MATH 1080: Calculus of One Variable II Fall 2018 Textbook: Single Variable Calculus: Early Transcendentals, 7e, by James Stewart Unit 2 Skill Set Important: Students should expect test questions that require a synthesis of these skills. Additionally, some test questions may require a synthesis of material learned in the current unit and previous units. Note: Technology is not allowed on any exams. Limits of integration for definite integrals should be written as exact values. The values for definite integrals should also be written as exact values. Section 7.4: Integration of Rational Functions by Partial Fractions, pp. 484 493 WebAssign: 3, 12, 15, 19, 23, 29, 33, 35, 39 Write out the form of the partial fraction decomposition. 1 6 Expand an integrand by partial fractions and evaluate the integral. 9, 12, 19, 25, 26, 30, 31, 34, 35 Use long division and then partial fraction decomposition if necessary to evaluate an integral. 7, 8, 15, 16, 21 Use a substitution to express an integrand as a rational function and then evaluate the integral using partial fraction 39, 45, 47, 48 decomposition. Exploration. 56 Applications involving integration and partial fraction decomposition. 64 66, 72 Section 7.5: Strategy for Integration, pp. 494 500 WebAssign: 1, 5, 7, 9, 15, 19, 25, 27, 37, 41, 49, 63 Evaluate the integrals 1 82 (except #53) Exploration. 83, 84
Review Section 4.4: Indeterminate Forms and L Hospital s Rule, pp. 301 309 Determine the form of the limit. 1 6 Indeterminate Forms: 0 9, 12 16, 20, 22, 31, 32, 0 35, 38 Indeterminate Forms: 0 and 40, 46, 47, 49, 50, 54 Indeterminate Powers: 1, 0 0, and 0 56 58, 60, 62 Exploration 73, 74 Section 7.8: Improper Integrals, pp. 519 529 WebAssign: 1, 7, 9, 13, 15, 23, 27, 29, 35, 39, 49, 51, 55 Identify improper integrals of Type I ( ff(xx)dddd). Identify improper integrals of Type II ff(xx) dddd or ( ff(xx) dddd), where there is a discontinuity at some point,, or cc where < cc < ). Determine whether an integral is convergent or divergent. Evaluate convergent integrals. Use the Comparison Theorem to determine whether the integral is convergent or divergent. Use an improper integral to find an area or volume (if the area or volume is finite) and other applications. Find values of pp for which an integral converges. (Example 4, p. 522) 1, 2 7, 10, 15, 16, 18, 21, 23, 26, 34 36, 38, 39, 40 49, 50 3, 44, 45, 63, 71, 75,76 57 Chapter 7 Review Problems, pp. 530 532 Exercises 1 50, 72 76
Section 8.1: Arc Length, pp. 538 544 WebAssign: 7, 11, 13, 17, 31, 33 Use the Arc Length formula LL = 1 + [ff (xx)] 2 dddd to find the length of the curve yy = ff(xx), xx, given ff (xx) is continuous on [, ]. Or for xx = gg(yy), cc yy dd, given gg (yy) is continuous on [cc, dd], dd LL = 1 + [gg (yy)] 2 dddd cc Use the Arc Length Function, ss(xx) = 1 + [ff (tt)] 2 dddd. 34, 35 Additional Concepts 31, 32 Applications of Arc Length 29, 33 xx 1, 2, 10, 13 15, 17, 19 20
Section 8.2: Area of a Surface of Revolution, pp. 545 551 WebAssign: 7, 11, 13, 15, 25 Use the formula for the surface area of the solid obtained by rotating the curve about the xx-axis: yy = ff(xx), xx, given ff (xx) is continuous on [, ], SS = 2ππππ(xx) 1 + [ff (xx)] 2 dddd = 2ππππ 1 + dddd 2 If the curve is described as xx = gg(yy), cc yy dd, use a formula defined in terms of yy for the surface area of the solid obtained by rotating the curve about the xx-axis: dd SS = 2ππππ 1 + dddd 2 cc Use the formula for the surface area of the solid obtained by rotating the curve about the yy-axis: yy = ff(xx), xx, given ff (xx) is continuous on [, ], SS = 2ππππ 1 + dddd 2 If the curve is described as xx = gg(yy), cc yy dd, use a formula defined in terms of yy for the surface area of the solid obtained by rotating the curve about the yy-axis: dd SS = 2ππππ(yy) 1 + dddd 2 cc 8, 9 6, 11 16 15 Application of Surface Area 25, 28, 31, 33
Chapter 8 Review Problems, pp. 575 576 Exercises Problems from the Text Concepts: 1, 2 Exercises: 1-4, 7, 8 Section 11.1: Sequences, pp. 690 702 WebAssign: 1, 5, 11, 15, 21, 25, 27, 29, 31, 33, 35, 41, 43, 45, 47, 49, 73, 75 Concepts 1, 2, 71 List terms of a sequence. Find a formula for the general term of a sequence. 3 11 odd, 13 18 Determine whether a sequence is convergent or divergent. If a sequence is convergent, find the limit. Determine whether a sequence is increasing, decreasing, or not monotonic. Determine if a sequence is bounded. 23 56 72, 76, 78 Section 11.2: Series, pp. 703 714 WebAssign: 1, 3, 5, 15, 19, 23, 27, 31, 35, 43, 47, 53 Define an infinite series. Explain what it means for an infinite series (versus a sequence) to be convergent or divergent. Interpret infinite series notation. Explain the relationship of a series to its sequence of partial sums. Determine whether a geometric series is convergent or divergent. If it is convergent, then its sum is rr nn 1 nn=1 = 1 1 rr, rr < 1. If rr 1, the geometric series is divergent. Determine whether a series is convergent or divergent. If it converges, find its sum. Express the series as a telescoping series and determine if it converges or diverges by using the definition of a series (pp. 705). Express a number as a ratio of integers. 1, 2, 15, 16 3 8 17 25 odd, 49 27 42 43 47 odd 51 55 odd
Section 11.3: The Integral Test and Estimates of Sums, pp. 714 722 WebAssign: 5, 7, 11, 15, 23, 25, 27, 29, 37, 39 Problems from the Text Concepts 1, 2, 27, 28 Use the Integral Test. nn is convergent if and only if ff(xx) dddd is convergent, nn=1 1 given that ff is a continuous, positive, decreasing function on [1, ) and nn = ff(nn). 3 8 Determine if the series is convergent or divergent. 9 25 odd Find the values of for which the series is convergent. 29, 31 Find the sum of a series. 35, 36 Estimates of sums using the Remainder Estimate for the Integral Test. 38 40