3.2 Rolle’s Theorem and the Mean Value Theorem Objective: Understand and use Rolle’s Theorem and the Mean Value Theorem. Miss Battaglia AP Calculus AB/BC. Exploration.
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AP Calculus AB/BC
Let f be continuous on the closed interval [a,b] and differentiable on the open interval (a,b). If
then there is at least one number in c in (a,b) such that f’(c)=0.
Find the two x-intercepts of f(x)=x2-3x+2 and show that f’(x)=0 at some point between the two x-intercepts.
Let f(x)=x4-2x2. Find all values of c in the interval (-2,2) such that f’(c)=0.
If f is continuous on the closed interval [a,b] and differentiable on the open interval (a,b), then there exists a number c in (a,b) such that
Given f(x)=5-(4/x), find all values of c in the open interval (1,4) such that
Two stationary patrol cars equipped with radar are 5 miles apart on a highway (please draw!). As a truck passes the first patrol car, its speed is clocked at 55 mph. Four minutes later, when the truck passes the second patrol car, its speed is clocked at 50 mph. Prove that the truck must have exceeded the speed limit (of 55 mph) at some time during the 4 minutes.