Created by T Madas Question 191 (****) y y = f ( x) O P ( −1,0 ) Q x The figure above shows a curve with equation y = f ( x ) The curve meets the x axis at the points P ( −1, ) and Q , and its gradient function is given by f ′( x) = x3 − x2 , x ≠ a) Find an equation of the tangent to the curve at P b) Find an expression for f ′′ ( x ) c) Determine … i … an equation of the curve ii … the coordinates of Q C1A , y = −9 x − , f ′′ ( x ) = + x −3 , y = x + − , Q ,0 x ( ) Created by T Madas Created by T Madas Question 192 (****) y A M y = x (6 − x) O x The figure above shows the curve C with equation y = x (6 − x) , x ∈ » The point M is the maximum point of C and the point A has coordinates ( 0,12 ) Find the exact area of the shaded region, bounded by the curve, the y axis and the straight line segment from A to M SYN-G , area = 27 Created by T Madas Created by T Madas Question 193 (****) y 2x The figure above shows the design of a window which is the shape of a semicircle attached to rectangle The diameter of the semicircle is 2x m and is attached to one side of the rectangle also measuring 2x m The other side of the rectangle is y m The perimeter of the window is m a) Show that the total area of the window, A m , is given by A = x − ( + π ) x2 b) Given that the measurements of the window are such so that A is maximum, show by a method involving differentiation that this maximum value of A is 18 +π MP1-Y , proof Created by T Madas