Review Sheet for MATH 129 at UA
Review Sheet for MATH 129 at UA
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Date Created: 02/06/15
REVIEW SHEET FOR CALCULUS I 1 De nition The function f is continuous at z c if f is de ned at z c and if gig fltzgt 1 2 De nition For any function f we de ne the derivative function f by WHO 7 at 7 7 f 7 Rate of change of f at 0 7 flan h Sometime we denote f by 3 You may interpret f as a distant function of an object respect to time The derivative function f can be iterpreted as a instantaneous velocity of the object 4 Some basic facts about rst derivatives o If f gt 0 on an interval then f is increasing over that interval o If f lt 0 on an interval then f is decreasing onver that interval 5 Some basic facts about second derivatives o If f gt 0 on an interval then f is increasing so the graph of f is concave up there o If f lt 0 on an interval then f is decreasing so the graph of f is concave down there 6 De nition The function f is differentiable at I if hm 1 h 7 1 exists hgt0 h 7 Properties Some fundamental rules If f and g are differentiable and c is a constant then 39 ldml CHI lf1yrl f 19 1and Tilfl91l f 1 MI 0 The Product Rule fg f g fgh o The Quotient Rule g ngigfij The Chain Rule ym f 91 9 I 39 067190 8 Derivatives of elementary functions 0 For any constant n 21 nzn l o e e o a lnaa o For I in radians sin I cosz and cos I 7 sin I o For I in radians tan z o lnz o arctanz o arcsinz o coshz sinhz 9 10 11 12 13 14 15 16 o sinh z cosh z LlHopital7s rule If f and g are differentiable fa 9a 0 and g a 0 then lm a w Ha 9I y a De nition Left Sum Right Sum and De nite lntegral Suppose f is continuous for a S t S 12 The de nite integral off from a to 1 written 117mm is the limit of the lefthand or righthand sums with n subdivision of a S t S b as n gets arbitrarily large In other words I n71 dt lim Lefthand sum lim lt E ftiAtgt a ngtoo ngtoo 110 and b ft dt lim Righthand sum lim 2 ftAtgt a ngtoo ngtoo i1 Each of these sums is called a Riemann sum f is called the integrand and a and b are the limits of integration Property The De nite lntegral as an Area When 2 0 and a lt b we have 1 Area under graph off and above zaxis between a and b dzl If you interpret the continuous function as a velocity of some moving object at time t then the de nite integral f dt can be interpreted as a distance which the object travel during the time a and the time 12 The Fundamental Theorem of Calculus the rst version If f is continuous on the interval ab and Ft then b ftdt Fb 7 Fai Theorems about De nite lntegral If a b and c are any numbers and f is a continuous function then ffzdz fba eme f fedefffede game If f and g be continuous functions and let 0 be a constantl ffltfltzgt em dz ff fltzgt dz i Liam dry fjcfzdz er mode De nition If the derivative of F is f we call F an antiderivative of f Some properties of antiderivatives 17 18 19 20 o If Fz 0 on an interval then Fz C on this interval for some constant C o F and G are both antiderivatives of f on an interval then Fz Cz C All antiderivatives of are the form of Fz C We introduce a notation f dz for the general antiderivative and is called the inde nite integral fz dz Fm a Properties of Antiderivatives o ffzdzifgzdzi o fcfzdzcffzdzi Antiderivaties o fkdzkzCi o fzndz C forn7 ill 0 fidzlnlzlCi o fefdzeCi o fcoszdzsinzCi o fsinzdz7coszCi Fundamental Theorem of Calculus the second version If f is a continuous function on an interval and if a is any number in that interval then the function F de ned as follows is an antiderivative of f Fm mm mm m In other words we have
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