Rolle, l'Hôpital and Maclaurin series
Find where the tangent is horizontal, find the point of the mean value theorem, take a 0/0 limit with derivatives and find a coefficient of the Maclaurin series of cosh x.
- UK: Year 13
- Iran: grade 12
- Afghanistan: grade 12
- Palestine: grade 12
Guide
Rolle's theorem: if f is continuous and differentiable and f(a) = f(b), somewhere between a and b we have f′(c) = 0: a horizontal tangent. Mean value theorem: even if the ends differ, somewhere the tangent has the same gradient as the chord: f′(c) = f(b) − f(a)b − a.
L'Hôpital's rule: if the top and bottom both go to zero, the limit of fxg(x) equals the limit of f′xg′(x). Differentiate the top and the bottom separately, not the whole fraction.
cosh(x) = (ex + e−x) ÷ 2 and sinh(x) = (ex − e−x) ÷ 2 are hyperbolic functions. The derivative of cosh(x) is sinh(x) and the other way round. The coefficient of xn in the Maclaurin series is fn(0) ÷ n!.