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.
- فلسطين: الصف ١٢
- إيران: الصف ١٢
- أفغانستان: الصف ١٢
- بريطانيا: السنة ١٣
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الدليل
Rolle's theorem: if f is continuous and differentiable and f(a) = f(b), somewhere between a and b we have f′(c) = ٠: 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 نهاية 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) ÷ ٢ and sinh(x) = (ex − e−x) ÷ ٢ 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(٠) ÷ n!.