11.16 - Section 4.2 The Mean Value Theorem(continued Mean Value Theorem Let f be a function on the interval[a,b(with a < b that satisfies 1 f is

Info iconThis preview shows pages 1–3. Sign up to view the full content.

View Full Document Right Arrow Icon
Section 4.2: The Mean Value Theorem (continued) Mean Value Theorem : Let f be a function on the interval [ a , b ] (with a < b ) that satisfies 1. f is continuous on the closed interval [ a , b ] 2. f is differentiable on the open interval ( a , b ). (We assume a < b .) Then there is a number c in ( a , b ) such that f ( c ) = ( f ( b )– f ( a ))/( b a ). To see why the Mean Value Theorem is intuitively reasonable, note that ( f ( b )– f ( a ))/( b a ) is the slope of the secant joining ( a , f ( a )) and ( b , f ( b )). This secant line touches the curve y = f ( x ) in at least two points. If we imagine sliding the secant line up or down, there ought to be some position where it becomes an actual tangent line. If we write the intersection of the tangent line with the curve as ( c , f ( c )), then this is exactly the point c that we need. A special case of the Mean Value Theorem is Rolle’s Theorem: Let f be a function that satisfies 1. f is continuous on the closed interval [ a , b ], 2. f is differentiable on the open interval ( a , b ), 3. f ( a )= f ( b ). Then there is a number c in ( a , b ) such that f ( c ) = 0.
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Rolle’s Theorem can be seen as a special case of the Mean Value Theorem. But you can also derive the Mean Value Theorem from Rolle’s Theorem, as in Stewart. Consequences of Mean Value Theorem: Theorem 5: If f is continuous on [ a , b ] and differentiable on ( a , b ) and the derivative of f on ( a , b ) is constantly 0, the function f is constant on [ a , b ]. Proof: Take
Background image of page 2
Image of page 3
This is the end of the preview. Sign up to access the rest of the document.

This note was uploaded on 02/13/2012 for the course MATH 141 taught by Professor Staff during the Fall '11 term at UMass Lowell.

Page1 / 7

11.16 - Section 4.2 The Mean Value Theorem(continued Mean Value Theorem Let f be a function on the interval[a,b(with a < b that satisfies 1 f is

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online