Sept 4 - Sept 20.pdf - Math 137 notes FALL 2019 September 4...

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Math 137 notes - FALL 2019: September 4 - September 20 Ian Payne Please read these rules and guidelines. 1. These notes are for academic purposes only . That means you are not permitted to use them to acquire any sort of compensation, financial or oth- erwise. 2. If you wish to share these notes with someone who is not enrolled in MATH 137, please ask me first. 3. You are not allowed to complain about these notes. If I get a complaint I will remove them and stop updating them. 4. I will post these in bulk occasionally. I have my own reasons for deciding when to update them. Asking me to post them early is a violation of rule #3. 5. These notes are merely what I write to prepare for lectures. They are min- imally edited at best and there is no guarantee that they reflect what was done in class (though they usually will). 6. If you see a typo or you think there is a mistake (mathematical or otherwise), please let me know! Of course, it’s your job to figure out how to report an error without violating rule #3 , [I’m pretty reasonable. Just be polite!] 7. This should follow immediately from rule #2, but do not post these notes on any website without my permission . To save you some time, you won’t get permission so don’t bother asking. 1
Not to be posted on any website without permission from the author September 4 Absolute Values and Distances For a real number x , what is meant by | x | ? x without the negative. The size of x . The distance from x to 0. The absolute value of x . The modulus of x . All of these are correct. The last two are terminology and the first three are definitions. Here are a few examples: • | 2 | = 2. • | - 33 . 945 | = 33 . 945. • | π | = π . • | 3 - 4 | = 1. Keep a close eye on the last one. In mathematics, we need formal definitions, lest all of our work fall apart on us later in life. Here is a more mathie looking definition: Definition. For a real number x , we define | x | by the piecewise function | x | = x if x 0 - x if x < 0 Not to be posted on any website without permission from the author
Not to be posted on any website without permission from the author It may look like we have complicated things unnecessarily, but those feeling will pass with time. At least, have a careful read of this definition. It says that when x is nonnegative, the absolute value function doesn’t do anything, and when x is negative, it negates it (negating a negative makes something positive, remember.) Here is a proof. It will probably be the easiest proof in the course. Proposition. For any real number x , | x | = | - x | . Sure, you already knew this. Here is a formal proof. Proof. FIrst, suppose x = 0. Then - x = x , so | x | = | - x | because all functions do the same thing to the same input. If x > 0, then | x | = x by the definition of the absolute value function. As well, - x < 0, so | - x | = - ( - x ) = x = | x | by the definition of the absolute value function. Finally, if x < 0, then - x > 0. Again using the definition of the absolute value function gives | x | = - x and | - x | = - x , so | x | = | - x | . Up to this point in your life, you have probably used the absolute value function as another example of a function you can graph and on which you can perform

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