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28B MVT Integrals 1 Mean Value Theorem for Integrals

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28B MVT Integrals

Mean Value Theorem for Integrals

28B MVT Integrals

Definition Average Value of a Function

If f is integrable on [a,b], then the average value of f on [a,b] is

EX 1 Find the average value of this function on [0,3]

28B MVT Integrals

Mean Value Theorem for Integrals

If f is continuous on [a,b] there exists a value c on the interval (a,b) such that

28B MVT Integrals

EX 2 Find the values of c that satisfy the MVT for integrals on [0,1].

EX 3 Find values of c that satisfy the MVT for

integrals on [3π/4 , π].

f(x)=cos(2x-π)

Page 5: Mean Value Theorem for Integrals - University of UtahMean Value Theorem for Integrals If f is continuous on [a,b] there exists a value c on the interval (a,b) such that. 28B MVT Integrals

28B MVT Integrals

Symmetry Theorem

If f is an even function, then .

If f is an odd function, then .

Theorem

If f is a periodic function with period p, then .

28B MVT Integrals

EX 4

EX 5

Page 7: Mean Value Theorem for Integrals - University of UtahMean Value Theorem for Integrals If f is continuous on [a,b] there exists a value c on the interval (a,b) such that. 28B MVT Integrals

28B MVT Integrals

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