MATH : Calculus II (42809) SYLLABUS, Spring 2010 MW 4-5:50PM, JB- 138

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1 MATH -: Calculus II (489) SYLLABUS, Spring MW 4-5:5PM, JB- 38 John Sarli, JB-36 O ce Hours: MTW 3-4PM, and by appointment (99) jsarli@csusb.edu Text: Calculus of a Single Variable, Larson/Hostetler/Edwards (ninth edition) Prerequisites: MATH, with a grade of C or better This course begins with the Fundamental Theorem of Calculus and continues with its applications and relation to di erential calculus. We will cover: Chapter 4 (the de nite Riemann integral), Chapter 5 (transcendental functions) and Chapter 8 (techniques of integration). An assignment syllabus will be provided for each chapter; suggested exercises for acquiring the skills expected on the exams will be provided. Certain exercises will be designated as Assignment exercises; these are to be written up and will be graded, accounting for 5% of your course grade. The remaining 85% of the grade will be from an in-class exam (%), a project assignment (5%) and the nal exam (4%). It may help to think of the course in terms of four main ideas: I. The de nite integral and the Fundamental Theorem of Calculus II. Transcendental functions III. Inverse Function Theorem IV. Techniques of integration Course grades will be determined, according to the above weighting, as follows: A 9 C A 86 9 C 6 65 B C 56 6 B 76 8 D B 7 75 F < 45 Some important dates: First day of class First Exam CENSUS (Last Day to Drop Without Record of Enrollment) CAMPUS CLOSED CAMPUS CLOSED Last day of class Final Exam

2 Notes: ) Mid-term exam dates are subject to change. ) Due dates for the graded exercises will be set as we approach the end of each chapter of the text. Assignment must be presented either in an 8 blue book or submitted to me by as a.pdf le. 3) The only attendance requirement is that you complete the MDTP CR test within the rst two weeks of class: Go to mdtp.ucsd.edu, click on On-Line Tests and select the Calculus Readiness Test. Submit the results to me either by or as a print out. The results do not factor into your course grade in any way. The purpose is to inform you of what review is needed and to inform me of what review skills need to be emphasized. If you cannot access the on-line tests I can arrange for you to take it in hardcopy form at a convenient time. 4) If you are in need of an accommodation for a disability in order to participate in this class, please contact Services to Students with Disabilities at UH-83, (99) ) Excluding very special circumstances, I will not approve petitions to drop the course after April 3. Please refer to the Academic Regulations and Policies section of your current bulletin for information regarding add/drop procedures and consequences of academic dishonesty. First Exam Study Guide: ) Antiderivatives/Inde nite Integrals ) Di erential Equations a) general solution b) speci c solution (given initial conditions) c) motion in a straight line under constant acceleration 3) Summation Notation a) approximating areas b) limits of sums and exact areas 4) Riemann Sums and De nite Integrals a) constructing a Riemann sum b) limits of Riemann sums for continuous functions c) using properties of de nite integrals (linearity, summability) to evaluate them, particularly from known integrals d) area interpretation for non-negative functions 5) Fundamental Theorem of Calculus a) R b f(x)dx = F (b) a F (a) b) MVT for integrals c) average value of f on [a; b] d) derivative of F (x) = Z u(x) x f(t)dt

3 The Suggested Exercises are your guide for the in-class exams. They are a representative sample of techniques required for basic mastery. To reinforce written communication skills the Graded Assignment should be clearly presented with complete explanation of solution steps. Chapter 4 Suggested Exercises (Larson/Hostetler/Edwards, 9th ed.): 4. (pages 55-58): 7, odd 5 49, odd 6; 73; 77; (pages 67-7): 3, odd 5; 3; 35; 37; 43 47; 49; (pages 78-8): ; 5; ; 8; 9 3; 39; 43; (pages 93-96): 5 57, odd 6; 67; 75; 8; 87; 89; 4.5 (pages 36-3): ; 3; 5 39, odd 5; 53; 59 64; 7; 93 First Graded Assignment: Due April 6 Pages 38-3: 8; ; 46; 64; 74 Complete solutions are required for full credit. 3

4 MATH - Spring/: First Exam Study Guide ) Antiderivatives/Inde nite Integrals ) Di erential Equations a) general solution b) speci c solution (given initial conditions) c) motion in a straight line under constant acceleration 3) Summation Notation a) approximating areas b) limits of sums and exact areas 4) Riemann Sums and De nite Integrals a) constructing a Riemann sum b) limits of Riemann sums for continuous functions c) using properties of de nite integrals (linearity, summability) to evaluate them, particularly from known integrals d) area interpretation for non-negative functions 5) Fundamental Theorem of Calculus a) R b f(x)dx = F (b) a F (a) b) MVT for integrals c) average value of f on [a; b] d) derivative of F (x) = Z u(x) x f(t)dt 4

5 Evaluate the de nite integral Z 4 x 4 dx by the limit de nition as follows: a) Find x for the regular partition with n subintervals. x = 4 n = 4 n b) Find an expression for c i, the right-hand endpoint of the i th subinterval. c) Set up the Riemann sum + ix = 4i n nx f(c i )x i. i= nx f(c i )x i = i= nx i= 4 4i 4 n n d) Write the sum in terms of n only using the formula = e) Find the limit as n! : nx 4 4i 4 n n i= 5 4 X n i 4 n i= nx i= = 4 3n 5 n (n + ) (n + ) 3n + 3n = 4 (n + ) (n + ) 3n + 3n 3 n 4 lim n! = (n + ) (n + ) 3n + 3n 3 n 4 = 4:8 i 4 = 3 n (n + ) (n + ) 3n + 3n : 5

6 Chapter 5 (Larson/Hostetler/Edwards, 9th ed.): The Suggested Exercises are your guide for the in-class exams. They are a representative sample of techniques required for basic mastery. 5. (pages ): 7 8 9; 3; 33; , odd 83; 87; 89; (pages 34-34): 39, odd 53; 59; 69; 75; (pages ): 9 3 9, odd 4; 45; 5; 6; 7; (pages ): 7; ; 5 8; 3; , odd 83; 87; 97; 5; 7; 9; 5; (pages ): 39; 4; 47; 49; 59; 7; (pages ): 5; 7; 9; 7; 33, odd 4 5, odd 6; 63; 75; 8; (pages ): ; 3; 5; 7; 5; 9; 3 39; 5; 6; (pages 398-4): 7 39, odd 43; 49; 6; 63; 85; 89; 7 Second Graded Assignment. Complete solutions required for full credit. Due: May 9. (Numbers in parentheses are from 8 th edition.) Page 359: 87 (Page 357: 73) Page 36: 5 (Page 357: 75) Page 44:; ; 3 (Page 4: 8; 9; ) Project Assignment. Do one of the following. Solution must be in print form. Due: June. Page 4: 4 (Page 398: 94) Page 43: 6 (Page 4: 6) 6

7 A function de ned by a de nite integral: F (x) = Z x sin t t dt lim x! F (x) = F (x) = sin x x F (x) = x cos x x sin x y x.5 F (x) = Z x sin t t dt and its derivative f(x) = sin x x : 7

8 Schedule of remaining topics: May : Di erentiation of Inverse Trigonometric Functions May : Integration of Inverse Trigonometric Functions May 7: Applications of Hyperbolic Functions May 9: Integration By Parts; Chapter 5 Assignment Due May 4: Trigonometric Integrands and Related Substitutions May 6: Partial Fractions June : Partial Fractions (continued); Project Due June 7: L Hôpital s Rule June 9: Improper Riemann Integrals June 4: Review; Chapter 8 Assignment Due June 6: Final Exam 8

9 Vertical motion with air resistance: If v = ; s(t) = s Assume 6t without taking air resistance into account. Then Thus, q p 3k tanh k since v = : It follows that Z 3 3 v v (t) = 3 + kv ; k > Z 3 kv dv = q Z p 3k dv = t k 3 v u Z du = t; u = dt r k 3 v q = t + C, where C = p 3k tanh v(t) = lim v(t) = t! r 3 p k tanh 3kt r 3 k k 3 v = For example, the terminal velocity for a skydiver is about ft= s. Find k. Further For example, if k = : s = s(t) = = r Z 3 p tanh 3kt dt k p k ln cosh 3kt + s then s(t) = when t 59: 777 Note: With k = ; s 6t =, when t = 5: 9

10 Gudermannian function: G(x) = Z x sech tdt y x 5.5 y = G(x) G (x) = sech x G (x) = sech x tanh x g(x) = arctan (sinh x) g (x) = = sech x and g() = G() so g = G cosh x sinh x+ h(x) = arcsin(tanh p x) h (x) = tanh x = sech x and h() = G() so h = G

11 MATH -: Spring The Suggested Exercises are your guide for the in-class exams. They are a representative sample of techniques required for basic mastery. To reinforce written communication skills the Graded Assignment should be clearly presented with complete explanation of solution steps. Chapter 8 Suggested Exercises (Larson/Hostetler/Edwards, 9th ed.): 8. (pages 54-56): ; 3; 3; 35; 4; 47; 49; 59; (pages ): 7; ; 5; 9; 7; 33; 35; 43 73; 83; (pages ): 5; 7; 9; 7 5; 43; 5; 7; (pages ): 5; 9; 5; 7; 9; 7; 35; 47; (pages 56-56): 9; ; 3; 5; 3; (pages ): (pages ): 9; 3; 5; ; 35 93; (pages ): 9; ; 9; 7; 33; 77; 9; 7 Third Graded Assignment. Complete solutions required for full credit. Due: June 4. (Numbers in parentheses are from 8 th edition.) Page 59: ; 4 (Page 589: ; 38) Page 59: 54; 7; 78 (Page 59: 5; 68; 76)

12 An application of partial fractions integration to probability: Let f(x) = p x 4 + y y = f(x); dashed curve is the standard normal density function y = p e x x which could be a probability density function for a certain random variable. This means that the probability of this variable falling with the interval [a; b] is given by Z b a f(x)dx Using the FTC we can evaluate these probabilities if we nd an antiderivative F for f. First, decompose x 4 + into partial fractions, which requires the factorization x 4 + = x + p p x + x x + and so which yields the identity x 4 + = Ax + B x + p x + + Cx + D p x x + x 3 (A + C)+x B + D A p + C p +x A + C B p + D p +(B + D)

13 and thus the system with the unique solution A + C = B + D A p + C p = A + C B p + D p = A = B + D = p 4 = C B = = D Z p Z Then p 4 F (x) = x+ p x + p dx + 4 x+ x+ x p dx = u(x) + v(x), and we x+ calculate u(x); v(x) after completing the squares in the denominators: x + p x + = x + p + x p x + = x p + For u(x), let w = x + p. Then dx = dw and u(x) = Z p w + 4 dw w + p For v(x), let w = x p. Then dx = dw and v(x) = Z p w 4 dw w + p It follows that u(x) = p arctan x p + + p ln x + x p v(x) = p arctan x p p ln x x p and so F (x) s x + x p + x x p + + arctan x p + + arctan x p A+C If we let C = then lim x! F (x) =, and F becomes a cumulative distribution function for the probability density function f: 3

14 y y = F (x) x As an example, suppose the random variable measures the distance east (negative) or west (positive) of a station on a railroad track from which a hazard report might come in. What is the probability that the report will originate within mile of the station? This probability is given by the de nite integral Z f(x)dx = F () F ( ) = ln p :78 55 so there is a 78% probability that the report will occur within mile of the station. Exercise: The standard deviation of this random variable is de ned by = Z x f(x)dx: Find and calculate the probability of a measurement falling within standard deviation on either side of x = : 4

15 MATH - Spring : Final Exam Review Guide Integration by substitution Z Z Inde nite integrals f(g(x))g (x)dx = f(u)du = F (u) + C De nite integrals Z b Natural log function: ln x = Domain, range, base e d dx ln u(x) = u(x) u (x) Logarithmic di erentiation a f(g(x))g (x)dx = Z x Z g(b) t dt; x > g(a) f(u)du d u Antiderivatives: dx ln juj = u udu = ln juj + C Integrands involving trigonometric functions so Z Inverse Function Theorem If f has an inverse function g and f (g(x)) 6= then g (x) = f (g(x)) : Exponential functions f (x) = e x where Z f(x) = ln x d dx eu = e u du dx ; e u du = e u + C a x = e x ln a ; a > log a x = ln a ln x Exponential growth and decay Inverse Trigonometric Functions Domains and ranges Derivatives Hyperbolic functions. Domains and ranges Derivatives and antiderivatives Inverse functions Integration by parts. Recognizing u and dv Multiple step applications Trigonometric integrals and substitutions. Integrands with powers of sin, cos, tan and sec as factors Integrands involving p a u ; p u a ; p u + a 5

16 Partial fractions. Decomposing rational functions into terms with powers of linear and quadratic polynomials in the denominator L Hôpital s Rule. Limits resulting in the indeterminate forms or Reduction of other indeterminate forms to these types Improper integrals. Determining convergence of de nite integrals with in nite limits of integration and/or in nite discontinuities in the interval 6