Created by T Madas Question 264 (****+) y y = x − 27 y =1 O x The figure above shows the curve C with equation y = x − 27 , x ∈ » The finite region R , shown shaded in the figure, is bounded by C , the y axis and the straight line with equation y = When the lengths are measured in m , R models the design of a yacht rudder Show that the area of the yacht rudder is 11 m SYN-B , proof Created by T Madas Created by T Madas Question 265 (****+) The curve C has equation y= ( x3 x x − 128 x ) , x∈» , x>0 dy d y d3y a) Determine expressions for , and dx dx dx3 b) Show that the y coordinate of the stationary point of C is −k , where k is a positive integer c) Evaluate d2y at the stationary point of C dx Give the answer in terms of d) Find the value of MP1-M , d3y dx3 at the point on C , where d2y dx =0 dy d2y d3y − 12 2 , = 20 x3 − 320 x , = 60 x − 480 x = 120 x − 240 x , dx dx dx3 k = 3072 , 960 , 360 Created by T Madas Created by T Madas Question 266 (****+) Use differentiation to establish the number of real solutions of the equation x − x3 − 12 x + 15 = You are not expected to solve the equation MP1-N , solutions Created by T Madas