Created by T Madas Question 31 The curve C has equation y = f ( x ) given by f ( x ) = ( x − 2) , x ∈ » a) Sketch the graph of f ( x ) b) Find an expression for f ′ ( x ) The point P ( 3, ) lies on C and the straight line l1 is the tangent to C at P c) Find an equation of l1 The straight line l2 is another tangent at a different point Q on C d) Given that l1 is parallel to l2 show that an equation of l2 is y = 6x − f ′ ( x ) = x − 24 x + 24 , y = x − 16 Created by T Madas Created by T Madas Question 32 The point P ( 2,9 ) lies on the curve C with equation y = x3 − x + x + , x ∈ » , x ≥ a) Find an equation of the tangent to C at P , giving the answer in the form y = mx + c , where m and c are constants The point Q also lies on C so that the tangent to C at Q is perpendicular to the tangent to C at P b) Show that the x coordinate of Q is 6+ y = 2x + Created by T Madas Created by T Madas Question 33 The volume, V cm3 , of a soap bubble is modelled by the formula V = ( p − qt ) , t ≥ , where p and q are positive constants, and t is the time in seconds, measured after a certain instant When t = the volume of a soap bubble is cm3 and at that instant its volume is decreasing at the rate of cm3 per second Determine the value of p and the value of q p = 4, q = Created by T Madas