First Derivative Test Second Derivative Test

Hi! We are back with another topic. We will be discussing the applications of derivatives, mainly in identifying critical numbers of a function. Let's see how the First Derivative Test (FDT) and Second Derivative Test (SDT) are used in the examples below.

In the example above, we are tasked first to find the first derivative of the function. After finding the first derivative, we equate the equation to 0 to find the critical numbers. After which, we will be making a table of signs (as seen on the second photo). This is to predict the behavior of the graph on the intervals. If the f'(x) turned out to be positive, the graph is increasing on the interval, while if it is negative, the graph is decreasing on the interval.

In the example above, we used the Second Derivative Test to find the concavity of the function at the given intervals as well as the points of inflection. After finding the first derivative, we are to solve for the second derivative to find the critical numbers. After finding the critical numbers, we will also be making a table of signs just like in the previous example. If f''(x) is positive, the graph concaves upward, but if f''(x) is negative, the graph concaves downward. Moreover, we concluded that at each critical number, there is a point of inflection since the concavity of the graph changes at each interval. We find the y-values of the points of inflection by plugging in the critical numbers into the original function.

Reflection (Week 2)

I am glad that I have understood what the First Derivative Test and the Second Derivative Test were. I was kind of confused at first especially with the table of signs, but when I consulted supplementary materials and videos, I was able to understand how it works. I think it is truly important to consult other references whenever I am having a hard time understanding a topic. Generally, I think I did well this week. I took the quizzes early and was satisfied with my scores. I will try my best to maintain this habit until the end of the quarter.

so I messed up the video involving the 4d Mandelbrot set because I accidentally clicked one button meaning its not true. The main math behind these values are still the Mandelbrot set, but when multiplying the complex parts I clicked a button which forced the values to always be less than 1 for the ‘y’ value. But if instead I clamped all complex numbers I get these images which look really interesting and I'm still trying to wrap my head around the form of these (probably more math than I could handle though).

Ill try to get videos out at some point (takes a while to render, even with a new pc) and higher res/better quality images some time soon.

Hi!

You all know me as the weird artist guy, but did you know I'm also a mathematician now? I was given the opportunity to give a talk for the UO directed reading program on any topic, and I chose to talk about the Fermat-Torcelli problem! My talk was 13 minutes and is preserved for all time on youtube, if you want to watch it. There are no prerequisites required and my abstract is in the description. Thanks!

“The road to the unified field theory is littered with the corpses of failed attempts”

~ Mathematical physicist Freeman Dyson.

Term Lengths

George Washington: 2865 days

John Adams: 1460 days [average: 2162.5 days]

Thomas Jefferson: 2922 days [avg: 2415.66667]

James Madison: 2922 [2542.25]

James Monroe: 2922 [2618.2]

John Quincy Adams: 1461 [2425.33333]

Andrew Jackson: 2922 [2496.28571]

Martin Van Buren: 1461 [2366.875]

William Henry Harrison: 31 [2107.33333]

John Tyler: 1430 [2039.6]

James K. Polk: 1461 [1987]

Zachary Taylor: 492 [1862.41667]

Millard Fillmore: 969 [1793.69231]

Franklin Pierce: 1461 [1769.92857]

James Buchanan: 1461 [1749.33333]

Abraham Lincoln: 1503 [1733.9375]

Andrew Johnson: 1419 [1715.41177]

Ulysses S. Grant: 2922 [1782.44444]

Rutherford B. Hayes: 1461 [1765.52632]

James A. Garfield: 199 [1737.2]

Chester Alan Arthur: 1262 [1714.57143]

Grover Cleveland: 1461 [1703.04546]

Benjamin Harrison: 1461 [1692.52174]

Grover Cleveland: 1461 [1682.875]

William McKinley: 1654 [1681.72]

Theodore Roosevelt: 2728 [1721.96154]

William Howard Taft: 1461 [1712.29630]

Woodrow Wilson: 2922 [1755.5]

Warren G. Harding: 881 [1725.34483]

Calvin Coolidge: 2041 [1735.86667]

Herbert Hoover: 1461 [1727]

Franklin D. Roosevelt: 4422 [1811.21875]

Harry S. Truman: 2840 [1842.39394]

Dwight D. Eisenhower: 2922 [1874.14706]

John F. Kennedy: 1036 [1850.2]

Lyndon B. Johnson: 1886 [1851.19444]

Richard Nixon: 2027 [1855.94595]

Gerald Ford: 895 [1830.65790]

Jimmy Carter: 1461 [1821.17949]

Ronald Reagan: 2922 [1848.7]

George H.W. Bush: 1461 [1839.24390]

Bill Clinton: 2922 [1865.02381]

George W. Bush: 2922 [1889.60465]

Barack Obama: 2922 [1913.06818]

Donald Trump: 1461 [1903.02222]

Joe Biden: pending

More than 2 terms: 1/46

2 full terms: 12/46

Between 1 and 2 terms: 7/46

1 full term: 15/46

less than 1 term: 10/46

TBD: 1/46

Washington served 2 full terms, but he was sworn in as president two months after he was supposed to start, so his 2865 is less than the full 2922. His immediate successor John Adams served one full term, but because the year 1800 was not a leap year, his 1460 is one short of the regular 1461. William McKinley would have served 2921 days for the same reason had he not been assassinated. 1800 and 1900 were not leap years, but 2000 was, so George W. Bush served a full 2922. Grover Cleveland served two terms, but they are counted separately as single terms here because they were non-consecutive; he served once as the 22nd president and once as the 24th.

As of 2021, the average length of term for the president of the United States is 1903.022 days, or 5.21019 years (5 years, 77 days).

If we graph out term lengths for the first 45 presidents, we can see that, on average, they're lasting longer than before:

Line of best fit: 10.047036x + 1649.718182 (x = president number)

If we instead look at average term length over time, we'll see that while term length has trended down overall since Washington (the blue line), it has actually been trending up since reaching a local minimum at McKinley:

Line of best fit: -13.131288x + 2230.558258

If we look only at the presidents since Teddy Roosevelt, we get this:

Line of best fit: 9.677111x + 1475.141058

We can use this to try and extrapolate for Joe Biden's presidency: if we looked only at the term lengths of his predecessors, we could predict he would serve for 10.047036x46 + 1649.718182 = 2111.88184 days, or 5.78202 years (5 years, 286 days), which is considerably higher than the average of 5.21019.

Looking at how it has changed over time, we would expect the average term length after 46 presidents to be -13.131288x46 + 2230.558258 = 1626.51901 days, which is too far below the actual value to be useful. The actual average was 1903.022222 after 45 presidents, so the only way it could drop to 1626.51901 after 46 would be for Biden to serve for -10816.12554 days. Negative 11 thousand days! Negative 30 years! We'll assume that's highly unlikely, so let's instead look at the figures since Roosevelt. 9.677111x46 + 1475.141058 = 1920.288164 days. For the average to come up from 1903.022222 to 1920.288164 means Biden would be expected to serve 2697.255544 days, a much more reasonable 7.38468 years (7 years, 141 days).

In the first half the presidency, presidents died in office like clockwork. It was so common that people started believing the office was cursed; William Henry Harrison was a general known for slaughtering natives, especially at the Battle of Tippecanoe in 1811, and he became the first president to die in office in 1841. From then on, every single president elected in a year divisible by 20 also died in office; 1840 Harrison, 1860 Lincoln, 1880 Garfield, 1900 McKinley, 1920 Harding, 1940 Roosevelt, 1960 Kennedy. It was known as the Curse of Tippecanoe, or Tecumseh's revenge, and when Ronald Reagan survived his assassination attempt in 1981 people believed he had "broken" the curse. For the record, I don't believe there ever was a real curse, it was just a coincidence that failed to account for Zachary Taylor who was elected in 1848. The only significance the "curse" had was that it meant few presidents survived their full terms, meaning the vice president would assume office and serve out the remainder.

In 1974, Richard Nixon became the first president to leave office early through resignation rather than death, cutting his second term short, leaving Gerald Ford to fill out the last 2 years and some change. Since then, every single president has survived their full terms, and most of them have successfully been re-elected (save for Carter, Bush Sr. and Trump). A president is more likely to be re-elected now than at any other point in American history; the first half saw tons of single-term and partial-term presidents, bringing the average way down before FDR made it spike with 4 terms (of which he served 3 and a month before dying), followed by relatively stable lifespans ever since.

With the introduction of the 22nd Amendment during Truman's presidency, it will be near impossible for any future presidents to serve more than 2 full terms, 8 years, 2922 days. It's still theoretically possible, just extremely unlikely; the amendment says you cannot be elected president more than twice, and if you serve more than two years of a term to which someone else was elected, then you can't be elected more than once. This means that if the president died or resigned 2 years and a day into their term, their VP could serve out the remainder and still be illegible to run for two full terms of their own, for a total of 3652 days (almost 10 years exactly), so the graph will likely asymptote somewhere between 4 and 8 years (closer to 8, as re-election is much more common).

That said, who knows? Maybe we're overdue for an irregular transition and five of the next ten presidents might not make it through their full terms. That would actually be normal for America; the last 50 years of stability have been the outlier. Regularity is irregular here.

I like how I didn’t know about the dream speedrun scandal thing for a while, because I think of myself as not that much of a nerd. Turns out, I’m more of a nerd than dream fans. I found out about it because Numberphile(a math youtuber) did a video on it, which is quite possibly the nerdiest way to find out about one of the nerdiest things I’ve heard about in the last year or so. So ya.

I have- so much work to do-

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Why is there so much power creep in math?

I have nothing to offer but would anyone be interested in doing my Calculus hw?

oooo, i did a math project on infinity and talked a bit about pascal's triangle

i don't know much about it, as it was just an example that i used, but it's so cool!!!!

Ahh that's fun!! Pascal's triangle is cool, infinity is cool, all of this is cool.

Did you know there are different types of infinity? Now that is cool.

Me: in Middle School using a scientific calculator to do the old B00B5/B00B135

My friend: PFFFFT

Me: Bow before me the god of inappropriate variables

do you ever do so much math that your brain gets really hot and your neck starts to sweat and then you finish doing math and get cold because your brain isn’t hot anymore or are you normal

Gonna read through a few chapters of this!

DAY 2/30 15 april 2021, thursday,

Ugh I hate today! Started really good, but by noon i got too lazy to do anything. Reactivated instagram for some stupid reason, regretted it, a friend helped me out by changing the password for me and now I’m free from that shit. Couldn’t study a lot, not even little, just a few questions of coordination chemistry and THAT’S IT! HATE IT! Thinking of resuming my workout regime from tomorrow to eliminate negative vibes and laziness. Hope so i will. I know todY wasn’t good but I won’t let it affect my tomorrow (I’ll try my best). All the best to me and to you if you’re also struggling with something similar. I believe we can make it through. Okbyeyeyeye.

Quote of the day: “जो लोग मेहनत का हाथ नहीं छोड़ते, क़िस्मत उनका साथ नहीं छोड़ती है।”

1^2 = one squared

1^3 = one cubed

1^4 = one hypercubed

...

1^∞ = one sphered

Classes Recap (2): Harmonic Analysis

This is a really strange class, it's meant to give an introduction to the theory of C*-algebras and describe the Gelfand transform as a generalization of the more popular Fourier transform over the usual Banach spaces. Despite having "analysis" in its name, right now everyting is being done very algebraically which is why I've decided to attend this course.

It's also a bit of a shame though because I'm afraid we won't see any "concrete" application of this stuff and since I barely know any analysis I probably won't remember much after the exam. Conversely the lecturer assumes we are all good analysts that haven't hear the words "quotient" and "ideal" since undergrad, so he's going extremely slow in some points while skipping ahead so fast in those analytical details and motivations, oh well!

Right now we are done with the more abstract topics though and we're going to see some Fourier analysis over groups, so Haar measure and this kind of stuff which I always wanted to learn about! Nothing too spectacular so far but it's a pretty cool class.

Bitch Tf??

Basic Math Symbols want to fuck King Boo!