Bài tập CALCULUS 10

Created by T Madas Question 38 (**+) y y = x2 − x + A y =5 B O x The figure above shows a quadratic curve and a straight line with respective equations y = x2 − x + and y = The points A and B are the points of intersection between the straight line and the quadratic curve a) Find the coordinates of A and B b) Calculate the exact area of the finite region bounded by the quadratic curve and the straight line, shown shaded in the above figure MP1-R , A (1,5 ) , B ( 4,5 ) , area = Created by T Madas Created by T Madas Question 39 (**+) The curve C with equation y = f ( x ) has gradient function dy = x2 + , x ≠ dx x The point A ( −1, −1) lies on C Find an equation for C y = x3 − Question 40 −5 x (**+) y = 2x + x2 , x ≠ Find the coordinates of the stationary point of y and determine its nature C2A , ( 2,6 ) Created by T Madas Created by T Madas Question 41 (***) 24cm x 64 cm x x figure figure An open box is to be made out of a rectangular piece of card measuring 64 cm by 24 cm Figure shows how a square of side length x cm is to be cut out of each corner so that the box can be made by folding, as shown in figure a) Show that the volume of the box, V cm3 , is given by V = x3 − 176 x + 1536 x b) Show further that the stationary points of V occur when x − 88 x + 384 = c) Find the value of x for which V is stationary (You may find the fact 24 ×16 = 384 useful.) d) Find, to the nearest cm3 , the maximum value for V , justifying that it is indeed the maximum value x = 16 , Vmax ≈ 3793 Created by T Madas