## 170 – Final homework set

April 27, 2013

This set is optional. If you turn it in, you may ask for feedback but no grade, or for grade (and feedback). It is due May 10 at the beginning of lecture. I will have it graded by Monday, so you can have it back on time before your Wednesday final.

• Whitman College Calculus. Section 8.1. Exercises 11, 17, 18. Section 8.2. Exercises 4, 7, 8, 10.
• Spivak, Chapter 19. Problems 1, 2. Chapter 18. Problem 29.
• In the book (Chapter 18, page 344) it is explained how Riemann sums can be used to show that $2. Use the same approach to prove that $2.5.

## 170 – Final

April 26, 2013

The date of the Final Exam has been changed. It will now be Wednesday, May 15, 12:00 – 2:00 pm. (Solutions.)

Here is a copy of Quiz 2, from March 15.

Do not forget Quiz 3 is due Monday, April 29, at the beginning of lecture, and Quiz 4 is on Wednesday, May 1, at the beginning of lecture.

## Niece

April 22, 2013

Laura Emma, born April 10.

## 170 – Homework 6

April 22, 2013

This is the last homework set of the term, due Friday May 3, at the beginning of lecture.

• Spivak, Chapter 13, Problems 5, 8, 16, 17. Chapter 14, Problems 4, 5. Chapter 15, Problem 12. Chapter 18, Problems 6, 8, 13. (The definitions of the hyperbolic functions are in Chapter 18, Problem 7.)
• Whitman College Calculus, Section 7.1, Exercises 1, 7. Section 7.2, Exercises 3, 6, 11, 16, 20, 21.

## 403 – Homework 3

April 15, 2013

This set is due Monday, April 29, at the beginning of lecture.

1. Suppose $A$ is an $n\times n$ matrix with complex entries such that $A^2=I$. Verify that $e^{iAx}=\cos(x)I+i\sin(x)A$ for $x$ real.
2. Suppose that $N$ is nilpotent. Verify that $I+N$ is invertible.
3. Use linear algebra to find a closed form expression for the terms of the sequence $\{a_n\}_{n\ge0}$ given by the recurrence relation $a_0=1$, $a_1=-1$, $a_{n+2}=3a_{n+1}+10a_n$ for all $n\ge 0$.
4. (Norman Biggs, Algebraic Graph Theory, Proposition 2.3) Let $\Gamma$ be a (finite, simple) graph with $n$ vertices, and let $A$ be its adjacency matrix. Suppose that the characteristic polynomial of $A$ is ${\rm char}_A(x)=x^n+c_1x^{n-1}+c_2x^{n-2}+\dots+c_n$. Verify that:
1. $c_1=0$.
2. $-c_2$ is the number of edges in $\Gamma$.
3. $-c_3$ is twice the number of triangles in $\Gamma$.

## 170 – Homework 5

April 5, 2013

First of all, Zach Teitler pointed me to this game that you may find interesting. A human ship (that you control) is trying to avoid a plasma cannon operated by a robot. There are three levels, depending on whether the robot only has access to your position, or to your position and velocity, or to your position, velocity, and acceleration.

Now, this set is due Monday, April 15, at the beginning of lecture.

• L’Hôpital’s rule: Compute the following limits:

$\displaystyle \lim_{x\to0}\frac{e^x-\sin x-1}{x^2}$, $\displaystyle \lim_{x\to+\infty}x\cos\bigl(\frac\pi2+\frac1x\bigr)$, $\displaystyle\lim_{x\to1}(2-x)^{\tan(\pi x/2)}$, and $\displaystyle \lim_{x\to\frac\pi2^-}\frac{\tan x}{\ln\left(\frac\pi2 -x\right)}$.

• Use the mean value theorem to solve Problem 47 in Chapter 11  from Spivak’s book.
• Apply Newton’s method to solve (with six digits of accuracy) the equation $x^3+3x-2=0$.
• Recall that in Newton’s method, we produce a sequence $x_1,x_2,x_3,\dots$ of approximations to a solution $\alpha$ of the equation $f(x)=0$, where each new approximation $x_{n+1}$ is related to the previous one $x_n$ via $\displaystyle x_{n+1}=x_n-\frac{f(x_n)}{f'(x_n)}$. Recall as well that $\displaystyle |\alpha-x_{n+1}|=\frac12\left|\frac{f''(c_n)}{f'(x_n)}\right| |\alpha-x_n|^2$ for some $c_n$ between $x_n$ and $\alpha$.

Now consider the particular case where $f(x)=x^2-10$, so $\alpha=\sqrt{10}$. As first approximation, take $x_1=3$.

Begin by checking that $x_n\ge 3$ for all $n\ge 1$.

Find a number $A>0$ such that ${}|f'(t)|\ge A$ for all $t\ge3$. In particular, for any $n$, this means that ${}|f'(x_n)|\ge A$.

Find a number $B>0$ such that ${}|f''(t)|\le B$ for all $t\ge 3$. In particular, for any $n$, this means that ${}|f''(c_n)|\le B$. Conclude that $\displaystyle \frac12\left|\frac{f''(c_n)}{f'(x_n)}\right|\le \frac{B}{2A}$. Note that the right hand side is a constant $C$, it does not depend on $n$ at all.

Check that we could have picked $A,B$ so that this $C\le 1/2$. If the particular $A,B$ you first used do not satisfy this inequality, check that you can pick different values so that this is now true. (In fact, I expect you will get that $C$ is significantly smaller that $1/2$, though $1/2$ or even $1$ works for the next part.)

Conclude from this that each $x_{n+1}$ essentially duplicates the numbers of accurate digits. How large should $n$ be if we want $x_n$ to agree with $\sqrt{10}$ for at least $100$ digits?

• Use that $f(x)\approx f(a)+(x-a)f'(a)$ to approximate $\sin(.1)$ and $\sqrt{9.1}$. Compare with the actual values.

April 5, 2013

(Spoilers.)