Final Exam InformationMath 221December 1999Your final grade will be computed as follows: (1) We will add the curveson the exams. to produce an exam curve where 800 points is perfect. (2) Wewill calculate a preliminary grade for each student by adding the scores onthe exams and using this curve. (3) Each TA will decide a quiz grade curvefor his/her sections. It takes into account the preliminary grades for his/herstudents. (4) The TA adds the quiz grade curve to the exam curve to producea final curve (where 1000 points is perfect) for his/her sections.The final will be worth 300 points. It is guaranteed that a score of 240 isworth a B on the final. It is guaranteed that 240 points worth of questions onthe final will be essentially chosen from the following:(1) The homework assigned this semester. (I have reproduced the list of prob-lems below in case you have lost your syllabus.)(2) The three exams given this semester.(3) The additional problems below.The qualifier essentially is added because I may change the wording of aproblem slightly to make it more appropriate for an exam. In problems whereI have given a hint, I might not give the hint on the exam.You will be given the formulassin(α + β) = sin α cos β + cos α sin β,cos(α + β) = cos α cos β − sin α sin β,if they are needed. You will not be given the formulas for the area of a circle(but see Problem 99 below), the circumference of a circle (but see Problem 46below), the volume of a sphere (but see Problem 282 below), the area of a sphere(but see Problem 47 below), the area of a sector (but see Problem 98 below),or the volume of a cone (but see Problem 286 below).For graphing problems you may be asked to determine (a) where f (x) isdefined, (b) where f (x) is continuous, (c) where f(x) is differentiable, (d) wheref(x) is increasing and where it is decreasing, (e) where f(x) is concave up andwhere it is concave down, (f) what the critical points of f(x) are, (g) where the1points of inflection are, (h) what (if any) the horizontal asymptotes to f(x) are,and (i) what (if any) the verical asymptotes to f(x) are.For proofs the question will be carefully worded to indicate what you mayassume in your proof. (See Problem 10 for example.) In this document youmay use without proof any previously asserted fact. For example, you may usethe fact that sin0(θ) = cos(θ) to prove that cos0(θ) = −sin(θ) since the formerquestion precedes the latter below. (See Problems 12 and 13.) You may alwaysuse high school algebra (like cos(θ) = sin(π/2 − θ)) in your proofs.1 Homework assigned this semester0.1: 1,3,4,,8,10,11,15,25,39,41,51-55,61,63,87,88,89,99. 1.2: 1,3. 0.4:2,3. 1.1: 2, 6, 8. 1.2: 1, 9, 15, 16, 21-23. 1.3: 1, 13, 15, 19, 27,34, 35, 36, 39, 45=49,59, 60, 61, 65, 71. 1.5: 1,3, 8, 19-21, 37, 39, 45, 59.1.6: 1-3, 8, 9, 11, 13, 15, 23, 31, 34, 44-46. 1.7: 1, 3, 15, 17, 18. 1Review: 17, 18, 41.2.1: 1, 3, 30, 32, 45, 51, 57, 59, 60, 61, 62, 64. 2.2: 1, 3, 5, 7, 9, 15, 17, 31,35, 37, 39, 41, 49, 51, 53, 60, 61, 75, 76. 2.3: 1, 7, 9, 10, 13-16, 22. 2.4:1, 7, 9, 11, 20, 32, 35, 45, 57. 2.5: 1, 3, 5, 17, 35, 53, 56, 57, 64, 66. 2.6:1, 5, 17, 21, 25, 27, 45. 2.7: 1, 2, 5, 10, 15, 39, 41-43, 52, 53. 2.8: 1-4,7, 10, 11, 16,19, 21, 23, 26, 29, 30, 34. 2.9: 1, 3, 7, 13, 39, 51, 54, 55. 2Review: 1-12, 13, 35, 45-47, 58, 86, 90-94. 2 Problems Plus: 1, 2*, 3*,11, 22*, 25*.3.1: 1-7, 13, 14, 23, 25, 27, 29, 39, 43, 49, 52. 3.2: 1-6, 7, 9, 1, 13, 15, 17,19, 21, 23, 25, 33, 41. 3.3: 1-21(odd), 22, 25, 35-41(odd), 47, 49, 63, 65, 71,75, 81, 83. 3.4: 1, 7, 9, 11, 13, 35, 37, 45, 47, 49.66, 67. 3.5: 1, 5, 9, 11,12, 13, 19, 21. 3.6: 1, 3, 9, 10, 13, 15, 18, 21, 23, 31, 49, 50, 61, 65. 3.7:1, 2, 7, 8, 11, 12, 14, 23. 3.8: 1, 3, 5, 7, 21, 22, 29, 51, 57, 65, 71, 74-78.3 Review: 1-16, 27, 29.61, 62, 63, 111. 3 Applications Plus: 4.4.1: 1-10, 17, 21, 27, 39, 41, 43, 67-70. 4.2: 1, 5, 7, 11, 17-19. 4.3: 1,2, 3, 5, 7, 33, 35, 37, 41, 43, 45, 47. 4.4: 1, 2, 3, 5, 15, 23, 25, 29, 33, 34.4.5: 1, 2, 5, 9, , 11, 12, 45, 60, 62. 4.6: 1. 4.7: 1, 5, 7, 10-11, 15, 20,21, 23, 25, 26, 31, 33, 34, 37, 38, 43, 49. 4.9: 1, 9, 11, 17, 23, 41, 42, 61-65.4 Review: 1-16, 54. 4 Problems Plus: 4, 5.5.1: 3, 4, 10, 11, 21-31(odd). 5.2: 1-6, 11, 13. 5.3: 1-8, 21-24, 27, 29,31, 33, 34, 45, 47, 49, 51, 55, 57. 5.4: 3, 4, 5, 9, 11, 17, 18, 19, 25, 35, 49,50, 57, 59, 77, 79, 81, 83, 84, 95, 98, 100. 5.5: 1, 2, 3, 7, 9, 27, 29, 41, 43.5.6: 1. 5 Review: 1-20, 23, 39. 5 Applications Plus: 1, 3, 6, 7.6.1: 1, 5, 7, 9, 37, 39. 6.2: 1, 2, 5, 9, 11, 13-16, 33, 35, 49-51. 6.3: 1, 3,9, 11, 21, 23, 42. 6.5: 1, 3, 19. 8.4: 1-5, 10, 21-23. 6 Review: 1, 6,28, 33. 6 Problems Plus: 1, 3, 10, 13, 23.22 Additional Problems1. State and prove the Sum Rule for derivatives. You may use (without proof)the Limit Laws.2. State and prove the Product Rule for derivatives. You may use (withoutproof) the Limit Laws.3. State and prove the Quotient Rule for derivatives. You may use (withoutproof) the Limit Laws.4. State and prove the Chain Rule for derivatives. You may use (withoutproof) the Limit Laws. You may assume (as the proof in the Stewart text does)that the inner function has a nonzero derivative.5. State the Squeeze Theorem.6. Prove thatdxndx= nxn−1, for all positive integers n.7. Prove thatdxndx= nxn−1, for n = 0.8. Prove thatdxndx= nxn−1, for all negative integers n.9. Prove thatdexdx= ex.10. Prove thatlimθ→0sin(θ)θ= 1.You may assume without proof the Squeeze Theorem, the Limit Laws, and thatthe sin and cos are continuous. Hint: See Problem 98.11. Prove thatlimθ→01 − cos(θ)θ= 0.12. Prove thatd sin xdx= cos x.13. Prove thatd cos xdx= −sin x.14. Prove thatd tan xdx= sec2x.15. Prove thatd cot xdx= −csc2x.16. Prove thatd ln xdx=1x.17. Prove thatd sin−1xdx=1√1 − x2.318. Prove thatd cos−1xdx= −1√1 − x2.19. Prove thatd tan−1xdx=11 + x2.20. True or false? A differentiable function must be continuous. Explain.21. True or false? A continuous function must be differentiable. Explain.22. Explain why limx→01/x does not exist.23. Explain why limθ→π/2tan θ does not exist.24. Explain why limθ→π/2sec θ does not exist.25. Explain why limθ→0csc θ does not exist.26. Explain why limx→0sin(1/x) does not exist.27. Explain why limθ→∞cos θ does not exist.28. Let sgn(x) be the sign function. This function is given bysgn(x) =1, if x > 0,0, if x = 0,−1, if x < 0.Explain why limx→0sgn(x) does not exist.29. Explain why limy→021/ydoes not