DHS Differentiation & Its Applications (9758) Topical Revision

4. Differentiation and its Applications 1 4. Differentiation and its Applications (I) Tangents and Normals (Direct/Implicit Differentiation) 1. RI/2011Prelim/I/4 (a) Differentiate ( ) 1sin 3 x x− with respect to x, giving your answer as a single fraction. [3] (b) Given that cos 2 3 xyx = , find the exact value of dy dx at x =  . [4] 2. NYJC/2011Prelim/I/9 (modified) The diagram shows a sketch of the curve e xyx −= . (i) By differentiation, find the range of values of x for which the graph of e xyx −= is decreasing. [3] (ii) Determine the range of values of x for which the graph of e xyx −= is decreasing and concave downwards. [3] (iii) The tangent to the curve at P ( ),ab meets the y-axis at R ( )0, h . Express h in terms of a, and find the greatest possible value of h. [6] 3. HCI/2009Prelim/I/3 The diagram shows the graph of f '( )yx= . The curve passes through the origin and has turning points at ( −3, 0) and (−1.5, −1). The x-axis and x = 1 are the two asymptotes of the curve. (i) Find the range of values of x for which the graph of f ( )yx= is strictly increasing and concave downwards. [1] (ii) State the x-coordinates of all the stationary points of the graph of f ( )yx= and determine the nature of each point. [2] (iii) Given that f (0) 1= , sketch the graph of f ( )yx= for 1x . Your sketch should indicate clearly all stationary points, asymptotes and intersections with the axes. [2] x y x=1 O −3 (−1.5, −1) y x

4. Differentiation and its Applications 2 4. ACJC/2009Prelim/I/7 A point ( )yx, lies on the curve with equation ayx 111 =+ , where a is a non-zero constant, 0x and 0y . (i) Express d d y x in terms of x and y, and explain clearly whether y is an increasing or a decreasing function. [3] (ii) State, giving a reason, whether there are any stationary points on this curve. [2] (iii) Find the equation of the tangent to the curve at the point ( )aa 2,2 and determine if the tangent cuts the curve again. [5] 5. DHS/2009Prelim/I/4 (a) Suppose the following facts are known about the function g and its derivative: g(5) 1, g '(5) 3.= =− If ( ) g( )f ( ) tan e xx = , find the value of f '(5) . [3] (b) Find, by differentiation , the range of values of x for which the function 2 21 xy x= − increases as x increases. [3] 6. JJC/2015Promo/6 A curve has equation . (i) Show that d 3 2 d 2 2 y x y x x y −= − . [3] (ii) Find the exact coordinates of the points on the curve where the tangent is parallel to the x-axis. [4] (iii) The normal to the curve at the point P with coordinates (0, 1), meets the curve again at point Q. Find the area of triangle OPQ, where O is the origin. [5] 7. RI/2013Promo/1 A curve C is given by the equation +=x y a , for 0, 0xy , where a is a positive constant. (i) Show that C has no stationary points. [3] (ii) What can be said about the tangents to C as 0→x ? [1]

4. Differentiation and its Applications 3 8. NJC/2015Promo/8 A curve L has equation ( ) 2 4 16 48.x y y− + = (i) Find d d y x in terms of x and y. [3] (ii) The tangent to L at the point P is parallel to the x-axis. State the equation of this tangent and the coordinates of P. [2] (iii) The tangent to L at the point Q is parallel to the y-axis. Find the equation of this tangent and the coordinates of Q. [4] (iv) The tangents in (ii) and (iii) intersect at the point R. Find the area of triangle PQR. [1] (II) Tangents and Normals (Parametric Equations) 9. SRJC/2015Promo/6 The parametric equations of a curve are e sintxt= , e costyt −= , for 0 πt (i) Show that 2d ed ty x −=− . The point P on the curve has parameter p. Find the equation of the normal to the curve at P. [5] (ii) The normal at P when π 2p= meets the x-axis and y-axis at points A and B respectively. Find the area of triangle OAB, where O is the origin. [2] 10. SAJC/2015Promo/8 A curve C is defined by the equations 2, 2x t y t== , t . (i) Find the equation of the normal to C at the point P with parameter p. [3] (ii) Given that the normal at P where 2p= cuts C again at the point Q, prove that the angle QOP is ( ) 1 2 3 π 4 tan .−+ [4] 11. YJC/2010Prelim/II/Q1 A curve has parametric equations ( )tx −= 13 , 3 1 ty = for 0t . (i) Find x y d d in terms of t and deduce that the curve is an increasing function. [3] (ii) Find the equation of L1, the tangent to the curve at the point      − 3 1,33 tt . Hence, find the coordinates of point P on the curve at which L1 passes through the origin O. [4] (iii) The line L2 is another tangent to the curve which is parallel to L1. Find the equation of L2. [3] (iv) The line L2 cuts the y-axis at Q. Find the exact area of triangle OPQ. [2]

4. Differentiation and its Applications 4 12. HCI/2011Prelim/II/3 A curve is defined by the parametric equations ,xu= 2 1 2yu u =− , where 0u . (i) Express d d y x in terms of u . [2] (ii) The tangent to the curve at 1x= meets the x -axis at A and the y -axis at C , while the normal to the curve at 1x= meets the x -axis at B and the y -axis at D . Show that AB CD= . [5] (iii) Given that u is increasing at a rate of 0.5 units per second, find the rate at which d d y x is decreasing when 2.u= [3] 13. TJC/2015Promo/8 A curve C has parametric equations etx= , lny t t=− , where 0t  . (i) Show that there is no tangent to C parallel to the y-axis and find the equation of the tangent to C that is parallel to the x-axis. [5] (ii) Describe the behaviour of the tangent to C as 0t→ . [1] (iii) Sketch C, showing clearly its asymptote and turning point. [2] 14. MJC/2015Prelim/I/11 A curve C has parametric equations 2cos , sin , for 0 2 .x t y t t = =   Show that the equations of the tangent and normal to C at the point P with parameter  are ( ) ( )cos 2sin 2xy += and ( ) ( )2sin cos 3sin cosxy   −= respectively. [5] (i) Show algebraically that the tangent to C at the point P does not cut the curve C again. [3] (ii) The normal to C at the point P cuts the x-axis and y-axis at points A and B respectively. The point F is the midpoint of AB. Find a cartesian equation of the curve traced by F as  varies. Hence give a geometrical description of this curve. [6]

4. Differentiation and its Applications 5 15. HCI/2015Prelim/I/6 A curve C has parametric equations e sintxt=+ , e costyt=− . (i) Describe the shape of C as t→− . [2] (ii) Find the Cartesian equation of the normal to C at the point ( )e sin , e cosP  +− , where 0  , giving your answer in the form y mx c=+ . [3] The normal to C at P meets the y -axis at the point D , and the curve C meets the positive x -axis at the point E that has integral coordinates. (iii) Find the coordinates of D and E . [3] (iv) Give a geometrical description of the path traced by the midpoint of DE as  varies. [2] (III) Rate of change and Maximisation/Minimisation problems 16. VJC/2009Prelim/I/1 A man made pond is constructed in the form of a circular cylinder of radius 4.5 m. The ideal depth of water in the pond is 1.2 m. However, after a storm, the depth of water in the pond is 1.9 m. Water is being pumped out of the pond at a rate of 0.8 m3s-1. Find (i) the rate