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Calculus III

by: Miss Emerald Langosh

Calculus III MATH 2243

Miss Emerald Langosh
GPA 3.94

Yi Lin

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Yi Lin
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This 3 page Class Notes was uploaded by Miss Emerald Langosh on Monday October 12, 2015. The Class Notes belongs to MATH 2243 at Georgia Southern University taught by Yi Lin in Fall. Since its upload, it has received 6 views. For similar materials see /class/222045/math-2243-georgia-southern-university in Mathematics (M) at Georgia Southern University.

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Date Created: 10/12/15
Notes for Chapter 7 71 H A function fx is one to one if and only if that fx1 7 fx2 whenever an 7 x2 a Horizontal line test A function is one to one if and only if it s graph intersects each horizontal line at most once b If a function is increasing or decreasing on an interval 1 then it is one to one N Let f be a one to one function lt s inverse function f 1 is defined by the following rule f 1xy ifandonlyif fyx foranyx intherangeoff U Suppose x is an one to one function and is differentiable at point 1 Then it s inverse function f 1 x is differentiable at point b u moreover 72 H The natural logarithmic function x1 lnxJ idt xgt0 1t a lnxgt0whenxgt llnl 0lnxlt0if1 ltxlt l 1 b 14 g c The number 6 is the unique number in the domain of the natural logarithm sat isfying lne l 2 Properties of Logarithms For any number a gt 0 and b gt O the natural logarithms satisfy the following rules 1 a Product Rule ln ab ln 1 ln b b Quotient Rule lng ln 1 7 lnb c Reciprocal Rule In 7 ln 1 d Power rule In aT T ln 1 where T can be any real number 3 If u is any differentiable function which is never zero JldulnlulC u 1These rule also hold for logarithmic functions with arbitrary base a 1 4 ftanudulncosul C lnlsecu C fcotudulnsinulC lnlcsculC 73 1 y ex is defined to be the inverse function of y In x As a consequence it s domain is 70000 and its range is 000 a 611 x x lnequot x b exlxl ex ex1 ext xix ceTze 1 d em xz 2W2 2 cfl equot equot fexdxex C 3 e lirnxgo xl 74 1 Suppose a gt O a 7 l y ax is defined to be the function e d 1 d a 701quot ln aaquot dx ax b faxdxiC ln 1 2 Suppose a gt O a 7 l y logax is defined to be the inverse function ofy 1quot a lo x 7 In g 7 ln 1 1 d b a logax m 76 1 Suppose both x and gx are positive for sufficiently large x Then a x grows faster than gx as x a 00 if and only if x lim 7 oo39 xaoo gX b x grows at the same rate as gx if and only if lil l l m L gt 0 xaoo gx is a finite number 41 42 43 H N 0 F U I H 0 F H N Suppose f x 0 for all x in Notes for Chapter 4 Let f be a function with domain D The f has an absolute maximum minimum value on D at a point c if x S e x 2 c for all x in D Let f be a function with domain D The f has a local maximum minimum value at an interior point c of its domain D if x c x 2 c for all x in an open interval containing 0 Theorem Suppose f is a continuous function on a closed interval 1 b Then f has an absolute maximum value and an absolute minimum value in 1 b Theorem If f has an absolute maximum or minimum value at an interior point c of its domain and f c exists then f c O An interior point c is a critical point of a function f if f c 0 or f c does not exist Suppose that f be a continuous function on a closed interval 1 b and that x has a global maximum or global minimum at a point c in 1 b Then 0 must satisfy one of the following conditions a c is a critical point b c is one of the end points 1 or c b Rolle s Theorem Suppose that f is continuous on a closed interval 1 b and is differ ential on 1b and that 1 b Then there exists 1 lt c lt b such thatf c O The mean value theorem Suppose that f is continuous on a closed interval 1 b and is differential on 1b Then there exists 1 lt c lt b such that f c w 7 1 1b Then x C for all x in 1b where C is a constant Suppose that f x g x for all x in where C is a constant 1b Then x gx C for allx in 1b Suppose that f is continuous on a closed interval 1 b and is differentiable on 1 b lff x gt 0 on 1b then x is increasing on 1b iff x lt 0 on 1b then x is decreasing on 1 b First derivative test for local extrema Suppose that x is differentiable on 1b except for a point 1 lt c lt b and that c is a cricial point of x Moving across 0 from left to right a If f changes from negative to positive at c then f has a local minimum at c b If f changes from positive to negative at c then f has a local maximum at c c If f does not change sign at c then f does not have a local extremum at c


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