Fall 2010 Final
Student-provided questions and solutions to the Fall 2010 final exam.
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1Question 1¶
(i) The Archimedean property states that the set of natural numbers $\mathbb N$ is unbounded in $\mathbb R$. Assuming this property, show that $\inf \{1/n: n \in \mathbb N\} = 0$. Conclude that if $t > 0$, there exists some $n \in \mathbb N$ with $\frac{1}{n} < t$, and if $y > 0$, there exists a natural number $n$ such that $n-1 \leq y < n$.
(ii) Use (i) to show that for any two real numbers $x$ and $y$ with $x < y$, there exists a rational number $r$ with $x < y < y$.
1.1Solution¶
Coming soon
1.2Accuracy and discussion¶
2Question 2¶
(i) Let $f : A \to \mathbb R$ be a function. Define what it means for $f$ to be continuous on $A$, and uniformly continuous on $A$. Comment on the difference between these definitions.
(ii) Show that if $f$ and $g$ are uniformly continuous functions on $A$, and they are both bounded on $A$, then their product $fg$ is uniformly continuous on $A$.
(iii) Show that if $A$ is the closed bounded interval $A = [a, b]$, then a continuous function $f: A \to \mathbb R$ must be bounded. State any theorems you use. Conclude that the product of two uniformly continuous functions defined on a closed bounded interval is uniformly continuous.
(iv) Give an example of a set $A$ and a uniformly continuous function $f: A \to \mathbb R$ which is not bounded, justifying your choice.
2.1Solution¶
Coming soon
2.2Accuracy and discussion¶
3Question 3¶
(i) Let $a \subset \mathbb R$ be a nonempty set. Define what it means for a real number to be the supremum of $A$. State the completeness axiom for $\mathbb R$.
Now let $I_n = [a_n, b_n]$, $n\in \mathbb N$ be a nested sequence of intervals, i.e., $I_{n+1} \subset I_n$ for each $n \in \mathbb N$.
(ii) Show that the set $\{a_n:n \in \mathbb N\}$ is bounded above, and let $a*$ be its supremum.
(iii) Show that $\displaystyle a* \in \cap_{n=1}^{\infty} I_n$.
(iv) If $f: [a, b] \to \mathbb R$ is a continuous function, with $f(a) < 0$ and $f(b) > 0$, show that there exists $x \in (a, b)$ with $f(x) = 0$. Hint: use part (iii).
(v) Show that the polynomial $f(x) = x^4 + 7x^3 - 9$ has at least 2 real roots. Describe how you would locate one of these roots with error less than 0.005.
3.1Solution¶
Coming soon
3.2Accuracy and discussion¶
4Question 4¶
Prove or disprove each of the following statements.
(i) If every subsequence of $(x_n)$ has a subsequence that converges to 0, then $\displaystyle \lim_{n \to \infty} x_n = 0$.
(ii) The sequence $(1 + (-1)^n)$ is a Cauchy sequence.
(iii) The sequence $((3n)^{\frac{1}{2n}})$ diverges to $\infty$.
4.1Solution¶
(i) I have no idea. That's interesting.
(ii) Disprove.
(iii) Disprove. It converges to 1 for some reason. Why???
4.2Accuracy and discussion¶
5Question 5¶
(i) If $0 < x < 1$, show that $x^n \to 0$ as $n \to \infty$.
(ii) If $0 < a <b$, find the limit of the sequence $\displaystyle \left ( \frac{a^{n+1} + b^{n+1}}{a^n + b^n} \right )$, stating any theorems that you use.
5.1Solution¶
(ii) This is straight out of the textbook. Coming soon.
5.2Accuracy and discussion¶
6Question 6¶
(i) Suppose $f: [a, b] \to \mathbb R$ with $c \in (a, b)$. Show that $f$ is differentiable at $c$ if and only if there exists a function $\phi: [a, b] \to \mathbb R$ such that $\mathbb \phi$ is continuous at $c$ and satisfies
$$f(x) - f(c) = \phi(x)(x-c)$$
whenever $x \in [a, b]$. Show that in this case, $f'(c) = \phi(c)$.
(ii) If $f(x) = x^3$, find $\phi(x)$ and deduce that $f'(c) = 3c^2$.
(iii) Suppose $f$ is strictly monotone increasing and continuous on $[a, b]$. Let $J = f([a, b])$ and consider the function $g: J \to [a, b]$ defined as $g = f^{-1}$. Explain why $J$ is a closed bounded interval. Show that if $f$ is differentiable at $c \in [a, b]$ and $f'(c) \neq 0$, then $f^{-1}$ is differentiable at $d = f(c)$ and $g'(d) = \frac{1}{f'(c)}$. Hint: use part (i). Then, using part (ii), if $f(x) = x^3$, find $g'(d)$.
6.1Solution¶
Coming soon
6.2Accuracy and discussion¶
7Question 7¶
Let $f$, $g$ be differentiable on $\mathbb R$ and suppose that $f(0) = g(0)$ and $f'(x) \leq g'(x)$ for all $x \geq 0$. Show that $f(x) \leq g(x)$ for all $x \geq 0$. State completely any theorems that you use.
7.1Solution¶
Coming soon