Unformatted text preview: The Mean Value Theorem Consider: Let be a function that is: 1. continuous on the closed interval 2. differentiable on the open interval 3. Rolle's Theorem Let be a function that is: 1. continuous on the closed interval 2. differentiable on the open interval 3. f then there is a number in such that case 1: then case 2: , where is a constant. everywhere next page
MTH 173 section 4.2 notes, page 1 case 2: there is at least one in such that . then has a maximum in . But since there is an such that maximum is not at an end point. The maximum occurs at an interior point since the derivative always exists, Fermat's thm tells us that case 3: there is at least one in such that . then again appeal to the Extreme Value thm and Fermat's thm to obtain , the , and, . We for some in Now consider: Let be a function that is: 1. continuous on the closed interval 2. differentiable on the open interval Is there a point on the graph between the end points at which the tangent line is parallel to the line connecting the end points (the secant line) ? MTH 173 section 4.2 notes, page 2 The Mean Value Theorem Let be a function that is: 1. continuous on the closed interval 2. differentiable on the open interval Then there is a number in such that or Proof: define the function this function gives the length of the vertical line segment we saw in the animations is continuous, applies: exists everywhere in , so Rolle's thm there is a in such that MTH 173 section 4.2 notes, page 3 10. Graph in , and line thru estimate where tangent line is parallel to the secant line and . determine the exact values of slope where this occurs. 12. show MVT hypotheses satisfied and then determine . continuous in differentiable in MTH 173 section 4.2 notes, page 4 18. Show that sin cos sin has exactly one real root consider since cos for all , where for all there is a in such that thus since for any cos or, cos cos now since and , or , we know that for all this tells us that, as the graph goes from left to right, the Now sin thus there is a zero between another zero. 24. Suppose sin sin values are always increasing and . Since the function always increases, there can't be for all . Show that for some in Consider but , so then, multiplying by , we obtain
MTH 173 section 4.2 notes, page 5 ...
View
Full Document
 Spring '09
 Bush
 Calculus, Topology, Derivative, Mean Value Theorem, open interval, Rolle

Click to edit the document details