MATHEMATICS · Multi-part · AP · Calculus AB · Differentiation: Definition and Fundamental Properties · Differentiation: Definition and Fundamental Properties
Problem
Let
h(x)=ax3−6x+4
for real a. Consider the interval [1,4].
(1) Compute the average rate of change of h on [1,4] and write your answer as a function of a. Show all algebraic steps.
(2) Find the instantaneous rate of change of h at x=1 in terms of a. Clearly indicate any derivative you compute.
(3) Determine all real values of a for which the instantaneous rate of change found in part (b) equals the average rate of change found in part (a). Justify your solution algebraically.
(4) For each value of a obtained in part (c), discuss whether there must exist at least one x in the open interval (1,4) at which the instantaneous rate of change of h equals that same common rate. Provide a clear justification based on properties of the function on the interval.