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2. Rolle’s Theorem and Mean Value Theorem Rolle’s Theorem Let f be a function that satisfies 3 hypotheses: f is continuous on the closed interval [a,b] f is differentiable on the open interval (a,b) f(a) = f(b) Then there is a number x=c in (a,b) such that f’(c)=0 This makes sense because if a function has the same y value in 2 places, it must turn around somewhere in between and therefore the derivative will be 0 Rolle’s Theorem Here are some functions that satisfy Rolle’s Theorem Application Rolle’s Theorem is often applied with motion problems such as throwing a ball into the air. You know that sometime in between when the ball is thrown and when it returns to the ground, it must turn around and therefore its derivative will equal 0 at its maximum height. Example 1 Determine if Rolle’s Theorem applies. If so, find the values of x guaranteed by the theorem for f ( x ) x 2 x on ( 2, 2) Note: Even if you given an open interval, when the theorem is invoked, we MUST check the conditions on a closed interval. 4 2 Mean Value Theorem French mathematician Joseph-Louis Lagrange first stated the next important theorem. Mean Value Theorem Connects concepts of average rate of change over an interval (secant line) with the instantaneous rate of change at a point within that interval (tangent line) Somewhere between points a and b on a differentiable curve, there is at least one tangent line parallel to secant AB. Mean Value Theorem (MVT) If f(x) is continuous at every point on [a,b] and differentiable on (a,b), then there is at least one point c in (a,b) where f '( c ) Slope of tangent line f (b ) f ( a ) ba Slope of secant line Rolle’s Theorem is the special case of the MVT where f(a) = f(b) Example 2 Show that f(x) = x2 satisfies the Mean Value Theorem on [0,2]. If so, find the value(s) guaranteed by the theorem. Example 3 Why does the mean value theorem not work on [-1,1] for the following functions? Not differentiable at x=0 x 1 x 3, 2 x 1, 2 x 1 Not continuous at x=1 x 1 Example 4 Calculator permitted. Determine all the numbers c which satisfy the conclusion of the MVT for the function f ( x ) x 2 x x on [-1,2]. 3 2 Example 5 The calculus cops have set up their elaborate speed trap on a busy stretch of road. A suspected pumpkin farmer who uses the road daily to haul his pumpkins to market is suspected of chronic speeding well beyond the posted limit of 55 mph. The calculus cops aim to finally ticket this unlawful transporter of seasonal gourds. The calculus cops set up 5 miles apart from each other. Calculus cop A spots the farmer with his truck loaded down with would-be jack-o-lanterns. As the farmer passes cop A, he is clocked at a paltry 50 mph. Cop A could have sworn the farmer waves at him as he drives by. Cop A immediately radios Calculus cop B, whereby cop B starts his timer. Four minutes later, cop B clocks the farmer cruising by at only 55 mph. He clearly sees the farmer wave at him with a giant grin that would make a jack-o-lantern jealous. After a quick calculation on his field-issued TI-84 calculator, he pulls out with his lights on to issue a speeding ticket to the ornery pumpkin farmer. For what speed can calculus cop B ticket the farmer? Would this ticket hold up in court? Why or why not? Example 6 Let f(x) be a function that is differentiable for all x. a. Suppose that f(0)=-3 and f’(x)≤ 5 for all values of x. How large can f(2) possibly be? b. Suppose that f(5)=2 and f’(x)≥ -3 for all values of x. How large can f(1) possibly be? MVT The MVT can help establish some basic facts of differential calculus, such as If f’(x)=g’(x) for all x in an interval then f and g are “parallel” on that interval