We Calculus!!!

3.2 Rolle’s Theorem and the Mean Value Theorem

Rolle’s Theorem

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 numberc in (a, b) .

a bc c

Ex. Find the two x-intercepts of and show that at some point between the two intercepts.

x-int. are 1 and 2

Rolles Theorem is satisfied as there is a point atx = 3/2 where .

Let . Find all c in the interval (-2, 2)such that .

Since , we can use Rolle’s Theorem.

Thus, in the interval(-2, 2), the derivativeis zero at each of thesethree x-values.

8

If f (x) is continuous over [a,b] and differentiable over (a,b), then at some point c between a and b:

f b f af c

b a

Mean Value Theorem for Derivatives

The Mean Value Theorem only applies over a closed interval.

The Mean Value Theorem says that at some point in the closed interval, the actual slope equals the average slope.

y

x0

A

B

a b

Slope of chord: f b f a

b a

Slope of tangent:

f c

y f x

Tangent parallel to chord.

c

A function is increasing over an interval if the derivative is always positive.

A function is decreasing over an interval if the derivative is always negative.

A couple of somewhat obvious definitions:

y

x0

y f x

y g x

These two functions have the same slope at any value of x.

Functions with the same derivative differ by a constant.

CC

C

C

Example 6:

Find the function whose derivative is and whose graph passes through .

f x sin x 0,2

cos sind

x xdx

cos sind

x xdx

so:

f x could be cos x or could vary by some constant .C

cosf x x C

2 cos 0 C

Example 6:

Find the function whose derivative is and whose graph passes through .

f x sin x 0,2

cos sind

x xdx

cos sind

x xdx

so:

cosf x x C

2 cos 0 C

2 1 C 3 C

cos 3f x x

Notice that we had to have initial values to determine the value of C.

HW Pg. 176 1-7 odd, 11-27 odd, 33-45 odd