MATHEMATICS · Multi-part · AP · Calculus AB · Differentiation: Definition and Fundamental Properties · Differentiation: Definition and Fundamental Properties
Problem
Let f be the function defined for x>0 by f(x)=xr, where r=1.
Suppose that the average rate of change of f on the interval [1,4] equals the instantaneous rate of change at x=2.
Parts:
(a) Write an algebraic equation that expresses the condition described above. Your equation should involve r only.
(b) Solve the equation from part (a) to determine all real values of r that satisfy the condition. Justify each step in your solution, including any restriction you impose.
(c) For the value(s) of r found in part (b), determine whether f is concave up, concave down, or neither on the interval (1,4). Provide complete justification using second-derivative analysis.
(d) Interpret, in the context of the graph of f, why the equality of the average and instantaneous rates at the specified points is or is not consistent with the concavity you identified in part (c).