MIT 18.01 Single Variable Calculus Problem Set 4, Study notes of Mathematics

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18.01 Single Variable Calculus

For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms. Fall 2006

18.01 Problem Set 4

Due Friday 10/20/06, 1:55 pm This is all of Problem Set 4 (not split into 4A and 4B). Although it is not due until after Exam 2, you should do all the Part I exercises through Lecture 16 and all the Part II problems through Problem 4 before the exam, in order to prepare for it. Practice exam problems and an actual past exam will be posted on line as usual. Part I (20 points) Lecture 14. Fri. Oct. 6 Mean-value theorem. Inequalities. Read: 2.6 to middle p. 77, Notes MVT Work: 2G-1b, 2b, 5, 6 (Columbus Day Holiday. No classes Mon and Tues, Oct 9 and 10) Lecture 15. Thurs. Oct. 12 Differentials and antiderivatives. Read: 5.2, 5.3 Work: 3A-1de, 2acegik, -3aceg Lecture 16. Fri. Oct. 13 Differential equations; separating variables. Read: 5.4, 8.5 Work: 3F-1cd, 2ae, 4bcd, 8b Lecture 17. Tues. Oct 17 Exam 2 Covers Lectures 8–16. Lecture 18. Thurs. Oct. 19 Definite integral; summation notation. Read: 6.3 though formula (4); skip proofs; 6.4, 6. Work: 3B-2ab, 3b, 4a, 5 4J-1 (set up integral; do not evaluate) Lecture 19. Fri. Oct. 20 First fundamental theorem. Properties of integrals. Read: 6.6, 6.7 to top p. 215 (Skip the proof pp. 207-8, which will be discussed in Lec 20.) Work: assigned on PS 5 Part II (36 points + 10 extra credit) 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. (Lec 14, 10pts: 2 + 2 + 2 + 2 + 2)) a) Use the mean value property to show that if f (0) = 0 and f �(x) ≥ 0, then f (x) ≥ 0 for all x ≥ 0. b) Deduce from part (a) that ln(1 + x) ≤ x for x ≥ 0. Hint: Use f (x) = x − ln(1 + x). c) Use the same method as in (b) to show ln(1 + x) ≥ x − x^2 / 2 and ln(1 + x) ≤ x − x^2 /2 + x^3 / 3 for x ≥ 0. 1