Homework 1 - Norbert Shao Tuesday 10am Homework 1 1...

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Norbert Shao 1/25/17 Tuesday 10am H omework 1 1. Definitions a. Intermediate value theorem- If a function, f, is continuous on [a,b] and takes values f(a) and f(b) at each end of the interval then it also takes any value between f(a) and f(b) at some point within the interval. b. Mean value theorem- if a function, f, is continuous on [a,b] and differentiable on (a,b) then there exists point c in (a,b) such that f’(c)=(f(b)-f(a))/(b-a). c. Rolles theorem- if f is continuous on [a,b] and if f’ exists on (a,b) and if f(a)=f(b) then f’(j)=0 for some point j that lies in (a,b). d. Mean value theorem for integrals- if f is defined on [a,b] and is differentiable on (a,b) then for x and c in the closed interval [a,b] f(x)=f(c)+f’(j)(x-c) where j is in between c and x. e. Weighted mean value theorem for integrals- let u and v be continuous real-valued functions on [a,b] and suppose v is greater than or equal to 0. Then there exists a point j in [a,b] such that the integral from a to b of u(x)-v(x) dx= u(j)* the integral from a to b of v(x).
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