Step i 1990 question 9 solution

Therefore the equation of the tangent at the point  is 

The co-ordinates of C are at the intersection of  and 

Using the expression for the area of a trapezium:

Area of ACC'A' = 

Area of CBB'C' = 

Sum of Areas = 

As the gradient is constantly increasing (becoming 'less' negative), the area under the curve between 1 and b is greater than the sum of the areas of ACC'A' and CBB'C:

To prove the left-hand inequality, it is sufficient to show that 

Which is clearly true.

The inequality  follows immediately from the fact that  by the Maclaurin expansion of .

Alternatively, note that the area of the trapezium ABB'A' is greater than the area under the curve between 1 and b.


Solution by Dystopia.