- Rosenlicht Chap. 6, exercises 18, 22. No trig functions allowed in #18. The point of the problem is to show directly that the two integrals are equal to each other, not that they are equal because they both yield tan-1x. Due date: Fri. 2/6/09. Hand in only #22.
- Rosenlicht Chap. 6, exercise 24a. Due date: Mon. 2/9/09.
- Read this handout on Improper Integrals and do all the exercises through #16 except as noted in the handout. These problems are numerous but short. Due date: Mon. 2/9/09 except for #14, which is due Wed. 2/11/09. (You may want to do Rosenlicht #25, assigned below, before #14 on the handout.) Hand in 11-14 only. In doing any of these problems, you may assume the truth of all results stated in earlier problems in the handout (e.g. for #14 you may assume the truth of anything stated in #1-#13). Important note added 2/20/09: There was a mistake in #13 as originally written. This has now been corrected. Bravo to Alex Stopnicki for finding the mistake.
A minor typo on p. 7 was also corrected on 2/16/09; depending on when you printed the handout (the interval in the first line of the proof should be (a,b], not [a,b)), your version may or may not have this typo in it. My thanks also go to the sharp-eyed student who pointed this one out to me.
Later in the semester, you will be assigned exercises 17-19, but you don't yet have a tool you'll need to do these.
- Rosenlicht Chap. 6, exercises 23-26 (minus 24a, which was due 2/9/09). Due date: Wed. 2/11/09. In #25, Rosenlicht's "hence that" is there for a reason: you are supposed to figure out how the first part of the problem implies the limit-statements in parts (a), (b) , (c). Do not use l'Hôpital's Rule, which is entirely unnecessary for the computation of these limits, contrary to the impression you may have received in Calculus 1. l'Hôpital's Rule serves only to obscure the reason why these limits are zero.
You should also not use l'Hôpital's Rule to help with 24cde. In any of these, wherever you think you'd like to use l'Hôpital's Rule, check whether the definition of "derivative" would achieve the same purpose. (It should, unless you're doing something wildly wrong.)
- Rosenlicht Chap. 7, exercises 2-5. In these exercises, Rosenlicht's "converges", for a sequence of functions, is my "converges pointwise" (and similarly for "convergent"). Due date: Mon. 2/16/09. Hand in only #3.