MATHEMATICS · Multi-part · AP · Calculus AB · Differentiation: Definition and Fundamental Properties · Differentiation: Definition and Fundamental Properties
Problem
Let p(x)=x2 and q(x)=5x+7.
(i) Derive a simplified expression for the average rate of change of p between x=a and x=b.
(ii) Show that this expression can be written as a+b, and hence comment on whether it is constant for all intervals.
(iii) Find the average rate of change of q between x=a and x=b and prove that it is independent of a and b.
(iv) Using your results in parts (ii) and (iii), state two conclusions that distinguish linear and quadratic functions with respect to their rates of change.