Cubic equation for each interval
Two unknows are second derivatives at the end of each interval, so we can determine them from
the following equation
Remember, the second derivatives at the end knots are
zero
𝑓𝑓 ′′ 𝑥𝑥0 = 𝑓𝑓 ′′ 𝑥𝑥𝑛𝑛 = 0
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Example
The cross-sectional area A of a gutter with equal base and edge length of 2 is given by
A = 4sinθ (1+ cosθ )
a) Find the angle θ which maximizes the cross-sectional area of the gutter. Using an
initial interval of[0,π / 2], find the solution after 2 iterations.
b) Use any software to find the maximum angle θ (a sufficiently small number = xu-xl <
0.05)
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θ=1.0456 rad (or 59.9°) so area is 5.196
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