Calculus and Analytic Geometry
Calculus and Analytic Geometry MATH 222
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This 6 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 Staff in Fall. Since its upload, it has received 27 views. For similar materials see /class/205291/math-222-university-of-wisconsin-madison in Mathematics (M) at University of Wisconsin - Madison.
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Date Created: 09/17/15
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1 10 V A Y 4L 3 l l ugtc u l J i cw Va C quotCA jri m G l X A Q TT QNN 9L 4L 9 Second Midterm Exam July 24 200 This exam will include material we have covered since the rst exam from the latter part of chapterl 10 se tions 105 108 and all of chapter 11 As before you might want to look at old exams for ideas as to what kinds of problems and how dif cult I think it is practical to put on an exam ll have old exams at my website wwwmathwiscedu wilson and there are also old exams froml other professors at the math library site As before do realize that in the summer particularly thd material covered on the i gtcond exam might not be the same as in some other semes er but lookingj at the topics belov and the textbook i ons included you could tell whether some problem wasl or x 39as not relevantl You are allowed to bring in two sheets ordinary notebook sheets you can use both sides of notes with whatever you think will be useful written on it There will M be a sheet of formulasl included as there was on the rst exam You are allowed to use a calculator including scientific and graphing calculators but you w show your work leading to an answer An answer that just a ea without ex lanation will not 39 d39 n be useful in wor inU out numeric values but should not be the source of your answers And for numbers something like at is much preferred to 314 It is exact and unambiguous 0 Chapter 10 the sections on polar coordinates Be able to convert coordinates for a point between Cartesian and polar 7 Graph functions given in polar form You should know a few of the named curves Clearly lines and circles and it might be convenient to recognize cardioids and roses 7 Find the slope of the graph and the tangent line at a point 011 a polar graph Determine intersection points for curves given in polar form 7 Set up and e Valuate integrals to compute area for regions described in polar coordinates This might be the area inside of one curve or the area between two curves There willl not be problems on arc length or surface area i For conic sections in olar coordinates NothinU more than recognizin the functior 396 k as describing a conic with eccentrity e and identifying whether that gives anl 1cos9 e 1pse para o 0 Chapter 11 Clemlencequot Series of Numbers and Power Serie 7 Know the difference between a sequence and a series 7 Know what the limit of a sequence lini a7 means Be able to do arithmetic as in quotADC Theorem 1 with sequences Be able to use Theorems 3 and 4 to nd limits of sequences Know some useful sequence limits given in Theorem 5 I won39t ask anything requiringi Theorem 6 m 7 Know how the sum of an in nite series 2 a is de ned using the sequence of partiall 711 sumsl Re oUnize and work vs l1 eometi 39 39 39 d ii 1din the sum if the series does converge th Be able to use the 77 term test including knowing what it does not say Be able to go back and forth between rational fractions and repeating decimals e g 1 7l and 0142857142857 Be able to use at least the integral t t the comparison test and the ratio test in deter l mining whether a series of non negative terms converg Any positive s appearing on the exam will succumb to one of these tests If you also know the other tests inl 1l3 115 you welcome to use them so long as you do so correctly and clearly statd what you are doing Be able to use Leibniz Theorem the Alternating Series Test to decide Whether anl alternating series converges and ii if so estimate and give the sign of the error in using just the sum of some nite munber of terms instead of the whole series Tell Whether a series converges ab Olutelv ronditimmllv or not at all Be able to usel absolute convergence to determine convergence Theorem 16 Know what a power series is Be able to nd power series for functions or what functionl a given power series represents using geometric series Know the pattern of convergence that any power series must have Corollary bottoml of page 798 and be able to nd the radius and interval of convergence and to test fort convergence at the ends of the interval Be able to use term by term differentiation and integration together with geometric oil other known series to nd a series representation You will not need Theorem 21 Know the de nitions of Taylor and llaclaurin Series and how to nd them Be able to nd the Taylor Polynomial of order n representing a function at some center point a Be able to use Taylor s Theorem and the remainder term IL a to tell when thd series converges to the function and b to estimate error in approximating a function byi a Taylor Polynomial I recommend rising R as in Example 1 rather than the formality of Theorem 23 but do Whatever works best for you If you do use Theorem 23 be surel and what A means and how to nd it not nec 39ily evaluating at the endl Be able to use the remainder term to work backwardsquot Rather than nding the erron in using some given polynomial approximation determine what polynomial ie whatl order is needed to get an approximation with prescribed error bounds 1l10 won t appear on the exam But reading it carefully will give you practice in thd error estimation mentioned in the previous two items 1lll won t be on the exam It is very useful material for many applications eg inl engineering and economics but if u are going that far you will probably take somethingi