1013 Lecture 11 PDF

Lecture 11 due Oct 31 Bonus assignment webwork problems is solins to posting coming PopQuit works Implicit differentiation...

Lecture 11 due Oct 31 Bonus assignment webwork problems is solins to posting coming PopQuit works Implicit differentiation by x for some that y y remembering x and applying function of role which differentiation any in it ateran with y we diff time chain rule rule B product rule c quotient difference rule D sum f constant rule f o always  A function is called one to one 1 1 É it never takes the same value twice if X Xa then f x fix L f x 2 3 with domain IR ex a line is 1 1 we can see this FI visually by using 1 nmaitim i if and int seismograph more than once domain IR 1 1 Is gas x with ex it i t the HLT is by at anemic approach x2 suppose g x g 13 XP x2 x X2  1 is 1 1 gix works as we showed This approach it 942 and after following g mathematical roles assumed this we conclude ftp.bra X Xz this statement is Divalent to g Egg if A then B is the same as InotBthennotA. Q are all functions 1 1 fix x with the domain AR fails A no use HLT on the graph fy solid algebraic approach suppose fix fixal 22 xi xi 0 notice that 3 x x2 x x2 fr27 4 X 4 0 OR XiXz an 4 x xz oRx f 2 f 2 so we have this indicates that 2 2 b I can have fix fix is not t 1 function even if Xz  what does that have to do with inverses ok have a function that you suppose describing bacteria growth Populationof input Et N fit bacteria 100 toutput this function tells 9 168 that at time us 2 2 59 t we have 3 358 N bacteria 4 4,45 if I want to know given a What at what time bacteria population of this did it reach population a function that goes would need we direction the in opposite F N time at which we reach popn a we could make this second table because 358 3 each time had a 4 445 unique population i 1 so for each N we only had anechoic for the time these t t scenarios we can undo the In  AKA invert the process original operation Let f be a 1 1 function with Thtain A and range B Then it has an inverse f and f has domain B hangppy apiyame Lingges This picture describe what we called f ft the cancellation equations ITB f f x for all B f f X for all XE A and x we read cancellation egins as f return start with apply f apply back to of f f undoes the operation fat with domain 112 This says Hand inverse have an does not we will address this after an example it most have an inverse f is 1 1 suppose How do we get  Follow the following algorithm write y fax Steph solve for x if possible steps Guy to x express swap f the as role a of y function and x of inverse fix x 2 ex Find the of 3 2 sold y 7 x y 2 3 y 3 x 2 for EIR f x wrote if possible on step 2 see we you R is functions where step There are many the not feasible since at stage of the inverse we are actually defing finding it to address this so how defin inverse functions we use the of the relationship in particular ffiffforayes  e what is its inverse Let y ex Let's the algorithm sold try ex y no where to go solve choice is to use our only Recall y ex has domain IR 0 6 has range from here we know FonÉaYn e has 186 range IR now we know eo 1 so inverse must send 1 to 0 note gas so it's inverse most have as 0 note as inverse X ex x inouse most have it is built we so the once A it a name gives inverse to give we call in this case it ha x  In e for all EIR Takeaway e'nm for all x cop 2 solve where e 1 ex lack to both sides sold first apply 2 en e en i x2 2 0 52 4 52 0 x 52 or x 52 b bso b 1 ex what is inverse of y x to undo we use the function logy self b b for all Xe IR that is log 510901 for all E IR Properties 5 b 5 5 5 7 5 5 5 Xy log x logbly and log  log Xy log x logbly y log x x logs Remarked This is much more common approach inverses when compared for finding to the algorithm using to to this question we want go back Remarly and its inverse ix about y x that a function is not The here is key its we also only determined by need aldomains Remember fax with domain in not the same is function as with domain e gas 0,8 restriction we with this x is 1 1  1 we have an inverse IT is the inverse with domain cop then of me domain 0,6 key 52 for all range 019 1 x ECON X for all O X function to be 1 1 force our Of we can I we can find the domain by restricting inverses even for bad functions