TASK 1

In the early 18th century a Scottish mathematician, called Colin Maclaurin, developed a way of finding an approximate value of a function. It is in the form of an infinite series and is shown below:

[a] Take f(x) = (1 + x)n where n is positive and in the form where a and b are integers and b not equal to 1. Find f'(x), f''(x) and f'''(x). [Note: the differentiation of (a + x)b is b(a + x)b-1 and follows the same pattern as the differentiation of powers of x.]

[b] Find numerical values of f(o), f'(0), f''(0) and f'''(0).

(c) Choose a value of x (which lies between -1 and 1) and use your calculator to work out the value of (1 + x)n with your chosen values of x and n.

[d] Work out and then What do you notice?

[e] Now choose a different value of x (which also lies between -1 and 1) and repeat part [d). What else do you notice?