Mathematics ideas #1: convex, concave functions and averages by mathowl

steemstem · @mathowl · Mar 8 '19 (edited)

$5.34

Mathematics ideas #1: convex, concave functions and averages

I haven't written a math post in a while. I thought might be a nice idea  to write about some random mathematical ideas. Today a slightly  technical one relating concave, convex properties of function to average. Actually, this comes from an engineering text book, PDEs for Scientists and Engineers by Farlow. It is a pretty nice read if you don't think existence uniqueness proofs are relevant :P

So given a smooth one dimensional function f on the real line. Then we have that 

<center>
![00.png](https://cdn.steemitimages.com/DQmabU58YwDjys8ULPGVhoro2PR1eEkHCKXb1dSGDRwqjFf/00.png)</center>

With some basic calculus this can&nbsp; be rewritten as 


<center>
![0.png](https://cdn.steemitimages.com/DQmX9z2K2VA13J74qGJoajbduAng5Da73AAKBTLWoEuYNDS/0.png)
</center>


So  the left term inside the brackets is something like an average of f over [x-Δx,x+Δx] when considering only the endpoints. When f(x) is greater than this average then f''(x)<0 and when f(x) is smaller than this average then f''(x)>0. Or if you are too lazy to think about what this means here is a picture:

<center>
![rect5477.png](https://cdn.steemitimages.com/DQmamyLahmMmUzJDV2jA77xQxowiyBSjpsx72RhjuqgWV1c/rect5477.png)
</center>

 In the left graph f''(x)<0 the graph lies below the line so it is convex, in the right graph f''(x)>0 the graph lies above the line so it is concave. This is exactly what the earlier equation indicated so that is pretty neat. 
:o)


---

<center>
## Merchandise :D

![2018-06-07 (2).png](https://ipfs.busy.org/ipfs/QmSutAZm2v3GyfttdAW8KfJWKLUaD4mh8jugQG47c5dnaK
)
There is a [MathOwl shop](https://www.redbubble.com/people/MathOwl/shop) which sells my artsy fartsy stuff. If you got some spare money head over there.  I have really cheap stuff available like these [stickers](https://www.redbubble.com/people/mathowl/shop/stickers?accordion=product&cat_context=u-stationery&ref=artist_shop_product_refinement&asc=u) They are an absolute hoot.

---

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mathowl

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2019-03-08 20:29:18

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@minnowsupport · Mar 8 '19

<p>Congratulations!  This post has been upvoted from the communal account, @minnowsupport, by mathowl from the Minnow Support Project. It's a witness project run by aggroed, ausbitbank, teamsteem, someguy123, neoxian, followbtcnews, and netuoso. The goal is to help Steemit grow by supporting Minnows.  Please find us at the <a href="https://discord.gg/HYj4yvw"> Peace, Abundance, and Liberty Network (PALnet) Discord Channel</a>.  It's a completely public and open space to all members of the Steemit community who voluntarily choose to be there.</p> <p>If you would like to delegate to the Minnow Support Project you can do so by clicking on the following links: <a href="https://v2.steemconnect.com/sign/delegateVestingShares?delegator=&amp;delegatee=minnowsupport&amp;vesting_shares=102530.639667%20VESTS">50SP</a>, <a href="https://v2.steemconnect.com/sign/delegateVestingShares?delegator=&amp;delegatee=minnowsupport&amp;vesting_shares=205303.639667%20VESTS">100SP</a>, <a href="https://v2.steemconnect.com/sign/delegateVestingShares?delegator=&amp;delegatee=minnowsupport&amp;vesting_shares=514303.639667%20VESTS">250SP</a>, <a href="https://v2.steemconnect.com/sign/delegateVestingShares?delegator=&amp;delegatee=minnowsupport&amp;vesting_shares=1025303.639667%20VESTS">500SP</a>, <a href="https://v2.steemconnect.com/sign/delegateVestingShares?delegator=&amp;delegatee=minnowsupport&amp;vesting_shares=2053030.639667%20VESTS">1000SP</a>, <a href="https://v2.steemconnect.com/sign/delegateVestingShares?delegator=&amp;delegatee=minnowsupport&amp;vesting_shares=10253030.639667%20VESTS">5000SP</a>. <br><strong>Be sure to leave at least 50SP undelegated on your account.</strong></p>

properties (22)

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@pangoli · Mar 9 '19 (edited)

$0.03

Hey Mathowl, loving the SteemSTEM banner at the end of this post. It's lovely. thanks. 
I recognize those formula from my early days in college. Calculus was a tough mountain to climb at first but i got used to it and i loved it.

👍 mathowl

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@mathowl · Mar 9 '19

it is unfortunate that a lot of undergrads die on the calculus mountain. The view from the top is quite nice :o)

👍 votes4minnows, elviento

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@astrophoto.kevin · Mar 9 '19

$0.03

I really have respect for you mathematicians cracks!
I recently helped with some homework... Just the formulas for the exponential function cost me over an hour. :-D

👍 mathowl

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@mathowl · Mar 9 '19

https://i.redd.it/kbo23zdomfk01.jpg

👍 dustbunny, cuddlekitten, astrophoto.kevin, votes4minnows
👎 atempt, stimialiti, slowgrow, fersher

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@steem-ua · Mar 9 '19

#### Hi @mathowl!

Your post was upvoted by @steem-ua, new Steem dApp, using UserAuthority for algorithmic post curation!
Your **UA** account score is currently 3.426 which ranks you at **#7137** across all Steem accounts.
Your rank has dropped 2 places in the last three days (old rank 7135).

In our last Algorithmic Curation Round, consisting of 189 contributions, your post is ranked at **#139**.
##### Evaluation of your UA score:

* You're on the right track, try to gather more followers.
* The readers like your work!
* Try to improve on your user engagement! The more interesting interaction in the comments of your post, the better!


**Feel free to join our [@steem-ua Discord server](https://discord.gg/KpBNYGz)**

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@steemstem · Mar 9 '19

re-mathowl-mathematics-ideas-1--1552076959-20190309t190936033z

<div class='text-justify'> <div class='pull-left'> <br /> <center> <img width='85' src='https://i.postimg.cc/Bbrkt9rf/steemstem_utopian2.png'> </center> <br />  </div> <br /> <br /> 

 This post has been voted on by the **SteemSTEM** curation team and voting trail in collaboration with **@utopian-io**.  <br /><br /> 
 If you appreciate the work we are doing then consider [voting](https://www.steemit.com/~witnesses) both projects for witness by selecting [**stem.witness**](https://steemconnect.com/sign/account_witness_vote?approve=1&witness=stem.witness) and [**utopian-io**](https://steemconnect.com/sign/account_witness_vote?approve=1&witness=utopian-io)! <br /><br /> 
 For additional information please join us on the [**SteemSTEM discord**]( https://discord.gg/BPARaqn) and to get to know the rest of the community! <br /></div>

👍 ztz

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