# Recitations

The page references to exercises refer to the textbook by T. Apostol, Calculus, Vol. I Second Edition (1967).

RECITATION ASSIGNMENT # ASSIGNMENTS
1 1. Give an example of (a) a finite abelian group; (b) an infinite non-abelian group;
2. Ex. 1 (except Thms I.6, I.11) on p. 19;
3. Ex. 1 on p. 21;
4. Ex. 12 on p. 36-37, ex. 2 on p. 40, ex. 12 on p. 41.
2 1. Read subsections I.4.5 and I.3.11 in the book;
2. Ex. 1, 2, 3, 4, 5, 7, 8 on p. 28.
3 1. Ex. 1abchij on p. 43;
2. Ex. 2def, 4ab on p. 63-64;
3. Ex. 1bc, 5, 12 on p. 70-71.
4 1. Prove that if f(x) is bounded on [a, b] and monotonic on (a, b), then it is integrable on [a, b];
2. Give an example of a function on [a, b] which is monotonic on (a, b) and not integrable on [a, b];
3. Ex. 13, p. 45-46.
5 1. Ex. 9, 21, 25 on p. 83-84;
2. Ex. 17b, p. 94.
6 1. Review integration: ex. 19, 25, 27 on p. 105, ex. 8, 15 on p. 124, ex. 15 on p. 114;
2. Prove using the \epsilon-\delta language: f(x)=\sqrt(x) is continuous for all x>0;
3. Ex. 28, p. 139.
7 1. Read section 3.4 in the book;
2. Prove that if f(x)= x^2 sin(1/x) for x not equal to 0, and f(0)=0, then f(x) is continuous at x=0;
3. Ex. 1, 3, 11, 14, 16, 19 on p. 142;
4. Ex. 1, 5, 6 on p. 145.
8 1. Let f(x)=x^4 +2 x^2 +1 for x in [0, 2]. Show that  f(x) is strictly increasing and find the domain of the inverse function g(y). Find an expression for g(y);
2. Show by example that Intermediate Value Theorem can fail if f(x) is continuous only on (a, b) and bounded on [a, b];
3. Ex. 7, p. 155.
9 1. Ex. 12, 14, 15, 24, 35, 38 on p. 167-168;
2. Find f'(x) by definition (if it exists):
a) f(x) = x sin(1/x) for nonzero x, and f(0)=0;
b) f(x) = x^2 sin(1/x) for nonzero x, and f(0)=0.
Are the derivatives continuous at x =0?
3. Ex 4, 6, 7, 10, 14, 15 on p. 179-180.
10 1. Find the derivative dy/dx by implicit differentiation:
(a) a cos^2(x+y) = b, where a, b are nonzero numbers;
(b) x^3 + y x^2 + y^2 = 0;
2. Read sections 4.17, 4.18 (in particular, thm. 4.10 on convexity);
3. Ex. 9, 14 on p. 191; Ex. 5 on p. 186.
11 1. Ex. 3, 4, 6, 8, 12, 22 on p. 208;
2. Obtain an estimate for \pi: 3 < \pi < 2 \sqrt(3) by using estimates for the integral of cos(x) from 0 to (\pi)/6;
3. Ex. 2 to 17 on p. 216.
12 1. Ex. 2-9 on p. 220.
13 1. Show that (1 + 1/n)^n < e < (1 + 1/n)^{n+1} (it may be helpful to consider the integral of e^x from 0 to 1/n);
2. Ex. 10-20 on p. 236;
3. Ex. 9-16, 21-29 on p. 248-249;
4. Ex. 30-37 on p. 258.
14 1. Ex. 5, 11, 12, 19, 20, 21, 22 on p. 267;
2. Ex. 33-38 on p. 267.
15 1. Ex. 6-14, p. 291 (use Taylor's formula);
2. Ex. 3, 4, 5, 7, 8 p. 278;
3. Ex. 4 on p. 285;
16 1. Ex. 4-14, p. 295;
2. Ex. 1-7, 11, 12, p. 303.
17 1. Ex. 15-29, p. 291;
2. Ex. 14-25, p. 303;
3. Ex. 1-8, 24, 27, 30, p. 382;
4. Ex. 2, 4, 11, 12, 15, 17, 18, p. 391.
18 1. Review for the quiz.
19 1. Ex. 1, 2, 4-11, p. 402;
2. Ex. 1-14, p. 398;
3. Ex. 1-9, p. 409.
20 1. Ex. 13-21, 27-32 p. 409;
2. Ex. 49-51, p. 411.
21 1. Ex. 1-10, p. 420;
2. Ex. 1-10, p. 430.
22 1. Ex. 1-4, 6-10, p. 438;
2. Expand the function f(x)= x^3 -2x^2 -5x -2 in a series of powers of (x+4).
23 1. Ex. 11, 13-17, p. 439;
2. Expand the integral from 0 to x of sin(t)/t and find the radius of convergence;
3. Expand 1/(4-x^4) and find the radius of convergence.