Lecture 4: Sketching graphs

Problem of the day

This problem is due on Friday, January 22 along with POW1.
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.
Solution. Lets say the man set his clock at noon just before leaving. When he returns later, he will know how long he was gone. He also can calculate how long it took him to walk to his friends, because he noted the time when he arrived and the time he left. The time he was away from home is just twice the time it took to walk to his friends plus the time spent there. Subtracting the time spent at the friends and dividing the difference by 2 gives the one-way walking time. Adding this one-way walking time to the time he left his friends house gives the correct time.

Assignment

These are the problems you should work before January 29.
Review problems, page 75, 4n+1, for n=0..22;
Section 1.1, page 92, 4n+1, for n=0...24;
Section 1.2, page 102, 1-6, and 4n, for n=3...23;
Section 1.3, page 113, 4n+1, for n=0...19.

Brief review of the lecture

Outline for day 4

I Graphing of equations

Equations with one unknown

i. 2x=7

B. Equations with two unknowns

y=7-3x
y=x^2-2
(x-2)^2+(y-4)^2=9
y=|x|

III. Things to look for in a graph

x- and y- intercepts
Symmetry

x-axis symmetry
y- axis symmetry
symmetry with respect to the origin.

The material below is left from last semester. Its useful information for your section, however.

Continuation of absolute value problems. Consider the following problem. How many real numbers x satisfy |||x-10|-12|-15|=12? Again we saw conditioning at work. The equation has 6 solutions, -29, -5, 1, 19, 25, and 49.
We were also concerned with the problem of representing real numbers. There are many different ways to do this, but one of the best is the decimal representation. Not only is it available for EVERY real number, the form of the representation says a lot about the number. For our purposes, we talk about three types of decimal representations:
1. those which end in zeros from some point on. We don't write the zeros, but instead just the nonzero digits of the representation. 4.205 is an example. The convention is that no zeros are written after which all the digits are zeros. So 4.205 is the same as 4.2050000... with zeros extending forever.
2. those which repeat in blocks from some point on. For example, 2.99999.... which we usually write as 2.9 with a bar over the 9.
3. those which don't repeat. For example, 1.01001000100001....
The first two are the types we get representing RATIONAL numbers, and the third is the type IRRATIONAL numbers have. We are concerned today with the problem of converting those of type 2. (all such numbers are rational, remember) to the form a/b where a and b are integers. The solution is to give the representation a name, say x. Multiply it by ten raised to the power equal to the size of the repeating block, and subtract the former from the latter. The `tails' of the two numbers disappear and the difference is a decimal of the type 1. above. Some easy arithmetic will finish the job.

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