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# CALCULUS I MATH 165

ISU

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This 6 page Class Notes was uploaded by Ms. Helen Sipes on Sunday September 27, 2015. The Class Notes belongs to MATH 165 at Iowa State University taught by Staff in Fall. Since its upload, it has received 22 views. For similar materials see /class/214500/math-165-iowa-state-university in Mathematics (M) at Iowa State University.

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Date Created: 09/27/15

Math 165 F3 Review 1 Sections 11 167 21 27 in Varberg7 Purcell7 Rigdon 1 Limits a Limits as m a c m a 0 m a 0 i substitution ii 77factor and cancel iii The trig limits lim sum 1 and lim 0 maO maO 2 lim fm L gt lim fm limi fm L V The Squeeze Theorem b Limits as m a 00 i divide the numerator and the denominator by the highest power of m that appears in the denominator 1 ii lim 70ifagt0 maoo ma iii if fm is bounded as m a 00 and lim gm 07 then lim fmgm 0 c ln nite limits f 96 if limgm 0 and lim fm L 31 07 then lim does not exist in the nite sense7 mac mac mac 9 it can be foo or 00 ii if gm a OJr as m a c and limfm L gt 07 then limw 00 not not 9x i i 7 i f96 7 111 if gm a 0 as m a c and limfm a L gt 07 then lim a 700 mac mac 9 z 2 Continuity a m is continuous at c it Lint fm fc b C d the discontinuity at c is removable if lim fm exists in the nite sense otherwise the mac fm is right continuous at c if fm fc fm is left continuous at c if fm fc discontinuity is non removable e functions continuous in their domains i polynomial7 rational ii sinm7 cos m7 tan m7 cot m7 sec m7 csc m iii lml W f If f and g are continuous at c then so are kf f g f a g f g fg provided 90 31 07 f and W provided that fc 2 0 if n is even g If g is continuous at c and f is continuous at 90 then the composite f o g is continuous at c h Intermediate Value Theorem Let f be a function de ned on 1 b and let W be a number between fa and fb If f is continuous on 1 b then there is at least one number 0 between a and b such that f0W 3 Derivative fz h 7 fa a We 533 f b f c is equal to the slope of the tangent line to the graph of fx at the point z c c Rules i The SumDifference Rule g f g x7 ig f 791x ii The Product Rule gx fzg s f g VA f 96996 711909196 9W iv The Chain Rule fltgltzgtgti f gz we V DAM 0 vi mm mm vii The Power Rule DEW aza l for a 31 0 The Quotient Rule l viii Trigonornetric functions Dwsin s cos 7 Dmlcos x isin x Dmltan x sec2 x Dmlcot x 7 csc2 7 Dmlsec s tanxsec x Dmlcsc s 7cotcscm d Higher order derivatives7 notations f 96 f 96 WW WW f 96 V Dzy D39 D39 D39 D321 dy dzy dgy d4y d y dx dz dx3 dz4 39 39 39 d l39 e lrnplicit Differentiation Given Fzy 07 nd 3 i differentiate both sides of the equation Fzy 0 with respect to 7 ii solve for Si 1 Math 165 Ju Ming Spring72008 REVIEW NOTES FOR FINAL EXAM Before taking the nal exam7 you should know 1 The nal exam will consist of two parts7 just as midterm exam 2 For part 17 please focus on chapter 67 and the First Fundament Theorem Know how to use the chain rule for nding derivatives and the substitution rule for nding integrals involving exponential functions7 logarithmic functions and trig and the inverse of trig funs sample questions 43 2123 61 3791522 62 117157177277374163 1718247 68397437467557617 3 For part ll7 please focus on i monotonicity and concavity7 sample questions 63 252729 ii extreme values 7 sample questions 34 example 1 and the question like A box with an open top has the form of a rectangular parallelepiped with a square base If the material used in the bottom costs 5 per square foot7 and the material used in the sides costs 3 per square foot7 and the box is to have a total volume of 20 cubic feet7 what dimensions will minimize the cost of the box iiiuse de nition of de nite integral to nd the integrals Sample questions 42 exam ple 3 and the No1 problem in part II of previous nal exam iv the existence and the derivative of the inverse function f l Sample questions 62 7 737 740 v exponential growth and decay7 Newton7s law of cooling Sample questions 65 ex ample 27 example 4 virelated rates viidi erential equations section 39 example 27 problem set 59 4 memorize the derivative and integral formulas for some basic functions7 for instance7 trig funs7 inverse of trig funs7 exp funs cm or 7 log funs lnm and logaz 5 try the previous nal exam before you check answers 6 the nal exam can not be nished at home Math 165 Ju Ming Spring72008 REVIEW NOTES Chapter 6 I 1 Natural Logarithm Eun1lnx 1dt 2 1 t lnxl7 ldxlnlxlC 4 iln107 iilnablnalnb iiiln lna 7111b ivlnaT Tlna 2 Inverse Function 1if function x is oneto one function7 then there exists an inverse f 1 of f 7furthermore y x ltgt x f l 2Thm If f is strictly increasing or decreasing7 then f 1 exists 3 4 where y x Natural Exponential Fun 1 def the inverse of lnx 7denoted as y em in other words7 x ey ltgt y lnx 2 03 eh x7 lnem x 3 em em emdx em C 4 ex 57 General Exponential and Logarithmic Euns 1General Exponential Fun I1 call where a gt 0 and a 1 2 awry emey em y q 1 iamy away iiam y a aw ivabm ambm my 7 m a amlna amdx L C lna 1 4General Logarithmic Fun the inverse of general exponential fun ylogamltgtmay 5 1 In Oga 7 lna 6 logl 7 mlna 7logarithmic differentiation ex suppose y mm nd 3 Solution y lny lnzm zlnz 1 lny mlnzy lnmm7 lnm1 13 1 since lny 11 y 1 a 7y lnm1 y a y lnm 1y lnm 1zm 5 Exponential Growth and Decay 1 g kg y Cekm dm 6 First Order Linear Differential Equations 37 Pltzgty QM integrating factor 5f PM d d efPmdm efPmdem ltgt ltefPmdmy efPmdem efPmdzy gt y eifPmdz 7 The lnverse Trig Funs and their Derivatives1 msin 1yltgtysinz7 where 77r2gzg7r2 zcos 1gltgtycosz7 where0 z 7r mtan 1yltgtytanz7 where 77r2ltzlt7r2 zsec 1yltgtysecm7 where0 zlt7r2and7r2ltz 7r y sin 1 z domain 711 range in27172 y cos 1 x domain 711 range 07139 y tan lm domain 70000 range in27172 1 y sec 1 z domain foo 71 U 1 00 range 0 7r2 U 7r2 7r i 1 H 7 1 2sm z 7 W 22cos 1x 77W 7 1 7 1 222tan 1x 71m2 2Usec 1m 7 1 1 2 dz sin 71 c MU12 mzdz itan 1 1 c 444 7 7 71 222 2 azdz7asec aC

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