Lecture 3: Problem Session

Problem of the day 2

This problem is due on Friday, Sept. 4.
A man whose clock had stopped running wound it up but did not have access to the correct time to reset it. Leaving his home, he walked at a constant rate to a friend's home. The friend had an accurate clock, which he noted upon arrival. He stayed for a while at the friend's, then noted the time and walked back to his home at the same rate as before. Upon arriving at his home, he was able to reset his clock correctly. Explain how he was able to do this.

Quiz 1

Show that (x+1)(x+2)=x^2+3x+2 geometrically by demonstrating that a rectangle with dimensions x+1 by x+2 can be split into pieces of the right sizes. For a more attractive presentation and a solution

Assignment

By today, Friday, August 28, you should have worked

Section P.4, page 42, problems 6n+1, n=1...14; problems 91-96;

Section P.5, page 52, problems 8n+1, n=1..10;

Section P.6, page 62, problems 6n+1, n=1,...,10; and

Section P.7, page 70, problems 6n+1, n=1,...,11.

Brief review of the lecture 3

Problem session

The material below was part of the third lecture in previous semesters. It is left here because it may be of interest to you.
1. Representations of Real Numbers.
There are many ways to represent numbers, but one very convenient way is DECIMAL representations. We looked at what decimal representation means and how to move between the decimal representation of a rational number and the quotient of integers representation.
2. The real numbers. We constructed a venn diagram showing two ways of chopping up the real numbers: first by positive, negative and zero, and second by rational and irrational. We can classify a number as rational or irrational based on the nature of its decimal representation. If the number has terminating (ends with zeros from some point on) or repeating (the same block of digits repeats forever) decimal representation, then the number is rational.
3. Integer exponents.
4. Absolute value, and the graph of the absolute value function. How to solve equations with absolute values? There are two ways.
A. Solving by conditioning.
Solve |x|=4. First, ask `what could we say if we knew x was positive?' Of course, then x would have to be 4. Now what if we knew x was not positive? We could say that x was -4. We've just used conditioning on x to solve the equation. IE, we supposed x has some property, and made a conclusion, then we supposed that it had did not have that property and came to another conclusion.
B. Solving by geometric means
We defined the function f as follows:

f(x) = 3-2x if x satisfies x<=1, 2+x if 1< x <=5, and 2x-7 if 5 < x.
Then we asked the question, for what values of x is f(x)=6.
A. We used conditioning as follows. First, condition on x <= 1. In other words, assume the inequality is satisfied and ask if there are any of those x for which f(x) is 6. We found that -3/2 works. Then we conditioned on 1 < x <=5, and found another solution in that interval. Finally, we solved the equation 2x-7=6 and found yet another solution 13/2. We will say we conditioned on the value of x three times, one for each clause in the functions definition.
B. To solve the problem geometrically, we can sketch the graph of the function and locate the places where this graph touches the line y=6. This can be done either by hand or with the aid of a graphing calculator.

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