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# Calculus and Analytic Geometry MATH 222

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This 10 page Class Notes was uploaded by Zechariah Hilpert on Thursday September 17, 2015. The Class Notes belongs to MATH 222 at University of Wisconsin - Madison taught by Qian You in Fall. Since its upload, it has received 32 views. For similar materials see /class/205272/math-222-university-of-wisconsin-madison in Mathematics (M) at University of Wisconsin - Madison.

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Date Created: 09/17/15

Worksheet 15 Math 222 7 Lecture 1 7 Section 315 WES Monday7 Oct 137 2008 Some of these are from previous worksheets If everyone in your group has done them already7 feel free to skip them 1 2 Find an example of a series with non zero terms that satis es CO e lim an 0 and E an converges Hoe n1 n1 00 00 b Egan 73 f an 0 and 210 diverges 7L 7 00 g lim an 7 0 and 2a converges Hoe n1 00 h an 7 0 and 2 an diverges n1 Classify the following statements as true of false lf true7 explain lf false7 give a counter example a If the sequence an converges7 then the sequence 02 converges b If the sequence an diverges7 then the sequence 02 diverges C d If the sequence 02 diverges7 then the sequence 02 diverges If the sequence 02 converges7 then the sequence an converges 00 00 e If the series 2 an converges7 then the series 2 agn converges n1 n1 00 00 f If the series 2 an diverges7 then the series 2 agn diverges 1 n1 00 00 g If the series 2 agn converges7 then the series 2 an converges n1 n1 00 00 h If the series 2 agn diverges7 then the series 2 an diverges n1 n1 P 3 Sierpinski s Triangle The Sierpinski triangle also called the Sierpinski gasket or the Sierpinski Sieve is a fractal named after Waclaw Sierpinski You build it as follows 1 Start with a triangle of length I 2 Draw lines connecting the midpoints of the triangles edges and cut out the enclosed area 3 Repeat step 2 with each of the smaller triangles remaining This process should look something like this A t AA M m AAAA Annuu a The sum of the areas cut out in this process forms an in nite series What is it b Calculate this in nite sum nd the area of the total area removed c What is the area of Sierpinski7s triangle d ls every point actually removed Find some that remain how many are there Prof Nagel talked last Monday about the Fibonacci sequence 1 1 2 3 5 8 de ned recursively by M1 am l anrb d1 1 d2 1 He calculated that the sequence bn 01107L converges to L 1 a Does this limit depend on our choice of 01 and 02 For example what is linanoo bn if we instead e 01 d 7r b What happens if we de ne a new Fibonacciilike sequence de ne recursively by 00244 2anamib a1 1 02 1 Write down the rst few terms in this sequence What is liman 127 c What happens in either part if we chose 01 2 and 02 73 d Does the sequence de ned as follows converge or diverge 1 1 03972 Write out a few terms If it converges what does it converge to 04021 ZnH Worksheet 28 Math 222 7 Lecture 1 7 Section 315 WES Friday Nov 14 2008 1 Geometric properties a For each of the conic sections parabola ellipse hyperbola write down as much 0 C d information as you can about them Given a graph or other geometric informa tion how do you go about constructing the equation Or what about the other way around give equations geometric descriptions and properties what the graphs look like etc Find the equation of the set of points z y such that the sum of distances of z y from i13 0 is 30 Before you do it perhaps try seeing what these points look like graphically Find the equation of all points z y such that the distance between zy and the point 40 is equal to the distance between z y and the line z 74 Suppose you had a perfectly elliptical billiard table You place a ball on one of the foci and hit it with enough force to keep it bouncing off the edges of the table forever What can you say about the path of the ball 2 Shifting coordinates 7 a Do It Yourself kit a A T V A O V A CL V Write down an equation for a circle of radius 2 centered at the origin Shift it up 2 and left 5 Draw the graph of this shape Put your new equation in standard form Write down an equation for a parabola with focus 0 14 and directrix y 714 Shift it down 1 and left 1 Draw the graph of this shape Put your new equation in standard form Write down an equation for a parabola with focus 18 0 and directrix z 718 Rotate counterclockwise 7r6 Draw the graph of this shape Put your new equation in standard form Write down an equation for a ellipse with semimajor axis of length 4 and semimi nor axis of length 2 and centered at the origin Shift it up 1 and right 2 Rotate the axes counterclockwise 7r4 Draw the graph of this shape Put your new equation in standard form Write down an equation for a circle of radius 3 centered at the origin Shift it up 1 and right 2 Rotate the axes counterclockwise 7r4 Draw the graph of this shape Put your new equation in standard form 3 F Start from standard form Take the following equations and graph them by moving and rotating the axes a x24xy27120 b x25y24x710 e d e 96y f xy7y7x10 x24xy4y273x760 From last time A whole different kind of series Taylor series are great for approximating functions which behave like polynomials because they re built like polynomials A lot of people those who study electrical engineering acoustics optics signal processing image processing uid dynamics harmonic analysis to name a few actually care more about functions which are periodic and have a lot more to do with sine and cosine waves than they do with polynomials In section 84 we showed that functions of the form x asinmx or x acosmx are almost all pairwise orthogonal on the interval 0 g x g 27139 see problems 45 and 46 This tells us that if we could write x as 00 x a0 2 an cosnx bn sinbx n1 then the coef cients of this series would be of the form 1 2 1 1 27r x an 7 x cosnx bn 7 27r E 0 7r 0 7r 0 xs1nnx a0 Even if x can t be written exactly like this we can at least approximate it as such on a small interval familiar story no This is called the Fourier series 3 Sketch x 7 what is its period Calculate the coef cients a0 11012 1 b2 and write down the second degree Fourier approximation of f a De ne if Qk n39 x lt 2k 17r m if 2k717r xlt2k7r k07i17i27 39 b On the computer go to httpwwwfalstadcomfourier There s an applet which shows various approximations of periodic functions Play around with it What do the nodes at the bottom mean What happens if you move them What happens if you move the function What do these waves sound like Worksheet 3 Math 222 7 Lecture 1 7 Section 315 WES Monday Sept 8 2008 1 1 had a couple of friends in college who sat down together one day to try to calculate the integral fem sinzdm using integration by parts Brett who is no slow calculator is still there seven years later trying to nish simplifying the integral Colin being the tricky fellow that he is was instead able to use integration by parts twice and get the answer emsinm 7 cosm a Check Colin s answer b What happened to poor Brett c Brett did all his steps right so how was Colin able to get the right answer so quickly Hint use by parts twice then call fem sinzdm A and solve for A d Colin got an answer that worked but still failed Calculus Why to Calculate f sinm cosmdz ve ways a couple of them can be similar 1 7dz 1 let 1z gm 1 so fz 71z2 gz m ldmlm7m21dz1ldm z z z z This implies that 0 1 Oh no What went wrong OJ Consider the following integral Perform integration by parts Then obtain F If n and m are integers nd formulas for the following integrals a z sinnmdz b e cosnzdm Hint Try some speci c examples rst See if you need to break it into cases Does it matter if n is negative etc 01 You have a four ounce glass and a nineounce glass You have an endless supply of water You can ll or dump either glass It turns out that it s possible to measure six ounces of water using just these two glasses What s the fewest number of steps in which you can measure six ounces Worksheet 2 Math 222 7 Lecture 1 7 Section 315 WES Friday7 Sept 57 2008 With your group7 do as many as you can Feel free to skip around 1 Everything I know about integrals I learned from derivatives Some derivative rules we know are Some integration rules we know are 6M 6 12 D1 mndm C 71 yr 71 n gimme gltzgtgt 12 13 D2 fltgltzgtgtd fldz fltgltzgtgtdgltzgt lt12 35 my 7 12 dig D3 cfd 6 mm 13 gram ed f dgfw D4 m ew mm ow 14 W di D5 d flmdz fltzgtgltzgt e fzddx lt15 What name do each of D1 D5 go by What name do each of ID 5 go by Everything that we know about antiderivatives comes from something thats true about derivatives That means that every integration technique or rule is directly connected to some rule about deriva tives Match up each of ID 5 with something from D1 D5 Use the fundamental theorem of calculus to prove each of I27 I47 and 5 straight from its partner D7 Now7 for each of I27 I47 and 57 give an example of an integral you would use I to solve7 and solve it Then7 take the derivative of your solution What should you expect What derivative rule showed up when you differentiated the second function to 03 F U a For each of the following integrals use by parts to integrate Be explicit What s x f m gz g m Choosing carefully is what makes life easier Once you have a solution take it s derivative to check your answer a fmsinmdz b fzsemz dz c ftan 1mdz REMINDER 1 for example if x i but f 1m 7 lnstead f 1m means the inverse function of So em then f 1m lnz since 511 z and lnem z Sometimes you ll need to use tricks more than once Don t panic Just stay organized Forget what you ve done so far and think of each new line as a new problem Let s try one Calculate fmz sinzdm a First note x2 sinmdm fmgmdm where x 2 g m msinm Calculate f z and b Simplify fmz sinzdm by plugging stuff from part a into MicW fltzgtgltzgt e f zgzdz but DON T WORRY ABOUT THE RHS INTEGRAL YET c OK Forget for a second that you did part b and lets just worry about the integral you get by plugging your answer from part a into ff mgmdz Solve it d You should now be able to combine your answers from b and c to solve fzz sinmdz e Take a derivative of your answer in d to check your work For each of the following integrals use by parts to integrate Be explicit At each step what s x f m gz g m Once you have a solution take it s derivative to check your answer a fzg sinzdm b f1355m2 dz 1 had a couple of friends in college who sat down together one day to try to calculate the integral fem sinzdm using integration by parts Brett who is no slow calculator is still there seven years later trying to nish simplifying the integral Colin being the tricky fellow that he is was instead able to use integration by parts twice and get the answer emsinm 7 cosm Check Colin s answer What happened to poor Brett a b c Brett did all his steps right so how was Colin able to get the right answer so quickly d Colin got an answer that worked but still failed Calculus Why Calculate a ImSEmdm fm50000000436md C fnezdm H OJ q 01 Worksheet 4 Math 222 7 Lecture 1 7 Section 315 WES Wednesday Sept 10 2008 Set up but do not solve the partial fractions decomposition for each of the following integrals Notice that each of the polynomials in the following exercises are polynomials we already factored in the warmupl dm dm dm b az25m6 m3m272 C m42m37z a Evaluate the following integrals using partial fractions decomposition dm 2m 5 d Z x2 1 5m 6 m x2 1 5m 6 b What is another way you can do the second integral in part a Consider the integral dm 137 7 1 If you were to try partial fractions what would the set up for it be without solving for the constants How many constants do you have to solve for Well get another technique that makes this one simpler Suppose that Pm is a degree 2 polynomial such that P0 1 and P 0 0 a Write down a general form for Pm using the information given to you PM Zdz b ls there such a Pm such that W i 1 is a rational function If so what are the possible Pm Imagine that you have a long long corridor that stretches out as far as the eye can see In that corridor attached to the ceiling are lights that are operated with a pull cord and theyre all off Somebody comes along and pulls on each of the chains turning on each one of the lights Another person comes right behind and pulls the chain on every second light thereby turning off lights 2 4 6 8 and so on Now a third person comes along and pulls the cord on every third light That is lights number 3 6 9 12 15 etcetera Another person comes along and pulls the cord on lights number 4 8 12 16 and so on Of course each person is turning on some lights and turning other lights off If this goes on forever and ever can you predict which ones will be on at the end of time Worksheet 27 Math 222 7 Lecture 1 7 Section 315 WES Monday Nov 10 2008 OK so what7s the big deal with all this series stuff 1 Nonelementary integrals a Notice we don t know how to calculate the integral fsinm2dz We do7 however7 know how to calculate its Maclaurin series First calculate the series expansion for sinm2 plug 2 into the series for sinz7 then integrate b Use this to approximate fol sinz2dm to within 001 c Repeat this process for the following integrals start with common series you know each time i 1 sinm H 1 m2 1 idz 11 5 dz 0 95 0 a Using error to bound ratios 2 Calculating limits i As a function of m what is the error of approximating cosz as 1 7 z22 How about the error of approximating 1 7 cosz as z22 ii Now7 recall the de nition ofa limit limmna fz L if for every 6 gt 0 there is a 6 gt 0 for which if 0lt lz7al lt6 then lfz7Ll lt8 Use the previous part to show that 1 7 cosz 1 33 m2 7 2 hint As a function of m how far away can 17m be from b Directly calculating Use the series expansion of cosz to show that 17cosm71 2 4 x2 7 2 4 6 Finally7 use this to calculate limmace c Use both methods to calculate the following limits 1 7 2 m 7 7w 5 ii lim 5 5 170 3 F Solving differential equations Remember our rst encounter with the series expansion of em We said that if it could be expressed as a series then that meant that it had to look something like x a0 alx ang ang and it had to satisfy the initial value problem x O 1 We used this information to solve for the coefficients a0 a1 12 Now instead of starting with a function and then writing down a differential equation we ll play the same game starting with just the differential equation Assume that the following differential equations have solutions of the form yaoa1za212a3x3 and use this to calculate the solution If your solution looks familiar ie you know what it converges to plug in and check your answer a y i 2 7 2y 07 240 17 MW 1 b y 107 0 y0 17 MW 0 A whole different kind of series Taylor series are great for approximating functions which behave like polynomials because they re built like polynomials A lot of people those who study electrical engineering acoustics optics signal processing image processing uid dynamics harmonic analysis to name a few actually care more about functions which are periodic and have a lot more to do with sine and cosine waves than they do with polynomials In section 84 we showed that functions of the form x asinmx or x acosmx are almost all pairwise orthogonal on the interval 0 g x g 27139 see problems 45 and 46 This tells us that if we could write x as 00 x a0 2 an cosnx 1 bn sinbx n1 then the coefficients of this series would be of the form 1 2 1 1 27r an 7 x cosnx bn 7 27r E 0 7r 0 7r 0 xs1nnx a0 Even if x can t be written exactly like this we can at least approximate it as such on a small interval familiar story no This is called the Fourier series a De ne 1 if Qk n39 x lt 2k17r 71 if 2k717r xlt2k7r k07i17i27 39 M Sketch x 7 what is its period Calculate the coefficients 010 011012 1 b2 and write down the second degree Fourier approximation of b On the computer go to httpwwwfalstadcomfourier There s an applet which shows various approximations of periodic functions Play around with it What do the nodes at the bottom mean What happens if you move them What happens if you move the function What do these waves sound like

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