Sir Arthur Lewis Community College

MAT109 CALCULUS ASSIGNMENT – DIFFERENTIATION

Assignment Three

1. The curve C has equation 
   1. 1. ³    8

                                                                ? =        ?³ − 9?2 +        + 30, ? > 0

1. 0. 2. ?
   1. Find ^?? .                                                                                                                               [4]

??

1. 2. Show that the point ?(4, −8) ???? ?? ?.                                                                  [2]
   3. Find an equation of the tangent to C at the point ?, giving your answer in the ???? ?? + ?? = ? = 0 ????? ?, ? ??? ? are integers to be determined 

                             [6]

2. A curve C has equation            ? = −2?³ + 3?² + 12 − 5

??

1. 1. Find (i)                                                                                                                                [3]

??

?²?

                               (ii) ₂                                                                                                                                                [2]

??

1. 2. Verify that C has  stationary points  at ? = 2 ??? ? = −1                                    [6]

              ( c) Determine the nature of the stationary points of C                                                                                                                                                                                                              [4]

3. A rectangular box with a square base, an open top, and a volume of 96?³ is to be constructed. What should the dimensions of the box be to minimize the surface area of the box? What is the minimum surface area?                                                               [8]

4. Sketch the graph of the curve ? = ?⁴ − 4? + 10                                                                 [7]

5. The length ? of a rectangle is decreasing at the rate of 4 cm/s and the width ? is increasing at the rate of 5cm/s. When ? = 8   ???  ? = 6 find the rate of change of
   1. perimeter and (b) are of the rectangle.                                                                    [8]

6. The volume of a cube is increasing at a rate of 9??³/?. How fast is the surface area

              increasing when the edge is 12cm?                                                                                                                                                 [6]

Total 50 marks