1. Express in partial fractions:

2. (a) Express

in the form

and then in the form

(b) Use the result obtained in (a) to sketch the graph of the rational function

3. (a) Sketch the graph of the rational function f(x) defined by

(b) Using (a) as a guide, sketch the graph of

where p and q are positive constants.

(c) Sketch the graph of

(d) The cost of eliminating a large part of pollutants from a lake is relatively small. However, to remove the remainder results in a significant increase in cost. If x represents the percentage of pollutants removed, and C(x) the cost (in thousands of pounds) for its removal, which of g(x) and h(x) would you choose to model C? If it costs £5000 to remove 10%, and £60,000 to remove 60% of the pollutants, use your model to predict the cost for complete purification.

(a) (d) 1 (x-2)(x+3) x-1 x - 4x-5 (b) (e) 1 x(x + 2) X x + x -2 (c) 3x+1 (x+1)(x-2) 1 (x+1)(x+2)(x+3)