mvt-hopitals-rule - Mean Value Theorem and...

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Unformatted text preview: Mean Value Theorem and L’Hˆopital’s Rule Jonathan L.F. King University of Florida, Gainesville FL 32611-2082, USA [email protected] Webpage http://www.math.ufl.edu/ ∼ squash/ 22 March, 2010 (at 17:35 ) Abbrevs. IVT; Intermediate Value Thm . MVT; Mean Value Thm . FTC; Fund. Thm of Calculus . Use cts for “continuous” and cty for “continuity”. The exponential fnc exp can also be written exp( x ) = e x . ( So log ◦ exp = Id R and exp ◦ log = Id R + . ) Use Nev Z to mean “never-zero”; e.g “exp() is Nev Z on R ”. Prolegomena. Use R for the extended re- als [ ∞ , ∞ ] . With DNE denoting “ Does Not Exist ”, adjoin a point to R to create R ~ := [ ∞ , ∞ ] t { DNE } . Use diff’able for “differentiable”. A fnc f : R → R is ext-diff’able ( for extended diff’able ) at point 6 if the lim x → 6 f ( x )- f (6) x- 6 exists in [ ∞ , ∞ ] . Use f LH for the lefthand ext-derivative , from lim x % 6 . Use f RH for the righthand ext-derivative , lim x & 6 . The result below apply to fncs on a closed bounded interval J . For specificity, I will use J := [ 4 , 6 ] and will use J ◦ := ( 4 , 6 ) for its interior. A fnc h : J → R has a ( global ) max-point P ∈ J if ∀ x ∈ J : h ( P ) > h ( x ); and the number h ( P ) is the “ max-value of h on J ” . Weaker, P is a local max-point of h ( on J ) if there exists a J-open set U 3 P , so that P is a global max-point of h U . Imagine analogous defns for min- point , min-value and local min-point of h . 1 : Tool. A continuous h : J → R has a max-point and a min-point. Proof. Interval J is compact, etc. ♦ 2 : Lemma. Suppose continuous h : J → R has a local-extremum at a point τ ∈ J ◦ . If h is extended- differentiable at τ , then h ( τ ) = 0 . ♦ Proof. WLOG τ is a local-min of h . So for all x > τ with x suff. close to τ , necessarily h ( x )- h ( τ ) > 0; thus h RH ( τ ) ∈ [ , ∞ ] . Similarly, h LH ( τ ) ∈ [ ∞ , ] . By hypothesis, h LH ( τ ) = h RH ( τ ). So h ( τ ) is zero. 3 : Rolle’s Thm. Suppose a continuous h : [ 4 , 6 ] → R is ext-diff’able on ( 4 , 6 ) . If h (4) = h (6) , then there exists a point τ ∈ ( 4 , 6 ) such that h ( τ ) = 0 . ♦ Pf. Courtesy(2), WLOG h () has no global-max in J ◦ , so its global-max on J must be the common value of h (4) = h (6). Thus h has a global- min in J ◦ ; pick one, and call it...
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mvt-hopitals-rule - Mean Value Theorem and...

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