Download 3.2 Rolle*s Theorem

3.2 Rolle’s Theorem
•Understand and use Rolle’s Theorem.
•Understand and use the Mean Value Theorem.
1. Sketch a rectangular coordinate plane. Label the
points (1,3) and (5,3). Draw the graph of a
differentiable function f that starts at (1,3) and
ends at (5,3).
a) Is there at least one point on the graph for which f’ =0?
b) Can you draw a graph where every point’s derivative is
zero?
c) Would it be possible to draw a graph where there is no
point for which f’ = 0?
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 is at least one number
c in (a,b) such that f’(c) = 0.
Determine whether Rolle’s Theorem can be applied
to the function. If it can find all values of c where
f’(c) = 0.
1. f(x) = x² - 5x + 4 [1,4]
2. f(x) = (x² -1)/ x [-1,1]
3. f(x) = sec(x) [ -π/4, π/4 ]
Mean Value Theorem
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
f ’(c) =
Two interpretations
1) Guarantees there is a tangent line that is parallel to
the secant line that goes through a and b.
2) Implies that there must be a point at which the
instantaneous rate of change is equal to the
average rate of change.
Remember that polynomial, rational and trigonometric
functions are all differentiable at all points in their
domains.