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# Calculus II MATH 116

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This 26 page Class Notes was uploaded by Edgar Jacobi on Monday September 7, 2015. The Class Notes belongs to MATH 116 at Kansas taught by Yasuyuki Kachi in Fall. Since its upload, it has received 5 views. For similar materials see /class/182380/math-116-kansas in Mathematics (M) at Kansas.

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Date Created: 09/07/15

Math 116 CALCULUS II REVIEW OF LECTURES 7 II June 19 Fri 2009 Instructor Yasuyuki Kachi Line 84287 0 The di 39erential rules 6 Last time I talked about 6 Today I want to talk about Today I also want to talk about derivatives Actually today I only want to talk about the derivative aspect7 of 6 So lets talk about derivatives first You know the derivatives You have seen the symbol d d d You remember that for the first time you were introduced to the symbol T at you were taught that this symbol alone does not make sense In your former calculus class you were taught that once it is attached to some quantity written in terms of at such as 2 then it will make sense So you write something like d 2 die in 39 39 d 2 39 39 u 39 39 and then it Will make sense As you know d x signi es the derivative of at 2 77 As you know this equals 215 d 2 7x 210 dx 0 Today I offer you a complementary perspective or a shift of paradigm The symbol alone itself is a mathematical object has a mathematical meaning d d is worth studying in its own at right 77 Today I give you a direction as to how to see or view properly from now on until the end of the semester Math 116 way This may be contrary to the 1 way you used to see it in the former calculus class It is important that we are on the same page before we start doing anything concrete that has to do with d at Here is the direction 0 Direction From now on until the end of the semester I want you to view d d as a mechanism7 that has input and output like a magic box You throw in something into that box and you expect something else will come out of that box In the above you threw in 2 Then 2 x came out So the input was 2 The output was 2 x This is nothing hard to chew This is nothing dif cult to understand This is just a matter of perception I am offering you the right way to perceive derivatives which will make your life easier What I am asking you is nothing more than just to make a small psychological shift on your part Some of you might say that you already knew this at least at the subconscious level Then l7d say you are already on the right track If you did not used to see it this way then I suggest you to get used to seeing it this way and I guarantee you it will help you a great deal 0 You remember ddac 2 2 at ddac 3 3 2 ddac 4 4 3 ddac 5 5 4 dd 6 6 5 Formula A 7 x ax lt 04 is a constant Here it is important that 04 is a constant So know that something like the following cannot be handled using the formula above d dx So far we do not know how to calculate this last derivative We will deal with it later Now conforming to what I said above I want you to see Formula A as follows We threw 0 as an input into the magic box called Then 04 xa l came out of the box as an output 0 The beauty of Formula A above is it is versatile7 First of all it covers the case 04 l 2 3 4 5 6 namely the case 04 is a positive integer as I listed before The truth is in Formula A 04 could be any real number not necessarily an integer Also 04 could be either a positive real number 0 or a negative real number In short 04 could be just an arbitrary7 real number For example you may set 04 to be 1 To read Formula A for 04 l d 1 1 0 dac at at Here we know 1 x and 0 1 So the above reads just d 7 1 die in dx We already knew this In fact you could write x as d So we realize at that Formula A for 04 l is exactly dx 7 dx 7 This last identity is true It is of form Nothing surprising or dramatic But this is just for 04 1 Formula A is true for an arbitrary constant a Formula A is more versatile than it looks 1 Another example of Formula A In Formula A you may set 04 to be To read 3 1 Formula A for 04 7 L 1 2 dx 7 le 1 L Of course as we all know x7 is the same as xx and x 2 is the same as 2y 1 Hence 7 x 2 is the same as Thus we may rewrite the same as 1 A spec1al case of Formula A a d 1 d x 7 2 This last highlighted formula should look familiar to you Once again I want you to see this last highlighted formula is as follows We threw V as an input into the 1 Then 7 came out of the box as an out ut 2 d x p magic box called In your former calculus class Math 115 you are taught a certain logic to math ematically deduce Formula A But today I want to put aside what logic it was that we used to pull a formula like this I am not saying the deduction is unimportant Today I just do not want to go to that direction 0 I want to stress one aspect of Formula A althoughl already mentioned it Formula A applies when 04 is a constant namely 04 does not involve an The same formula does not apply when 04 is not a constant namely when 04 involves at So we do not know yet how to go about die and such 0 Now suddenly out of the blue I give you the following formula This is actually today7s main theme 4 Formula B 0 Before I even explain what it is I ask you to memorize this Formula B No matter what it takes you remember Formula B This is not too hard to memorize If you don7t memorize Formula B then you are stuck and not going anywhere 0 First of all Formula B is about 6 Formula A was about 0 Agree that 6 and 0 are fundamentally different So know that Formula A and Formula B have nothing to do with each other Let7s forget about Formula A for now We have already moved on to Formula B Now lets see what Formula B says Let me test your memory What way is the best way to look at Formula B Yes the magic box We threw 6 as an input into the magic box called Then 6 came out of the box as an output at This is interesting The input is 6 and the output too is 6 The input and the output are identical Have we seen anything like this before True we have seen where 0 is the constant 0 This is true But other than that we have never seen anything like this before 7 the input and the output are identical So Formula B is very peculiar Formula B is very special See also Appendix I Once again I ask you to memorize Formula B You remember Formula B under any circumstance If you don7t then though I may sound harsh I do not see that there is any way you will understand my lecture for the rest of the semester By the way when I say like if you don7t understand this or that then you will be completely lost for the rest of the semester I am not joking and it is usually an accurate prediction of what will happen to you next This is because 7 good or bad 7 mathematics is a strictly hierarchical discipline What it means is that in mathematics everything has to be built from the complete scratch without relying on anything else but the notion of integers recall that integer7 is the of cial name for what you know as whole number Those parts constitute the lowest level or the ground level Then there are things that sit right above that level and then more things sit further above that level and so on so forth In particular most importantly mathematics does not rely on anything outside of mathematics Equally most importantly there is always the next level and there is no end to it See my Pitch7 Mathematicians are fond of the word self containedness They proudly use this word in describing this unique characteristic of mathematics as a science discipline to non mathematicians Repeat Mathematics is a hierarchical and self contained discipline by nature The consequence is that any basic subdiscipline of mathematics such as calculus always starts with the lowest level Then the level of dif culty rapidly increases as the semester progresses So often times the materials covered during the first week of the semester may appear to be deceptively easy and elementary Beware that if you feel that the content of today7s lecture is trivial it does not mean that you will feel the same way in tomorrow7s lecture However I can say that no part of my lecture is entirely trivial Some things are more easily understandable than others Making the subject matter understandable is one of my priorities Your job is to trust my expertise and to follow my direction Now I said Formula A is versatile How versatile is Formula B Formula B is just the derivative of one single object 6 How do we take care of something like d dx d 2 d m 290 m 6 7 6 7 6 7 die and such As for these we actually have the following variation generalization of Formula B Formula C Chain rule 7 I d emu emu fgc Here f is just any quantity written in terms of x In other words f is a function on x f means the derivative of at namely fac is another shorter f So I could have written Formula C as in 6 d way of writing d at Formula C Chain rule 7 I d fw fw x 6 6 dz f o ls this too complicated I hope not Formula C is extremely important Indeed it is extremely versatile So I ask you to memorize this Formula C ls it too hard Sure Formula C looks more complicated than Formula B But from my point of view Formula C is still not too terrible Let7s analyze Formula C It involves f That is the reason why it looks com plicated As for this mentally let7s think of f as Q9 Or you can choose any symbol you like Then you can write Formula C as Formula CC This one doesn7t look too bad does it This version you can easily memorize by heart7 Here let us keep in mind that Q represents any quantity that involves at 0 Formula CC is known as Chain Rule By the way Formula C or the same Formula CC is so important that it has a name Chain Rule7 that is To be a little bit more precise Formula C is a special case of the Chain Rule I will eventually tell you what the most general form of the Chain Rule looks like But for now just Formula CC is suf cient for us So at least for the time being whenever I say Chain Rule7 I refer to Formula C9 To repeat Whenever I say Chain Rule7 I mean Formula CC Okay let7s play some game using this last version of Chain Rule that is Formula C 7 Example In Formula CC Chain Rule let7s set Q to be 215 Then Formula CC reads d d 762m 62 Indeed I have substituted7 Q there are three spots all with 215 Since ddx 216 2 d we get the answer for d 62 Namely at d 2 7 e m 62 2 die You can write the same answer as in 2 62 which looks neater and is preferable In short 7 62 2 62 Lets do another example Example In Formula CC Chain Rule let7s set Q to be 715 Then Formula CC reads d e mie m d inc dx 7 dx Indeed I have substituted7 Q there are three spots all with 7 x Since ddx 716 71 6 m Namely d dx e m 6 9 71 You can write the same answer as in 7 6 m which looks neater and is preferable In short we get the answer for Lets do one more Example In Formula CC Chain Rule let7s set Q to be 2 Then Formula CC reads Indeed I have substituted7 Q there are three spots all with 2 Since ddx 2 216 d we get the answer for d 6 2 Namely at d 2 2 i w w 2 dac e e lt at You can write the same answer as in 2 x 6 which looks neater and is preferable In short So basically Formula C or its equivalent version Formula CC allows us to calculate the derivative of e raised to the power of some quantity written in terms of x 77 o Breakup Rules 7 Some old material Okay enough about Chain Rule for now Let us talk about something else You have actually seen this in the former calculus course Math 115 so this is an old material I want you to review this just in case so as to make sure we are on the same page Also with our new perspective namely to view as a magic box7 I want you to see how this old material looks Here is the old material It is called the Break up Rule7 First I state it verbally This way I can make the Rule more impressionable The Break up Rule actually has three parts i ii and iii Breakup Rules The magic box7 has the following capacity i dd breaks up the addition formation f g at at d ii d breaks up the subtraction formation f 7 915 at d iii die breaks up the constant multiple formation 0 o A quick remark before I proceed in case you are looking ahead What about the multiplication formation f g 7 The answer No never breaks up the multiplication formation f g As for the multiplication formation a completely different rule Leibniz Rule governs We Will cover it after the Break up Rules 0 To return to the original discussion 7 the Break up Rule The above descriptions i ii and iii are verbal and they best capture the spirit of the rules but I should also make those precise in terms of equations and symbols 0 Break up Rule more precisely means as follows d d ltfx Mm is calculated as or the same o Break up Rule ii more precisely means as follows d W M 7m is calculated as or the same 0 Finally Break up Rule iii more precisely means as follows ltcmgtgt Where 0 is a constant is calculated as or the same 0 Now it is worth repeating What I said in the above using the symbols Q and O Breakup Rulesvoz The following i ii and iii hold d d d i Wlt Ogt lt dac C9 lt dac 0 d d d ii Wlt iltgtgt lt dac Q9 7 lt dac 0 iii lt0 lt9 c d lt9 Here in iii 0 is a constant that is a concrete number For example the last one d 1 1 d day 79 7735 9 Now lets mix our new perspective7 into this old material Below is another equally legitimate and indeed a slightly more sophisticated or slick7 way to explain the substance of the Break up Rule for c 7 reads Break up Rule says that you first add Q and O and then throw the result in to the magic box Or you first throw Q and 0 each into the magic dac d box d and then add up the two outputs The two results are the same at Break up Rule ii says that you first subtract 0 from Q9 and then throw the Or you first throw Q and 0 each into the result in to the magic box dac d magic box and then subtract the second output from the first output The two results are the same 1 Break up Rule iii says that you first multiply a constant say to Q and Or you first throw Q into the l 2 then throw the result in to the magic box magic box and then multiply the same constant that is gt to the output The two results are the same 0 The term linear linearity For those who plan to take Math 290 Mathematicians isolate the very set of properties i ii and iii as a common set of properties as they know it is ubiquitous and it appears in so many differ ent mathematical contexts Mathematicians call the set of properties i ii and u iii as linearity7 So in mathematicians7 language possesses linearity die a Alternatively is a linear operator 12 o A combination of the break up formulas is useful For example we know we can simplify J v 30 as in 1 d d 7WQ 3WO Or we know we can simplify d dac V50 7 ltgt 61 Maw are 6w So far the above is something which we already knew but we may apply this to a new situation i 36 According to the Break up Rule we may take out the constant multiplier 3 sitting Example Let us find inside of to outside of d so you can write the above as at 3 d 9 dice 6 7 Yes by Formula B it is 6 Hence we have Now how much is obtained the answer Example Let us find d lt 1 6m 1 67m gt die 2 2 We use What we already know L L dac Taking these into accout we just apply the Break Up Rule as in oi em if oi 6m 2ltdeimgt 1 6m L we 2 2 1 1 7 0 Note that 7 6 7 e 9 lS alternatively written as 6 1 6 9 2 1 1 7 Also 7 6 7 7 e 9 lS alternatively written as ex 7 67m 2 Thus What we have obtained above is alternatively written as d 6 1 6 9 7 6 7 6 9 dac 2 2 Example Similarly let us find d lt 1 6m 7 1 67m gt die 2 2 This is entirely parallel to the previous one dlt16m7167mgt71ltdemgt71ltdeimgt dac 2 2 2 dac 2 dac l m l lt 7m 76 7 7 6 2 2 7 2 2 o What we have obtained above is alternatively written as d 6 6 9 6 e die 2 2 o Leibniz rule The above Break up Rule does not provide us with information as to how to handle d 2 6 dac for example For this matter the following Leibniz Rule7 which some call Product Rule is useful Leibniz Rule is calculated as or the same 0 To use Q and O Leibniz Rulevoz 6200 di lt9ltgtlt9ltgt 0 Please memorize this rule along With Formulas A B and C So this is the fourth item you memorize in the present lecture Example Let us find it 3 6m For this just apply the Leibniz Rule With Q z and O 6 In the above highlighted box you overwrite7 Q With at and O With 6 5 M dink wlt 5 690 This result is simplified as 1 an em lnshort 69 1 at 6 o Today7s lecture is packed With information Your brain might be a little saturated already You have done a great job to follow me So I Will do one more and call it the day This is actually something that is always paired With the Leibniz Rule like a tWin brothersister of the Leibniz Rule It is called the Quotient Rule7 o Quotient rule We cover one last formula the so called Quotient Rule For example we want to calculate For this matter we resort to the following Quotient Rule is calculated as or the same 0 To use Q and O Quotient Rulevoz 0 Please memorize this rule as the fifth and the last item to memorize in the present lecture Can you do it 17 Example Let us find For this just apply the Quotient Rule With Q x and O 6 7 l In the above highlighted box you overwrite7 Q With at and O With 6 7 l d an die 69071 This result is simpli ed as In short 0 Now proceed With Quiz 7 l problem Ill and also regular homework Section 54 Exercise Please do not skip this l 1 Use the Leibniz Rule to deduce 2 22gt Maw d 03 3 2 d QQ d 4 i 3 d dac 0 gt i 4 Q lt die 0 2 Use the Quotient Rule to deduce d 1 1 d dV77 2dac 39 3 Verify that What you have deduced in 1 and 2 is consistent With the fol lowing general formula called the Chain Rule d Q71 availlt d Q9 04 is a constant 1 Please memorize this Concretely write out this last rule for 04 0 Appendix 1 1 want to explain one thing Which is on the theoretical side Let us return to the example of it 36 36 What 1 said earlier about how we should read a line like this is as follows We threW Then 3 6 came out of the box as 36 as an input into the magic box d at an output Remember 1 called it an interesting situation Whenever I see that the input and the output are identical In our case the input is 36 and the output too is 36 So the input and the output are identical Remember Formula B Suppose the input d 6 is thrown into the box die then the box would give 6 right back What d we have here is that the same happens if 36 was thrown into the box 36 namely the box will give 36 right back I am basically talking about the rare7 situation where is true This is indeed rare it is not true most of the time However as we have eye witnessed both when Q 6 and when Q 36 this is true You can suspect if the same is true when Q 6 6 where 6 is any constant This is actually true To highlight d 7 lt0 6 6 6 6 a constant dac Now do we know of any other situation where d W is true You remember that l have formerly pointed out that d 0 i 0 die 7 is true In other words d 7Q Q die is true when Q 0 However this is not a separate case from the case Q 66 Rather this is viewed as a special subcase7 of the case Q 66 Namely the subcase7 6 0 The bottom line So far the only example of Q that satisfies d 79 Q die that we know of is Q 6 6 where 6 is an arbitrary constant If you are Inathernatically oriented or if you identify yourself as a future Inathernatician then here is the question you would ask yourself Does there exist any other Q that satisfies 20 d 7 7 dye Q9 Q9 The answer is known and the answer is no Later we will verify this fact that Q c 6 with c a constant is the only series of examples which satisfies d Q Q dac under the assumption that Q admits a so called Taylor expansion 0 Appendix 2 The two examples we did earlier are worth highlighting dltl m l 7mgti l m 179c dx 26 26 7 26 26 dltl m limgti 1m170c dx 26 26 7 26 26 We can actually write these results in a little more impressionable way Namely once you set H H l a 1m 173 i m 9 767 ltgt 26 26 we can rewrite the above two lines simply as From mathematicians7 point of view this last pair of equations is intriguing Meanwhile completely aside from the context of derivatives let us calculate Q92 and 02 each The answer 1 1 Q92 7 7 72m 4 2 4 6 1 1 1 2 i 7 2m 7 7 7290 O 4 e 2 4 e Here we have used 6 6 62 em 6 9 1 and so on In particular we have the following identity lt92 7 ltgt2 1 To summarize we obtain the following 0 Set 1 m 1 7m i 1 m 7 7 790 Q 76 76 ltgt 7 2 e 2 e These satisfy d a dye lt9 e ltgt d b KO Q9 and c Q92 7 02 1 Mathematicians find this last set of equations a b and c intriguing And this is not the end of the story 0 Characterizing hyperbolic version of trigonometric functions The of cial names for Q and 0 above are cosh x and sinh at respectively Thus 1 1 1 1 coshac 76 76 sinhac 6 7 7 7 6 9 2 o What mathematicians find even more intriguing is the following fact Let us completely forget what Q and 0 were Now let us suppose Q and O are unknown quantities but satisfy a b and c above Then Q and O are identi ed as Q a coshac b sinhac and O b coshac a sinhx respectively using some constants a and b satisfying a2 7 b2 1 hyperbola equation Later we will verify this fact under the assumption that Q and 0 both admit so called Taylor expansions 1n the above 1 said cosh x and sinh x are the of cial names7 By that lmean that 1 go to just any place anywhere in the globe where there is a circle of mathematicians and 1 use cosh x and sinh at then they will understand it and there is no room for misunderstanding o The original trigonometric functions Completely apart from the above you have probably seen cos at and sin at before These do not come with h7 Later in the semester we will cover these ex tensively 7 cos at and sin at These are called trigonometric functions 1 know you like to abbreviate trigonometric functions7 as trigs7 Mathematicians never do that We mathematicians never use trigs7 in our of cial mathematical documents and papers but some of us do only when communicating with non mathematicians Here let me just borrow your usage and use trigs7 To tell you the truth mathemati cians first came to recognize cos x and sin x Then they came to recognize cosh at and sinh x as analogs of cos x and sin x So the trigs cos x and sin x are there first which we would call the original trigs7 and then cosh at and sinh x are the hyperbolic version7 of trigs7 So you may call cosh at and sinh x as the hyperbolic trigs7 23 I am already looking ahead I may quit here but since you showed some intellectual curiosity let me go on In the above I showed you a set of equations a b and c that essentially characterizes the hyperbolic trigs7 The truth is there is a set of equations that essentially characterizes the original trigs7 ay by and cy below The intriguing fact is that such a set of equations shows some keen resernblance with the aforementioned set of equations a b and c that characterized the hyperbolic trigs7 Let me proceed Consider the following new set of equations d 7 7 3 W4 Q d 7 as W i 4 and cy 42 a 1 Compare these three equations with the previous a b and Once again Inathernaticians find the above set of equations ay by and cy as a whole very intriguing They find it intriguing because there is the following fact First of all of cos x and Q sin x satisfy the set of equations ay by and cy Second of all let us cornpletely forget about these Now conversely let us suppose Q and Q are unknown quantities but satisfy ay by and cy above Then of and Q are identified as Qacosxbsinz and Q ibcosxasinz respectively using some constants a and b satisfying a2 b2 l circle equation I have already stretched the subject matter quite a bit If you are Inathernatically oriented you would ask ls there any way you can write each of Q cos z and Q sin at using 6 24 That is an excellent question If you came up with such a question I would reward you just for having come up with such a question The answer is yes and no7 The answer is no if you want me to stay within our usual number system the real number system7 The answer to the same question will become yes if you allow me to use some sophisticated number system called the complex number system7 I am not going to tell you details but first of all the complex number system7 includes the real number system7 as a subsystem In other words the real number system7 constitutes a part and only a part of the entire complex number system7 The complex number system7 is a very versatile system Remember the real number system is identified with a straight line called the xv coordinate axis So it is one dimensional On the contrary the complex number system is two dimensional as it has two coordinate axes that transversally cross at the origin one the xv coordinate axis the real coordinate which is horizontal and the y coordinate axis the imaginary coordinate which is vertical If you use this system then I can write each of cos x and sin x as follows 1 l l l cosac 76 6 sinac 76 7 76 7 2i 2i Here the symbol i signifies a concrete complex number which lies at the vertical coordiate located exactly at the spot which is a unit length north7 of the coordinate origin It satisfies i2 71 Remember no real number squared equals 71 Since you are unfamiliar with the complex number system there is no point in trying to understand the above pair of expressions So don7t bother to try to understand the above pair of expressions lnstead please compare how these resemble l l l l coshac 76 76 sinhac 6 7 7 7 7m 6 Mathematicians value analogy and parallelism The analogy between the trigs7 and the hyperbolic trigs7 is one representative analogy that appears in calculus We will of cially introduce the trigs7 in the third unit of the semester Look forward to it 0 Appendix 3 We have calculated the derivative d an die 6 7 l 25 Later we consider the second the third the fourth derivatives of the same function Emil This particular series of derivatives has theoretical importance as we Will clarify later The key word is Bernoulli numbers

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