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