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Math_102_December_2006

# Math_102_December_2006 - December 2006 Marks[8 1...

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December 2006 Mathematics 102 Name Page 2 of 11 pages Marks [8] 1. Using the definition of derivative, find f 0 (2) where f ( x ) = 1 1 + x 2 . Continued on page 3

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December 2006 Mathematics 102 Name Page 3 of 11 pages [12] 2. Use implicit differentiation to find the points on the curve 3 y 3 - 2 x 3 - 6 x 2 y + 5 y = 0 where the tangent line to the curve is horizontal. Write your answers in the box in the form ( x, y ). Points: Continued on page 4
December 2006 Mathematics 102 Name Page 4 of 11 pages [14] 3. An ostrich 1.5 m tall is walking toward a street light 4 m above the ground at a speed of 5 m/s. How fast is the length of the ostrich’s shadow decreasing? At what speed is the tip of the shadow moving? If α is the angle between the light and the ground measured at the tip of the shadow, as shown in the picture, what is the rate of change of α when the shadow is 1.5 m long? α Length of shadow decreasing at: Speed of tip of shadow: Rate of change of α : Continued on page 5

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December 2006 Mathematics 102 Name Page 5 of 11 pages [6] 4. Suppose the number of bacteria in a colony doubles every 20 minutes. If there are 2 39 cells (about 5 . 5 × 10 11 ) after 12 hours, how many were there at the beginning? Initial number = [6] 5. The tide on a certain shore on the planet Outer Thebulon IV has a period of 36.5 hours, and the high tide level is 8 m above the low tide level. At t = 0 the water level is 2 m above the low tide level and rising. Using trigonometric functions, find a function to describe the height
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Math_102_December_2006 - December 2006 Marks[8 1...

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