A student claims: “Whenever a function is continuous on the closed interval and…
MATHEMATICS · Multi-part · AP · Calculus AB · Analytical Applications of Differentiation · Analytical Applications of Differentiation
Problem
A student claims:
“Whenever a function f is continuous on the closed interval [0,L] and differentiable on the open interval (0,L), there is at least one point c in (0,L) where the instantaneous rate of change of f equals the average rate of change of f on [0,L].”
(1) Provide a clear, logically ordered explanation that supports or refutes the student’s claim. Your explanation should reference appropriate properties of differentiable functions on closed intervals.
(2) Illustrate your reasoning with the specific function g(x)=x2 on the interval [0,5]. Identify every value of c in (0,5) that satisfies the student’s stated condition, and justify your findings.