Bài tập CALCULUS 17

Created by T Madas Question 68 (***) A curve with equation y = f ( x ) passes through the point ( 2,3) The gradient function of the curve is given by f ′ ( x ) = ( x − 3)( 3x − 1) a) Find an equation of the curve, giving the answer as a polynomial in its simplest form b) Show clearly that f ( x ) ≡ ( x + k )( x − 3) , where k is a constant to be found c) Sketch the graph of f ( x ) The sketch must show the coordinates of any points where the graph of f ( x ) meets the coordinate axes C1W , f ( x ) ≡ x3 − x + 3x + , k = Created by T Madas Created by T Madas Question 69 (***) h x x The figure above shows the design of a large water tank in the shape of a cuboid with a square base and no top The square base is of length x metres and its height is h metres It is given that the volume of the tank is 500 m3 a) Show that the surface area of the tank, A m , is given by A = x2 + 2000 x b) Find the value of x for which A is stationary c) Find the minimum value for A , fully justifying the fact that it is the minimum MP1-C , x = 10 , Amin = 300 Created by T Madas Created by T Madas Question 70 (***) f ( x ) = x + x + k , where k is a constant Find the value of k , given that  f ( x ) dx = k = −14 Question 71 (***) The point P ( −1, −1) lies on the curve C , whose gradient function is given by dy x3 − = , x ≠ dx x3 Find an equation for C y = 5x + Created by T Madas x2 +1