MATH 1080: Calculus of One Variable II Fall 2017 Textbook: Single Variable Calculus: Early Transcendentals, 7e, by James Stewart.

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1 MATH 1080: Calculus of Oe Variable II Fall 2017 Textbook: Sigle Variable Calculus: Early Trascedetals, 7e, by James Stewart Uit 3 Skill Set Importat: Studets should expect test questios that require a sythesis of these skills. Additioally, some test questios may require a sythesis of material leared i the curret uit ad previous uits. Note: Techology is ot allowed o ay exams. Limits of itegratio for defiite itegrals should be writte as exact values. The values for defiite itegrals should also be writte as exact values. Sectio 11.5: Alteratig Series, pp WebAssig: 1, 2, 3, 5, 7, 9, 13, 17, 23, 29 Cocepts 1 Use the Alteratig Series Test. If the alteratig series ( 1) 1 b = b 1 b 2 + b 3 b 4 +, b > 0 satisfies (i) b +1 b for all ad (ii) lim b = 0, the the series is coverget. 2 9, 11, 12, Estimatig Sums. Use the Alteratig Series Estimatio Theorem. If s = ( 1) 1 b is the sum of a alteratig series that satisfies (i) 0 b +1 b ad (ii) lim b = 0, the R = s s b +1 Fid the values of p for which the series is coverget. 32, 33

2 Sectio 11.6: Absolute Covergece ad the Ratio ad Root Tests, pp WebAssig: 1, 5, 7, 8, 11, 13, 15, 17, 21, 29, 35 Cocepts 1, 35 Use the Ratio Test to determie if a series is absolutely coverget, coditioally coverget, or diverget. If lim a = L < 1, the the series a is absolutely coverget (ad therefore coverget). If lim a a If lim = L > 1, or lim is diverget. a a =, the the series = 1, the Ratio Test is icoclusive; that is, o coclusio ca be draw about the covergece or divergece of a. Use the Root Test to determie if a series is absolutely coverget, coditioally coverget, or diverget. If lim a = L < 1, the the series a is absolutely coverget (ad therefore coverget). If lim a a is diverget. = L > 1 or lim a =, the the series If lim a = 1, the Root Test is icoclusive; that is, o coclusio ca be draw about the covergece or divergece of a. 2, 6, 8, 18, 24 20, 21, 22, 23 Use ay test to determie if a series is absolutely coverget, coditioally coverget, or diverget. 3, 4, 7, 9 13, 15 19, 26, 29, 30 Sectio 11.7: Strategy for Testig Series, pp WebAssig: Active Examples 1 6; 1, 5, 11, 13, 15, 23, 25, 31, 33 Test the series for covergece or divergece. 1, 3 28, 30, 31, 34, 35, 36, 38

3 Sectio 11.8: Power Series, pp WebAssig: 3, 7, 9, 11, 15, 17, 19, 23 Cocepts 1, 2, 29, 30 Fid the radius of covergece ad iterval of covergece of the series. 3 7, 9 18, Sectio 11.9: Represetatio of Fuctios as Power Series, pp WebAssig: 5, 7, 9, 11, 13, 15, 23, 27, 31 Cocepts 1, 2 Fid a power series represetatio for a fuctio ad determie the iterval of covergece. 3 10, Use partial fractios to express a fuctio as a sum of a power series. Fid the iterval of covergece. Use differetiatio to fid a power series represetatio for a fuctio. 11, 12 Evaluate a idefiite itegral as a power series ad fid the radius of covergece Use a power series to approximate a defiite itegral. 29, 31 13

4 Sectio 11.10: Taylor ad Maclauri Series, pp WebAssig: 5, 7, 13, 15, 33, 49, 51, 57, 61, 63 We do ot cover the Biomial Series. Cocepts 1, 3 Use the defiitio of a Maclauri series: f(x) = f (0) x! =0 = f(0) + f (0) x + f (0) x 2 + f (0) x 3 1! 2! 3! + to fid the Maclauri series for a give fuctio. Fid the associated radius of covergece. Use the defiitio of a Taylor series for f(x) cetered at a give value of a: f(x) = f (a) (x a)! =0 = f(a) + f (a) (x a) + f (a) (x a) 2 1! 2! + f (a) (x a) 3 + 3! to fid the Taylor series for a fuctio. Fid the associated radius of covergece. Prove that the series obtaied represets the fuctio (uses Taylor s Iequality). Use a Maclauri series i Table 1 i the text to obtai a Maclauri series for a give fuctio. Exploratio: Fid the Maclauri series for the fuctio. Compare the graphs of the fuctio ad the first few Taylor polyomials , 31, 33, Use series for approximatio. 43, 51 Evaluate the idefiite itegral as a ifiite series Use series to evaluate a limit. 55, 57 Use a Maclauri series i Table 1 i the text to fid the sum of the series. 63, 65, 69, 70

5 Sectio 11.11: Applicatios of Taylor Polyomials, pp WebAssig: 5, 9, 13, 19, 23, 27 We do ot cover the Applicatios to Physics i this sectio. Cocepts 1 Fid the third order Taylor polyomial for a fuctio cetered at a. Graph the Taylor polyomial ad the fuctio. Approximate a fuctio by a Taylor polyomial, T ( x) for a give fuctio f at a specific umber a, ad a specified. Use Taylor s Iequality to estimate the accuracy of the approximatio. Use Taylor s Iequality or the Alteratig Series Estimatio Theorem (parts a & b oly) 23, 25, 30 Chapter 11 Review Problems, pp Cocept Check 1 12 Exercises 11 22, 23 26, 27 32, 35, 36, 38, 40 43, 44, 45 52, 55, 57, 58, 59 Appedix C: Graphs of Secod-Degree Equatios, pp. A16 A23 WebAssig: 9, 15, 21, 26, 27, 30, 31, 32 Circles 4, 8, 10 Idetify the type of curve ad sketch the graph. Shift if ecessary. Completig the square may be ecessary. 15, 17, 23, 24, 26, 28 30, 32 Additioal practice idetifyig the type of coic sectio. Page 676: 25 30

6 Sectio 10.1: Curves Defied by Parametric Equatios, pp WebAssig: 4, 7, 9, 11, 13, 15, 25, 32 Sketch a curve defied by parametric equatios. Use a arrow to idicate the directio i which the curve is traced as x icreases. Elimiate the parameter to fid a Cartesia equatio of the curve. Give parametric equatios, elimiate a parameter to fid a Cartesia equatio of a curve. Sketch the curve. Match the graphs of two parametric equatios with the correspodig parametric curve. (No equatios will be provided oly graphs) Sketch a parametric curve from the graphs of x = f(t) ad y = g(t). 5 7, 9, , 18 24, Fid parametric equatios 31, 33, 34 Cocepts ad Applicatios 37, 38, 41, 42, 45

7 Sectio 10.2: Calculus with Parametric Equatios, pp WebAssig: 5, 7, 13, 17, 25, 31, 41, 61, 65 For parametric equatios x = f(t) ad y = g(t), fid the slope of the taget to the parametric curve F(x), give by F (x) = g (t) dy dy or f (t) dx = dt if dx 1, 2 dx dt 0. dt Fid the equatio of the taget to a parametric curve at a 3 8, 30 give poit. Fid dy ad d2 y dx dx2 for a parametric curve. Fid the values of t for which the curve is cocave up or dow. Fid the poits o a curve (aalytically) where the taget is horizotal or vertical or where the value of the derivative is specified. 11, 13, 15 17, 19, 25, 29 Fid the area of the regio eclosed by the give curves Use the formula β L = ( dx 2 dt ) + ( dy 2 dt ) dt α to fid the exact arc legth of a parametric curve, traversed exactly oce as t icreases from α to β. Fid the exact surface area formed whe a parametric curve is rotated about the x-axis usig the formula β S = 2πy ( dx 2 dt ) + ( dy 2 dt ) dt α 41 44, where x = f(t), y = g(t), α t β, f, g are cotiuous ad g(t) 0. Fid the exact surface area formed whe a parametric curve is rotated about the y-axis usig the formula β S = 2πx ( dx 2 dt ) + ( dy 2 dt ) dt α 65, 66 where x = f(t), y = g(t), α t β, f, g are cotiuous ad g(t) 0.

Math 113 Exam 3 Practice

Math 113 Exam 3 Practice Math Exam Practice Exam 4 will cover.-., 0. ad 0.. Note that eve though. was tested i exam, questios from that sectios may also be o this exam. For practice problems o., refer to the last review. This