Mathematics

### Homework

#### Assigned Jan. 18

Sec. 2.1, problem 11. Turn in Friday, Jan. 20.

#### Assigned Jan. 20

Sec. 2.1, problem 18.
Prove: If a>0, b>0, and a is not equal to b, then sqrt(ab) < (1/2)(a+b).
Suggestion. Assume that sqrt(ab) is greater than or equal to (1/2)(a+b). Square both sides and derive a contradiction.
Sec. 2.2, problems 5, 15.
Turn in Friday, Jan. 27.

#### Assigned Jan. 23

Sec. 2.3, problems 5, 6, 10. Turn in Friday, Jan. 27.

#### Assigned Jan. 27

Sec. 2.4, problems 6 (inf only), 7 (inf only).
Sec. 2.5, problems 3, 10.
Regarding problem 3: In the Nested Intervals Theorem, we let xi = sup(a_n), and we showed that xi is in every I_n. Similarly, if we let eta = inf(b_n), we could show that eta is in every I_n. This problem asks you to show that the intersection of all the I_n's, which by definition is the set of x's such that x is in every I_n, is exactly the closed interval [xi, eta].
Turn in Friday, Feb. 3.

#### Assigned Jan. 30

Sec. 3.1, problems 5b, 5d, 10. Turn in Friday, Feb. 3.

#### Assigned Feb. 3

1. Suppose lim(x_n)=x and c is fixed real number. Use the definition of limit (not theorems about limits) to show that lim(cx_n)=cx.
2. Sec. 3.2, problem 7.
3. Sec. 3.3, problem 8.
4. Sec. 3.3, problem 9. Assume that sup A is not an element of A.
Turn in Friday, Feb. 10.

#### Assigned Feb. 6

5. Sec. 3.3, problem 2.
Turn in Friday, Feb. 10.

6. Sec. 3.4, problem 12.
7. Sec. 3.4, problem 14.
Turn in Monday, Feb. 13.

#### Assigned Feb. 17

1. Sec. 3.5, problem 4: (x_n + y_n) part only.
2. Assume (1) lim (x_n ) = infinity and (2) the sequence (y_n) is bounded. Show that lim (x_n + y_n) = infinity.
3. Suppose (1) x_n > 0 for all n and (2) lim (x_n) = 0. Show that lim (1/x_n) = infinity.
Turn in Friday, Feb. 24.

#### Assigned Feb. 20

4. Sec. 4.1, problem 8. Problem should say: for any c greater than or equal to 0. See hint in back of book. On this problem it helps to multiply sqrt (x) - sqrt (c) by [sqrt (x) + sqrt (c)] / [sqrt (x) + sqrt (c)].
Turn in Friday, Feb. 24.

#### Assigned Feb. 22

5. Sec. 4.1, problem 14. For part (b) you will find Theorem 2.4.8 and Corollary 2.4.9 helpful. Turn in Friday, Feb. 24.
6. Sec. 4.1, problems 11a and 11c. Do not turn in.

#### Assigned Feb. 24

Sec. 4.2, problems 5, 6. Give epsilon-delta proofs. Turn in Friday, Mar. 3.

#### Assigned Feb. 27

Sec. 5.1, problems 3, 7, 9, 12. Do not turn in.

#### Assigned Mar. 1

Sec. 5.2, problem 7. Do not turn in.
Sec. 5.2, problems 8, 11.
On problem 8, prove that the answer is yes. Suggestion: Let x be an irrational number. Consider a sequence (r_n) of rational numbers that approach x.
Turn in Friday, Mar. 17.

#### Assigned Mar. 3

Sec. 5.3, problem 1, 2, 3. Do not turn in.
Sec. 5.3, problems 6, 11. Suggestion for 6: If f(1/2)=f(1), let c=1/2. Otherwise, show that g(0) and g(1/2) have opposite sign, and use Thm. 5.3.5. Suggestion for 11: Show that f(w)<0 and f(w)>0 are both impossible. Turn in Friday, Mar. 17.

#### Assigned Mar. 13

Sec. 5.4, problems 2, 6. Turn in Friday, Mar. 17.

#### Assigned Mar. 19

Sec. 6.1, problems 1b, 1c, 2, 4. Turn in Friday, Mar. 24.

#### Assigned Mar. 20

Sec. 6.2, problems 6, 13 (see definition on p. 171), 15. Turn in Friday, Mar. 24.

#### Assigned Mar. 31

Sec. 6.4, problems 11, 15. Problem 15 uses the Mean Value Theorem, not Taylor's Theorem. Turn in Friday, Apr. 7.
Sec. 7.1, problems 1a, 2b. Do not turn in.
Sec. 7.1, problems 6a, 11. Turn in Friday, Apr. 7.

#### Assigned Apr. 7

Sec. 7.2, problems 8, 11. Turn in Friday, Apr. 21.

#### Assigned Apr. 10

Sec. 7.3, problem 10. (Express G as the composition of two differentiable functions and use the chain rule.) Turn in Friday, Apr. 21.

#### Assigned Apr. 12

Sec. 8.1, problems 3, 4, 13, 14. (On 13, 14, leave the part after "but" to do after the class on Monday Apr. 17.) Turn in Friday, Apr. 21.

#### Assigned Apr 17

Sec. 8.2, problems 1, 4, 5, 7, 13 (first sentence).
Do not turn in.

#### Assigned Apr 21

Sec. 3.7, problems 5, 8, 11.
Sec. 9.1, problems 7a, 12, 13a.
Do not turn in.

#### Assigned Apr. 24

Sec. 9.2, problem 9. Sec. 9.3, problem 2. Do not turn in.