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fundamental theorem of calculus part ii assume fx is a continuous function on ab and also assume that fx is any anti- derivative for fx henceaintb
fundamental theorem of calculus part iif fx is continuous on ab sogx aintx ft dtis continuous on ab and this is differentiable on a b and asgprimex
proof of if fx gt gx for a lt x lt b then aintb fx dx gt gxbecause we get fx ge gx then we knows that fx - gx ge 0 on a le x le b and therefore
proof of int fx gx dx int fx dx intgx dxit is also a very easy proof assume that fx is an anti-derivative of fx and that gx is an anti-derivative
proof of various integral factsformulaspropertiesin this section weve found the proof of several of the properties we saw in the integrals section
rolles theorem assume fx is a function which satisfies all of the following1 fx is continuous in the closed interval ab2 fx is differentiable in
fermats theorem if fx has a relative extrema at x c and fprimec exists then x c is a critical point of fx actually this will be a critical point
proof of limqrarr0 cosq -1q 0we will begin by doing the followinglimqrarr0 cosq -1q limqrarr0cosq - 1cosq 1q cosq 1 limqrarr0cos2q - 1 q cosq
chain rule if fx and gx are both differentiable functions and we describe fx f gx so the derivative of fx is fprimex f primegx
quotient rule fg fg - fgg2here we can do this by using the definition of the derivative or along with logarithmic definitionproof here we do the
product rule f gprime f prime g f gprimeas with above the power rule so the product rule can be proved either through using the definition of the
power rule dxndx nxn-1there are really three proofs which we can provide here and we are going to suffer all three here therefore you can notice all
proof of the derivative of a constant dcdx 0it is very easy to prove by using the definition of the derivative therefore define fx c and the
proof of constant times a function cfxprime cf primexit is very easy property to prove using the definition given you a recall we can factor a
proof of sumdifference of two functions fx gxprime f primex g primex it is easy adequate to prove by using the definition of the
grimm plc grimm has the following transactionsa on 1st january 2010 grimm issued 400000 convertible pound1 6 debentures for pound600000 the
theorem from definition of derivative if fx is differentiable at x a then fx is continuous at x aproof since fx is differentiable at x a we
proof of various derivative factsformulaspropertiesunder this section we are going to prove several of the different derivative facts formulas orand
go back to the complex numbers code in figures 50 and 51 of your notes add code fragments to handle the following1 a function for adding two complex
proof of various limit propertiesin this section we are going to prove several of the fundamental facts and properties about limits which we saw
substitution rule for definite integralsnow we need to go back and revisit the substitution rule as it also applies to definite integrals at some
properties1 intbaf x dx -intba f x dx we can interchange the limits on any definite integral all that we have to do is tack a minus
substitution rulemostly integrals are fairly simple and most of the substitutions are quite simple the problems arise in correctly getting the
determine or find out the direction cosines and direction angles for a 2 1 -4solution we will require the magnitude of the vectora radic 4116
evaluate following integrals 1 - 1 w cos w - ln w dwsolutionin