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