Science:Math Exam Resources/Courses/MATH101/April 2012/Question 08 (b)/Solution 1

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The straight line below y=sin(x) that is selected for the inequality

Following the hint, we want to find a straight line that is below with minimal error.

To do this, we require the straight line to pass through at both and , which are the endpoints of the domain of integration.

Since at , we look for a straight line that passes through the origin with formula . Then we put , to find , so .

Note that the slope of the line , while the initial slope of is given by , so the linear function is really below in the interval Hence, we can deduce the following inequalities:

because and is a strictly increasing function. So by comparison, we obtain

because for any .