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Tush19
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how does the first step use mean value theorem? I don't get it , can anyone explain , thanks.
thanks but I couldn't find that mean value theorem statement anywhere ,all it shows that mean value theorem is the followingOrodruin said:The mean value theorem states that
$$
\int_x^{x+\delta x} f(s) ds = \delta x\, f(x^*)
$$
where ##x \leq x^* \leq x + \delta x##. Since ##f## is continuous, ##f(x^*) \to f(x)## for small ##\delta x##.
https://www.kristakingmath.com/blog/mean-value-theorem-for-integralsTush19 said:thanks but I couldn't find that mean value theorem statement anywhere ,all it shows that mean value theorem is the following
Wrong mean value theorem:Tush19 said:thanks but I couldn't find that mean value theorem statement anywhere ,all it shows that mean value theorem is the following
View attachment 295180
thank you so muchOrodruin said:Just to add: The mean value theorem for definite integrals is easy to obtain from the theorem you quoted. Just consider that
$$
(b-a) f’(c) = f(b) - f(a) = \int_a^b f’(x) dx
$$
and let ##g(x) = f’(x)##. You now have
$$
\int_a^b g(x) dx = (b-a) g(c)
$$
for some ##c## such that ##a\leq c\leq b##.