- Your fourth homework is due today.
- Your third homework is returned. Comments:
- I'll start with a general comment: you cannot prove
things with examples (except in special circumstances --
e.g. the four color problem). You can check conjectures with
examples, but we need to work more generally in proofs.
- 1: this one was relatively straightforward.
- 4: Here you need to use a property of the exponential
function
(and not make up any new ones!:).
- 5:
- Note: when the author says (about the two numbers in the product) that one has a "relative error of at most 1% in magnitude...", you may not write "RE(a)=.01".
- You really do need to use inequalities here, and the triangle inequality (again!) will be useful.
- Nonetheless, Eden did it without, with a good argument.
- 8:
- You'll need something, such as the MVT from calculus, to solve this. You can also use an integral from calculus and the MVT for integrals.
- It's good, however, to confirm the result graphically, which some of you did.
- Other potential strategies include Taylor Series expansions, and trig identities.
- Vicheth gave a nice picture for part b. The condition number near
illustrates this problem, as well.
- 14: This is one where you might look at the condition
number (or do the direct calculation of the limit, as is
suggested on p. 26).
Some of you interpreted this differently from me. The authors ask "for which values of
do tiny relative changes in
produce relative change in
much larger in magnitude?
You then talked about the derivative, essentially. This is legitimate in my view.
What's the difference?
The condition number is the "relative derivative": the ratio of the relative error in
over the relative error in
The derivative could be defined as the error in
In particular, condition is "dimensionless": if we change the units of the function
A useful example to think about is problem #8, above. If we look at the condition number of the function cos(x), it gets larger as we approach each root of cos(x): small change in x, big change in cos(x). But the condition number gets relatively larger as x gets bigger (and the derivative of cos(x) at the roots doesn't change). For a fixed error, the relative error in x is going to get smaller, whereas the relative error in the function won't change.
- 1, p. 34: Andres gave a very nice set of ideas:
- Pros:
- 12 has more divisors (1,2,3,4,6,12) than 10.
- with base 12 you can express most common fractions without repeating decimals.
- useful when we are expressing time
- Cons:
- We only have 10 fingers.
- It would cost a lot of money and time to change from one base to another.
- Pros:
- 3: the authors showed that terminating decimal expansions
(base 10) do not always have terminating binary
expansions (p. 35 -- 5.2 does not have a terminating
base 2 representation).
But because 2 divides 10.... Consider terminating binary decimal
for any integer
.
And because the sum of terminating decimals is terminating....
Some of you commented on other bases. It's possible, for example, that a terminating decimal in base 3 will not have a terminating decimal expansion in base 10. Can you give an example?
- 8:
- You need to first interpret the fraction (and some of you had trouble with that).
- Then you'll need the geometric series (that is, the Taylor series for
).
- I'll start with a general comment: you cannot prove
things with examples (except in special circumstances --
e.g. the four color problem). You can check conjectures with
examples, but we need to work more generally in proofs.
- Your first programming assignment is due this Thursday, 2/5.
- A fifth homework is due a week from today (2/10, and note that I moved some of assignment 4 to assignment 5).
We've pretty finished with chapters 1 and 2, and I think that we're ready to start into Chapter 3. We're going to be considering root-finding.
I will start, however, by asking what questions folks might have about chapter 2.
The most important equation in the world (Physicist Charles Shirkey, BGSU):
How would you do it?
(Put away your books!)
Today I want to lay out the problems and issues, and conditions under which we can solve this. Maybe even a strategy for solving some cases.
Then we'll see what our authors have to suggest. They have essentially three methods to propose. We'll look at those, and then one more -- Muller's method -- which will allow us to find complex roots of real-valued equations.