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.
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