{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

11.16 - Section 4.2 The Mean Value Theorem(continued Mean...

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

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.

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

View Full Document
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 ].
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}