ZOct 21 - Nov 8.pdf - Math 137 notes FALL 2019 October 14...

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Math 137 notes - FALL 2019: October 14 - November 8 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 my 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 October 21 The Derivative Function Definition. Let f ( x ) be a function and I be an open interval. We say that f ( x ) is differentiable on I if it is differentiable at every point on I . In this case, the function f 0 ( x ) is defined on I by f 0 ( a ) = f 0 ( a ) for each a . 1 Note that this implies f ( x ) is defined on all of I , and from a Theorem from last class, it implies it is continuous on I . There are similar ways of extending the notion of differentiability to closed intervals by taking some care with the end points, but I’m not going to do this. To be differentiable at a point, you really require the function be defined on an open interval containing it. Definition. If f ( x ) is differentiable on all of R , we say that f ( x ) is differentiable. I don’t think I mentioned this earlier, but the same goes for the word “contin- uous”. Notation. This notation is due to Leibniz: we say d dx f ( x ) or df dx means the same as f 0 ( x ). The creature d dx is called an “operator”, which is a fancy word for “function”, but we use a different word because it usually takes functions as its input. Indeed, you can think of d dx as a function which takes a function (whose variable is called x ) as input, and outputs another function (the derivative) whose input variable is x . If the function is called f ( t ), the operator would be written d dt . It is important to note that this is not a fraction. The notation is meant to remind us that derivatives are slopes, which are calculated as the limit of fractions. There are also ways, in particular when you get to integration, where df dx can be treated rather like a fraction.

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