12 of 33 Items .... Source: Curmudgeon

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Help me understand the fundamental definition of concavity.
Consider the function \(y = x^4 \)

Can we say that is concave up from \((-\infty, +\infty)\)
or do we need to say \( (-\infty,0) \cup(0,+\infty) \)?

Here is it, if you want to see a visual.

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Is money a discrete variable or a continuous one?

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It's not included in the PEMDAS Order of Operations ...

Should \(a^{b^c} = ({a^b})^c\) or should it be \(a^{b^c} = a^{(b^c)}\) ??

Does \(3^{2^0}\) equal 1 or 3?

Let's just consider easy numbers {1, 2, 3, 4} so we can explore. What's the probability that the two methods arrive at the same answer?

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An article about the Netflix price hike:

Netflix upped the rate of its "two-screen" plan from $8.99 to $9.99. This plan lets you watch two HD (or regular) streams simultaneously on different devices. Netflix's "one-screen" and "four-screen" plans remained unaffected.

Here's the thinking: If Netflix subscribers start dropping from the two-screen plans to one-screen plans, this will make them a lot less likely to share their password. No one wants a little offhand altruism to actually affect their viewing habits. That would be bad. If the resulting lack of password-sharers pushes more people to subscribe, then its worth the dollar per subscriber Netflix loses when a user downgrades because of the price increase.

Find the NetFlix subscription numbers to see how the price hike changed things. Was this a good move?

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You've all seen Plinko and its variations.  Here's one:




This one has a normal curve drawn on the background. Should it be a normal curve or more triangular like the arrangement of the pins?

Would it still be in this shape if the pins were arranged in a rectangle?

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Go to this page at Desmos, and hit play. Explain what you see and why it is behaving that way.













There is obvious symmetry in the figure at each value of a but should we be defining something new to describe the sequence of images? A dynamic symmetry, of sorts?

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This 1961-era 1MB hard drive is not large enough to store this picture before it was condensed and converted to a JPG. It is the size of a counter-top taoster oven.




This 28MB IBM 1301 machine cost $110,000.00, weighed half a ton, and was the size of three large refrigerators side by side
.



And now you have this to carry around in your pocket.



Go here:
http://www.jcmit.com/flashprice.htm

Can you compose a simple rule about prices over time?
Data copied here:


X (decimal date),Y (US$/MB)
2003.17,0.258
2003.58,0.191
2003.75,0.219
2004,0.191
2004.17,0.238
2004.33,0.266
2004.42,0.223
2005.5,0.0762
2006.33,0.0219
2006.67,0.0156
2006.83,0.0146
2007,0.0122
2007.17,0.0104
2007.33,0.0105
2007.5,0.00976
2007.67,0.00805
2007.75,0.00763
2007.83,0.00732
2007.92,0.00763
2008,0.00458
2008.08,0.00463
2008.33,0.00415
2008.5,0.00354
2008.58,0.00305
2008.67,0.00311
2008.83,0.00183
2008.92,0.00214
2009,0.00162
2009.08,0.00168
2009.25,0.00204
2009.42,0.00214
2009.5,0.00195
2009.58,0.00177
2009.75,0.00214
2009.92,0.00226
2010,0.00244
2010.08,0.00226
2010.17,0.00244
2010.33,0.00201
2010.5,0.00193
2010.58,0.00176
2010.75,0.00177
2010.83,0.00146
2010.92,0.00119
2011,0.001403
2011.08,0.001251
2011.33,0.001342
2011.42,0.001219
2011.67,0.001007
2011.75,0.001037
2012,0.000824
2012.08,0.000824
2012.25,0.000793
2012.33,0.000702
2012.58,0.00058
2012.62,0.00061
2012.83,0.00061
2013,0.000458
2013.08,0.000518
2013.33,0.00058
2013.42,0.000564
2013.58,0.000564
2013.62,0.000564
2013.75,0.000534
2013.83,0.000542
2013.92,0.000548
2014.08,0.000457
2014.17,0.000473
2014.25,0.000366
2014.42,0.000389
2014.58,0.000335
2014.67,0.000345

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R3
There are three red hats and two black hats on a hatstand.

Melba, Fran, and Brent line up in a single file line, and the mad Hatter places a hat on each person’s head and tosses the rest into a box where no one can see them.

The Mad Hatter takes off Brent's blindfold. Brent, who could see Fran’s hat and Melba’s hat, says, “I don’t know what color my hat is.”

The Mad Hatter takes of Fran's blindfold. Fran, who could only see Melba’s hat, says, “I heard what Brent said, but I don’t know what color my hat is.”

As the Mad Hatter moved to take off Melba's hat, she said, “I don't need you to take off my blindfold. I know what color my hat is.”

What color is Melba’s hat and how does she know?

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There's a neat game you can play with the English language: Place the word "only" anywhere in the sentence. How does it change the meaning?

Only she told him that she loved him.
She only told him that she loved him.
She told only him that she loved him.
She told him only that she loved him.
She told him that only she loved him.
She told him that she only loved him.
She told him that she loved only him.
She told him that she loved him only.

What would it look like if we were trying to create a math problem that worked the same way, except "only" is replaced by the equals sign?

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Which is true?

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What can we do with this? There doesn't seem to be enough information.


3

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What's the easiest way to fix this?