Bài tập CALCULUS 61

Created by T Madas Question 212 (****) y x The figure above shows a triangular prism whose triangular faces are parallel to each other and are in the shape of equilateral triangles of side length x cm The length of the prism is y a) Given that total surface area of the prism is exactly 54 cm , show clearly that the volume of the prism, V cm3 , is given by V = 27 x − x3 b) Find the maximum value of V , fully justifying the fact that it is indeed the maximum value c) Determine the value of y when V takes this maximum value C2O , Vmax = 27 , y = Created by T Madas Created by T Madas Question 213 (****) y x O R P y = x − 3x − The figure above shows part of the curve with equation y = x − 3x − , x > a) Show that an equation of the tangent to the curve at the point P , where x = , is given by y = − 2x The finite region R is bounded by the curve, the tangent to the curve at P and the coordinate axes b) Determine the exact area of R MP1-P , area = 29 12 Created by T Madas Created by T Madas Question 214 (****) The straight line with equation y = 2x + c is a tangent to the curve with equation y = x2 + x + Without using the discriminant, determine the value of the constant c and find the point of contact between the tangent and the curve c=3 , Created by T Madas ( −2, −1)