Math 409-300 Summer 2014
Assignments
Assignment 1 - Due Monday, June 9
- Read sections 1.1-1.6.
- Do the following problems.
- Section 1.2: 0, 3, 4(c), 6, 7(c), 10
- Problem 1.2.0: Let $a,b,c,d$ be real and consider each of the
statements below. Decide which are true and which are false. Prove
the true ones and give counterexamoles for the false ones.
- If $a < b$ and $ c < d$ then $ a c > c d $.
- If $a \le b$ and $c > 1$, then $|a+c| \le |b+c|$.
- If $a \le b$ and $b \le a+c$, then $|a-b| \le c$.
- If $a < b-\varepsilon $ for all $\varepsilon > 0$, then $a < 0$.
- The positive part of a real number $a$ is defined by
\[
a^+ = \frac{|a|+a}{2}
\]
and the negative part by
\[
a^- = \frac{|a|-a}{2}
\]
- Prove that $a=a^+ - a^-$ and that $|a|=a^+ + a^-$.
- Prove that
\[
a^+ = \left\{\begin{array}[ll] \\
a & a\ge 0\\
0 & a < 0
\end{array}\right.
\quad \text{and}\quad
a^- = \left\{\begin{array}[ll] \\
0 & a\ge 0\\
-a & a < 0
\end{array}\right.
\]
- Problem 1.2.4(c): Solve for all $x \in \mathb R$: $|x^3 - 3x + 1|
< x^3$
- Problem 1.2.6: The arithmatic mean of $a, b\in \mathbf R$ is
$A(a,b) = \frac{a+b}{2}$ and the geometric mean of $a,b \in
[0,\infty)$ is $G(a,b) = \sqrt{ab}$. If $0 \le a \le b$, prove that
$a \le G(a,b) \le A(a,b)$, and also prove that $G(a,b) = A(a,b)$ if and
only if $a=b$.
- Problem 1.2.7(c): Prove that $-3\le x \le 2$ implies that
$|x^2+x-6| \le 6|x-2$.
- Problem 1.2.10: For all $a,b,c,d\in \mathbf R$, prove that
$(ab+cd)^2 \le (a^2+c^2)(b^2+d^2)$.
- Show that ${\mathbb Z}_2$ (integers mod 2) is a field.
- Show that ${\mathbb Z}_4$ is not a field.
Assignment 2 - Due Monday, June 16
- Read sections 2.1-2.4.
- Do the following problems.
- Section 1.3: 0(a,b), 1(a,b,e) (Just state the answers.), 2, 6(a), 7(a), 11
- Section 1.5: 0(d), 2(a,d)
- Use your favorite software to plot the following sequences:
- $x_n = 1/2^n$, $n=1$ to $6$.
- $x_n= \frac{1-(-1)^n}{n+1}$, $n=1$ to $9$.
- $x_n=1+ \frac{(-1)^n}{n}$, $n=1$ to $10$.
- $x_n=(-1)^n+\frac{1}{n}$, $n=1$ to $10$.
Assignment 3 - Quiz, Friday, 6/20/2014
- Read sections 2.1-2.4.
- Know the definitions and be able to state and/or prove the theorems.
- Definitions:
- Supremum and infimum of a set $E\subset \mathbb R$.
- Completeness axiom.
- Sequence, subsequence, limit of a sequence, bounded sequence.
- Theorems:
- Squeeze theorem (2.9(a)).
- Be able to prove Theorem 2.11. (Statement will be given.)
- Comparison theorem (2.17).
Assignment 4 - Due Friday, June 27
- Read sections 3.1-3.4.
- Do the following problems.
- Section 2.3: 2, 7
- Section 2.4: 2, 6
- Section 3.1: 6, 7, 8
Assignment 5 - Due Wednesday, July 9
- Read sections 4.1-4.3.
- Do the following problems.
- Section 3.2: 0(d), 2(d), 5
- Section 3.3: 0(a), 1(d), 2(a), 4, 5
- Section 3.4: 0(b), 1(b), 4(b), 8
Updated 6/27/14 (fjn).