Aim: Rolle’s Theorem Course: Calculus Do Now: Aim: What made Rolle over to his theorem? Find the absolute maximum and minimum values of y = x 3 – x on.

Aim: Rolle’s Theorem Course: Calculus

Do Now:

Aim: What made Rolle over to his theorem?

Find the absolute maximum and minimum values of y = x3 – x on the interval [-3, 3]

Aim: Rolle’s Theorem Course: Calculus

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.

a c b

f

relative max.

d

Note: compare to Extreme Value Theorem: If f is continuous on a closed interval [a, b], then f has both a min. and a max. on the interval.

Aim: Rolle’s Theorem Course: Calculus

Model Problem

Find the two intercepts of f(x)= x2 – 3x + 2 and show that f’(x) = 0 at some point between the 2 intercepts.

f(x)= x2 – 3x + 2 = 0 Find intercepts

(x – 2)(x – 1) = 0

f(1) = f(2) = 0

f’(x) = 2x – 3 = 0

Rolle states there exists a least one c in the interval such that f’(x) = 0

x = 3/2

(1,0) (2,0)

f’(3/2) = 0

f(x)= x2 – 3x + 2

Aim: Rolle’s Theorem Course: Calculus

secant line

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

a c b

f

slope of tangent line = f’(x)

'

f b f af c

b a

tangent line

(a, f(a))

(b, f(b))

Aim: Rolle’s Theorem Course: Calculus

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 b f af c

b a

• “mean” refers to the average rate of change of f in the interval [a, b].

• geometrically, guarantees the existence of a tangent line that is parallel to the secant line thru (a, f(a)) & (b, f(b))

• implies there must be a point in the open interval (a, b) at which the instantaneous rate of change is equal to the average rate of change over the interval.

a c b

f

(a, f(a))

(b, f(b))

Aim: Rolle’s Theorem Course: Calculus

Model Problem

Given f(x) = 5 – (4/x), find all values of c in the open interval (1, 4) such that

4 1'

4 1

f ff c

The slope of the secant line thru (1, f(1)) & (4, f(4)) is

4 1' 1

4 1

f ff c

Because f satisfies the conditions of MVTD, there is at least 1 c in (1, 4) such that f’(x) = 1

2

4' 1f x

x x = ±2

c = 2

Aim: Rolle’s Theorem Course: Calculus

Model Problem

Because f satisfies the conditions of MVT, there is at least 1 c in (1, 4) such that f’(x) = 1

2

4' 1f x

x x = ±2

c = 2

4

3

2

1

2 4

secant line

tangent line

(2, 3)

(1, 1)

(4, 4)

f(x) = 5 – (4/x)

Aim: Rolle’s Theorem Course: Calculus

Model Problem

1( ) on the interval [ 2,2].

Find the MVTD.

f xx

1 112 2

'( )2 2 4

f c

'

f b f af c

b a

2

1'( )f x

x

2

1 1'( )

4f c

c

There is no value of c that satisfies this equation!

discontinuous at x = 0

[1,3]

2c i

Aim: Rolle’s Theorem Course: Calculus

Model Problem

Find the values of c that satisfy MVTD

for on the interval [-4, 4].1

( )f x xx

Evaluate f(-4) 1 17

( 4) 44 4

f

Evaluate f(4) 1 17

(4) 44 4

f

17 174 4 174 4

'4 4 4 4 16

f ff c

'

f b f af c

b a

Evaluate

Aim: Rolle’s Theorem Course: Calculus

Model Problem

Find the values of c that satisfy MVTD

for on the interval [-4, 4].1

( )f x xx

17 174 4 174 4

'4 4 4 4 16

f ff c

2

1' 1f x

x 2

1 17' 1

16f c

c

This confirms there’s no solution to this equation

2

17 1 11

16 16c

2 16c

Find f’(x) and evaluate and solve for c

4

3

2

1

-1

-2

-3

-4

-2 2 4

r x = 1+1

x

Aim: Rolle’s Theorem Course: Calculus

Model Problem

Two stationery patrol cars equipped with radar are 5 miles apart on a highway. As a truck passes the first patrol car, its speed is clocked at 55 mph. Four minutes later , when the truck passes the second patrol car, its speed is clocked at 50 mph. Prove the truck must have exceeded the speed limit of 55 mph at some time during the 4 minutes.

Aim: Rolle’s Theorem Course: Calculus

'

f b f af c

b a

Model ProblemTwo stationery patrol cars equipped with radar are 5 miles apart on a highway. As a truck passes the first patrol car, its speed is clocked at 55 mph. Four minutes later , when the truck passes the second patrol car, its speed is clocked at 50 mph. Prove the truck must have exceeded the speed limit of 55 mph at some time during the 4 minutes.

4 1

60 15t hr

s(t) = distance traveled as a function of times(0) = distance traveled at car 1 = 0 mi.s(1/15) = distance traveled at car 2 = 5 mi.

10

515' avg. vel. = 75mph

1 1 15015

s ss c