Let be a continuous function on and let

MATHEMATICS · Multi-part · AP · Calculus AB · Applications of Integration · Applications of Integration

Problem

Let ff be a continuous function on [a,b][a,b] and let k∈Rk\in\mathbb R. Define h(x)=f(x)+kh(x)=f(x)+k on the same interval. (i) State the definition of the average value of a continuous function on a closed interval. (ii) Prove that the average value of hh on [a,b][a,b] is equal to the average value of ff on [a,b][a,b] increased by kk. (iii) Take f(x)=x2f(x)=x^{2} on [0,3][0,3] and k=−5k=-5. Find the average value of h(x)=x2−5h(x)=x^{2}-5 on [0,3][0,3] and verify that your result is consistent with part (ii).

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