MATHEMATICS · Multi-part · AP · Calculus AB · Applications of Integration · Applications of Integration
Problem
For a family of functions fk(x)=kx2+1 defined on the interval [0,3], the numerical value M (where M>1) is prescribed to be the average value of fk on that interval.
(i) State the general formula for the average value of a continuous function on a closed interval.
(ii) Using the definition, derive an explicit equation involving k and M that must be satisfied so that the average value of fk on [0,3] equals M.
(iii) Without solving the equation obtained in part (ii), explain whether k is directly proportional to M or inversely proportional to M. Justify your answer algebraically.