons between two magnetic poles is given by th e formula 2 1 500F r= , where r m is their distance apart. Find the rate of change o f the force when the poles are 0.2 m apart and where the distance between them is increasing at a rate of 0.03 m s–1. [3] (b) Given lny x x=+ , prove that d d 2 y x x x xx += . [2] (c) Find the equation of the normal to the curve 32e xy += at the point where 1x=− , leaving your answer in terms of e. [4] [2008 ACS (Barker Road) AMaths P1] 2 If x + y = 82, using differentiation, find the stationary value of xy. Determine the nature of this stationary value. [4] [2008 ACS (Barker Road) AMaths P2] 3 Differentiate the following with respect to x. (i) y = tan3 (5x2 + 1), [2] (ii) y = 1 1 2 + − x x , leaving your answer as a single algebraic fraction. [3] [2008 ACS (Barker Road) AMaths P2] 4 The diagram shows a solid cone with base radius 24 cm and height 80 cm. A cone of radius r cm and height h cm is to be removed. (i) Express h in terms of r . [2] (ii) Hence show that the volume, V cm3, of the cone to be removed is given by 2380 10 39V r r =− . [2] (iii) Calculate the value of r for which V has a stationary value. Henc e, find the stationary value of V and determine whether it is a maximum or minimum value. [4] [2008 ACS (I) AMaths P2] r h 80 cm 24 cm

Differentiation & Applications_2023 KGH_DHS resources 2 5 A curve has the equation 53ln 25 xy x += − . (i) Find the gradient of the curve at the point where the curve meets the x-axis. [4] (ii) Show that 53ln 25 xy x += − has no stationary point for all real values of x. [2] [2008 ACS (I) AMaths P1] 6 P is the point (4, 7) on the curve 1562 +−= xxy . Find the equation of the normal at P. The tangent at another point Q is parallel to the normal at P. Calculate the x-coordinate of Q. [5] [2008 Anderson Sec AMaths P2] 7 The vertex of a pyramid is vertically above the centre of its horizontal base. It is given that the volume of the pyramid is 236 cm3 and that the side of the square base is 2x cm. (a) Show that if A is the area of the sloping triangular face, then x xA 61458+= . (b) Hence find the value of x for which A2 has a stationary value and determine whether this is a minimum or a maximum. [7] [2008 Anderson Sec AMaths P1] 8 Given that ( ) 23 41 e 2e x x y − − = , find d d y x . Calculate the rate of change of x at the instant when 3=x and y is changing at the rate of 5e− unit s −1. [4] [2008 Anglican High AMaths P1] 9 A piece of wire, 120 cm in length, is bent to form the figure as shown. Given that == 60EFGABC , == 90GHACDE , AB = BC =EF = FG = x cm and CD = DE = GH =HA = y cm. (a) Show that the area of the figure, P cm 2, is given by 900)60230(212 3 2 +−+      −+= xxP . [4] (b) Find the value of x for which P has a stationary value. [2] Determine whether the stationary value of P is a maximum or minimum. [2] [2008 Anglican High AMaths P2] y cm y cm x cm x cm 60 H G F E D C B A 60

Differentiation & Applications_2023 KGH_DHS resources 3 10 Show that the gradient function for the curve x xy cos tan= is x x 3 2 cos sin1+ . Hence deduce that the curve has no stationary points for 0 2x  . [6] [2008 Anglican High AMaths P1] 11 A piece of c hicken drumstick is removed from the refrigerator and is then allowed to thaw. Its temperature, x degrees Celsius after t minutes, is given by the formula 0.328 30e tx −=− . (i) Find the initial temperature of the drumstick. [1] (ii) Find the time taken for the temperature to reach zero degree. [2] (iii) Find the rate at which temperature is increasing when t = 5. [2] (iv) Sketch the graph of x against t. [3] [2008 Balestier Hill Sec AMaths P1] 12 The diagram shows part of the curve y = ln(2x − 1) which meets the x-axis at A. (i) Find the equation of the normal at A. [4] (ii) Find the coordinates of the point B on the curve such that the tangent drawn at B is parallel to the line 4y = x – 3. [3] [2008 Balestier Hill Sec AMaths P1] 13 A cylinder is placed inside a right circular cone of radius 12 cm and height 8 cm. The radius of the cylinder is r cm. (i) Express the height of the cylinder, h cm , in terms of r cm , hence show that its volume, V cm3 is given by V = 8 r 2 − 3 2  r 3. [3] (ii) Given that r varies, find the value of r for which V has a stationary value. [3] (iii) Find t he stationary value of V and determine whether it is a maximum or minimum value. [3] [2008 Balestier Hill Sec AMaths P1] 14 The equation of a curve is 3e xyx= . (i) Find an expression for d d y x . [2] The curve has a stationary point at M where 0x . (ii) Find the coordinates of M . [2] (iii) Determine whether this stationary point is a maximum or a minimum point. [3] [2008 Balestier Hill Sec AMaths P1] 8 cm 12 cm r cm O A y x y = ln(2x − 1)

Differentiation & Applications_2023 KGH_DHS resources 4 15 Given that 3e e 1xxy −= − + , find the rate of change of x at the instant when 1=y , given that y is changing at the rate of 4.0 units per second at this instant. [4] [2008 Catholic High AMaths P1] 16 Find the x-coordinate of the stationary point of the curve ( ) 1 1 3 + −= x xy , x > 0. By considering the sign of d d y x , or otherwise, determine the nature of the stationary point. [2008 Catholic High AMaths P1] 17 If 143 23 ++−= xxxy , show that y is an increasing function for all real values of x . Hence, state the minimum value of the gradient of this function. [5] [2008 Catholic High AMaths P1] 18 The diagram shows a roof in the shape of a right circular cone whose radius is r m and it s slant height l m. The sloping surface of the roof is covered with a sheet of thin metal whose area is 43  m2. [Curved Surface Area of Cone = rl where l is the slant height of the cone.] (a) Express l in terms of r and show that the volume of the cone, V cm3 , is given by 4483V r r=− . [3] (b) Given that r can vary, find (i) an expression for d d V r , [2] (ii) the value of r for which V has a stationary value. [3] [2008 Catholic High AMaths P1] 19 (a) Differentiate with respect to x (i) x2tan3 , [2] (ii) 12ln 2 x x −  + . [3] (b) Given that 23 += xy , show that 22 2 d d 1 0dd yy x x y += . [4] [2008 Catholic High AMaths P2] 20 A curve has the equation 12 )3(2 − += x xy . (i) Show that 3)12( )( d d − −= x BxA x y where A and B are integers. [4] (ii) Find the equation of the normal to the curve at the point where 5=x . [3] [2008 Cedar Girls’ Sec AMaths P1] l m r m

Differentiation & Applications_2023 KGH_DHS resources 5 21 Under a heating process, the length, x cm, of each side of a metal cube increases at a constant rate. Express the volume, 3 cmV , and the surface area, 2 cmA , of the cube in terms of x . Write down expressions for x V d d and x A d d . Given that V is increasing at the rate of /scm5.1 3 , find the rate of increase of (i) x , (ii) A , at the instant when x = 10 . [8] [2008 Cedar Girls’ Sec AMaths P1] 22 An object is heated until it reaches a temperature of T degrees Celsius. It is then allowed to cool. Its temperature T degrees Celsius when it is cooling for time t minutes is given by the equation 3 424 16e t T − =+ . Find (i) the value of t when T = 32, [2] (ii) the value of T when t = 4, [1] (iii) the rate at which T is decreasing when t = 12.