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 =