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Norbert Shao1/25/17Tuesday 10amHomework 11.Definitionsa.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 betweenf(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 integralfrom a to b of v(x).