Fact: All polynomial functions are continuous and differentiable everywhere.

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1 Dierentibility AP Clculus Denis Shublek ilernmth.net Dierentibility t Point Deinition: ( ) is dierentible t point We write: = i nd only i lim eists. '( ) lim = or '( ) lim h = ( ) ( ) h 0 h Emple: The grph o ( ) = ( ) 3 is dierentible = 3 since non-verticl tngent line to the grph cn be drwn t this point. Algebriclly, ssuming we live in pre-shortcuts to dierentition world, we cn compute the limit o the quotient to ind the precise vlue o the slope: 3 3 ( ) (3 ) ( 3 3)( 3) 3 3 '(3) = lim( 3 3) = 3 3 Fct: All polynomil unctions re continuous nd dierentible everywhere.

2 Dierentibility AP Clculus Denis Shublek ilernmth.net One-Sided Derivtives Deinition: ( ) is dierentible rom the right t Deinition: ( ) is dierentible rom the let t = i nd only i lim = i nd only i lim eists. eists. Emple: ( ) = ils to be dierentible t = 0, but both one-sided derivtives eist. From the let, the slopes rom the let nd right re 1 nd 1, respectively. Algebriclly, we veriy the one-sided derivtives. Note tht 0 implies tht > 0, so we cn write =. Also, 0 implies tht < 0, so we cn write =. We write: lim = lim = lim = 1 ( ) (0) lim = lim = lim = 1 ( ) (0) ' (0) = 1 ' (0) = 1

3 Dierentibility AP Clculus Denis Shublek ilernmth.net Theorem: I ( ) is dierentible t point Dierentibility Continuity =, then ( ) is continuous t =. The converse is lse. Continuity does not necessrily imply dierentibility. The bsolute vlue unction in the emple bove helps illustrte. There re three cses o continuous unctions tht il to be dierentible t point: shrp corner (bsolute vlue unction t the origin), cusp, or verticl tngent line. Emple: y = 3 hs cusp t the origin. Emple: y 1 = 3 hs verticl tngent line t the origin.

4 Dierentibility AP Clculus Denis Shublek ilernmth.net FACTS, THEOREMS, nd DEFINITIONS A Criticl Number = c is criticl number o ( ) i nd only i ( ) is continuous there nd '( ) 0 does not eist. A Locl (Reltive) Mimum ( ) ttins locl mimum t ( c, ( c )) i nd only ( c) ( ) = c. ( ner = c mens in smll open intervl contining c A Locl (Reltive) Minimum ( ) ttins locl minimum t ( c, ( c )) i nd only ( c) ( ) = c. A Globl (Absolute) Mimum ( ) ttins globl mimum t ( c, ( c )) i nd only ( c) ( ) the domin o ( ). A Globl (Absolute) Minimum ( ) ttins globl minimum t ( c, ( c )) i nd only ( c) ( ) the domin o ( ). Fermt s Theorem I '( c ) eists nd ( ) hs locl m or min t ( c, ( c )), then '( c ) = 0. 3 The converse o Fermt s Theorem is lse. Consider ( ) = t = 0. Etreme Vlue Theorem c = or or ll vlues ner = ) or ll vlues ner or ll vlues in or ll vlues in I ( ) is continuous n closed intervl [, b ], then ( ) ttins n bsolute mimum nd n bsolute minimum in [, b ]. The bsolute etrem could occur nywhere in [, ] b, in the interior or t the endpoints. EVT does not show how to determine these points; it simply gurntees their eistence.

5 Dierentibility AP Clculus Denis Shublek ilernmth.net Men Vlue Theorem I unction ( ) is dierentible (nd thereore continuous) on n intervl [, b ], then there eists t lest one number = c (, b ) such tht '( c ) =. ( b) ( ) b In plin English, the instntneous rte o chnge must equl the verge rte o chnge on the given intervl t lest once. Rolle s Theorem [ Specil Cse o MVT ] Suppose ( ) stisies the ollowing conditions: i. ( ) is continuous on [, b ] ii. ( ) is dierentible on (, b ) iii. ( ) = ( b ) Then there eists t lest one number = c (, b ) such tht '( ) 0 Corollries c =. I. I '( ) = 0 or ll, then ( ) is constnt unction: ( ) II. I '( ) = g '( ) or ll, then = g c. = k. Closed Intervl Method Gol: Determine the bsolute etrem (m nd min) o continuous unction whose domin is restricted to closed intervl [, b ]. Check tht ( ) is continuous on the intervl [, b ] Find ll criticl numbers c 1, c,... in (, b ) Evlute 1 ( ), ( ), ( ), ( ),... b c c The lrgest nd the lest y vlues rom the previous step re the globl (bsolute) etrem.