MIT 18.01 Single Variable Calculus Problem Set 8B, Study notes of Mathematics

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18.01 Single Variable Calculus

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18.01 Problem Set 8B

Due Friday 12/08/06, 1:55 pm Part I (20 points) Lecture 34. Fri. Dec 1 Indeterminate forms; L’Hospital’s rule, growth rate of functions. Read: 12.2, 12.3 (Examples 1-3, remark 1) Work: 6A-1befgj, 5, 6c Lecture 35. Tue. Dec. 5 Improper integrals. Read: 12.4, Notes INT Work: 6B-1,2,7afkm, 8c Lecture 36. Thurs. Dec. 7 Infinite series; simple convergence tests Geometric series; harmonic series. Read: pp. 439-442(top) Comparison tests. pp. 451-3 (skip proof in Example 3) Integral test. pp. 455-457(top) Work: 7A-1abc; 7B-1abf 7B-2acde Lecture 37. Fri. Dec. 8 Taylor series. Read: 14.4 through p. 498 (bottom); skip everything involving the remainder term Rn(x). Differentiation and integration of series. Read: 14.3-p.490(top); Examples 1-5. Work: see handout with remarks about the final exam Lecture 38. Tues. Dec. 12 Final Review. Part II (17 points + 5 extra) Directions: Attempt to solve each part of each problem yourself. If you collaborate, solutions must be written up independently. It is illegal to consult materials from previous semesters. With each problem is the day it can be done.

1. (not until due date; 3 pts) Write the names of all the people you consulted or with whom you collaborated and the resources you used, or say “none” or “no consultation”. (See full explanation on PS1).
2. (Lecs 34-36, 6 + 5 pts: 1 + 2 + 2 + 1 + (5 extra)) a) Use L’Hospital’s rule to evaluate lim xm^ e−x x→∞ b) Use part (a) and limit comparison to show that the improper integral ∞ xn^ e−xdx converges 0 for n ≥ 0. (Do not integrate by parts.) c) Denote A(n) = ∞ xn^ e−x^ dx. Use integration by parts to find the constant cn for which 0 A(n + 1) = cnA(n) (Explain what happens at the infinite limit using part (a).) 1