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MAT 1234 Calculus I. Section 3.2 The Mean Value Theorem. http://myhome.spu.edu/lauw. No Homework!!! Take time to review problems from section 2.8 and/or Start lab 04 -- be sure to read the info in the PPT. Maple Lab tomorrow. Preview. Rolle’s Theorem The Mean Value Theorem

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mat 1234 calculus i

MAT 1234Calculus I

Section 3.2

The Mean Value Theorem

http://myhome.spu.edu/lauw

slide2

No Homework!!!

  • Take time to review problems from section 2.8 and/or
  • Start lab 04 -- be sure to read the info in the PPT.
  • Maple Lab tomorrow
preview
Preview
  • Rolle’s Theorem
  • The Mean Value Theorem
  • Consequences of the Mean Value Theorem
rolle s theorem
Rolle’s Theorem

Suppose f satisfies the following 3 conditions:

1. f is continuous on [a,b].

2. f is differentiable on (a,b).

3. f(a) = f(b)

Then there is a number c in (a,b) such that

example 1
Example 1*

Prove that has exactly one real root.

example 1 q a
Example 1* (Q&A)

Why do we need to show it when it is obvious from the graph?

we know
We know…

If f(x)=C on (a,b), then f’(x)=0 on (a,b)

t or f
T or F

If f’(x)=0 on (a,b), then f(x)=C on (a,b)

the mean value theorem
The Mean Value Theorem

Let f be a function satisfies the following conditions:

1. f is continuous on the closed interval [a,b].

2. f is differentiable on the open interval (a,b).

Then there is a number c in (a,b) such that

or, equivalently,

theorem consequence
Theorem (Consequence)

If f’(x)=0 for all x in an interval (a,b), then f is constant on (a,b).

Why?

possible exam questions
Possible Exam Questions
  • State the Rolle’s Theorem
  • State the Mean Value Theorem
  • Explain why the following is true