Rolle's Theorem and x The Mean Value Theorem y Tangent line is B parallel to chord AB A x a c b If f is continuous on the closed interval [a, b] and differentiable on the open interval (a, b) there exists a number c in (a , b) such that VERBALLY SAYS: Instantaneous rate of change = average rate of change GRAPHICALLY SAYS: Tangent line is parallel to the secant line. MORE THOUGHTS: Mean Value Theorem guarantees that at some point on the closed interval [a,b], the tangent line will be parallel to the secant line for that interval. Since parallel lines have the same slopes, there is always some place on the interval where a continuous function is changing at exactly the same rate it's changing on average for the entire interval. 1 EX #1: Find the number c that satisfies the conclusion of the Mean Value Theorem for f on [1, 3]. Given: 2 EX #2: Finding a Tangent Line Given , find all values of c in the open interval (1,4) such that Find the slope of the secant line: Does f satisfy the conditions of the Mean Value Theorem? Find x such that f '(x) = f ' (c) 3 EX #3: At what x­value(s) on the interval [–2, 3] does the graph of f (x) = x2 + 2x – 1 satisfy the M.V.T.? Step 1: Find f ' (x) Step 2: Find f (–2) and f (3) Step 3: Apply definition of Mean Value Theorem 4 SPECIAL NOTES: y = |x| on [1, 1] Fails at a point! The function is differentiable on interior (–1, 1) except at x = 0, therefore graph has no tangent parallel to chord AB. A B –2 2 5 M.V.T. allows us to identify exactly where graphs rise or fall. positive increasing Functions with derivatives are functions. negative decreasing EX #4: Find local extrema and intervals on which 2 y = 5x – x is increasing and decreasing. Step 1: Find critical numbers of f Step 2: Set up intervals determined by critical numbers and test sign changes of f ' (x) Step 3: Determine behavior of f Step 4: Find local extrema 6 Rolle's Theorem Let f be continuous on the closed interval [a, b] and differentiable on the open interval (a , b). If f (a) = f (b) then there exists at least one number c in (a, b) where f ' (c) = 0. VERBALLY SAYS: If these conditions hold true, then there is at least 1 number between a and b so that the tangent line is horizontal. GRAPHICALLY SAYS: There is at least one point where the slope of the tangent line is zero on the interval. 7 EX.#5: Determine whether Rolle's Theorem can be applied. If so, find c. If not, explain why. [–2 , 2] STEP 1. Continuous and differentiable?? ? STEP 2: f(–2) = f(2) STEP 3: Find c. 8 EX #6: Find the two x­interceps of and show that f ' (x) = 0 at some point between the two intercepts. 9.