Let's say you want to find the intermediate value [tex] \xi [/tex] of the mean value theorem of the integral calculus for:(adsbygoogle = window.adsbygoogle || []).push({});

(a)[tex] \int^b_a 1 \ dx [/tex]

(b)[tex] \int^b_a x \ dx [/tex]

(c)[tex] \int^b_a x^n \ dx [/tex]

(d)[tex] \int^b_a \frac{dx}{x^2} [/tex]

Using the Mean Value Theorem we know that [tex] a \leq \xi \leq b [/tex] and [tex] \mu = f(\xi) [/tex] following from:

[tex] \int^b_a f(x) \ dx = \mu(b-a) [/tex] where [tex] m \leq \mu \leq M [/tex] and [tex] m, M [/tex] are the least and greatest values of [tex] f(x) [/tex].

Hence [tex] \int^b_a x \ dx = (b-a) f(\xi) [/tex] because [tex] dx = b-a [/tex]. So [tex] f{(\xi) = \frac{(b-a)(b+a)}{2} \times \frac{1}{b-a} = \frac{a+b}{2} [/tex] Is this right? I am not sure because I think this is for [tex] f(\xi) [/tex] but I want [tex] \xi [/tex]. Or are they the same thing?

Thanks for any help

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# Homework Help: Finding the Intermediate Value

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