1. A steel pipe is being carried down a hallway that is 3 m wide. At the end of the hall there is a right-angled turn into a narrower hallway, 2 m wide. What is the length of the longest pipe that can be carried horizontally around the corner. 2 m 2. Let f be a differentiable function. Use the definition of deivatives to show that drif ( x ) = 2rf (1) + 2 f (x) . 3. Express the following as a definite integral fo f (x) dx and find its exact value. 72 lim ME n-+00 13 + n3 i=1 4. Show that r+3 (a) 1T' + 5x + 7 de = = Ina+ (tan-1b - tan 'c) , where the num- bers a, b, c are to be determined. 1 In A (b ) 1 + 27 -d.x In B' where the numbers A, B are to be deter- mined. 5. The base of a solid is a circular disk with radius 30 cm. Find the volume of the solid whose parallel cross sections perpendicular to the base are right-angled isosceles triangles with hypotenues lying on the base