Academic Work 2018-2019

• For social science this year I took two AP courses:  Macroeconomics and Microeconomics and studied a lot of economic theory and history as well. I received a score of 5 on both exams.

• I spent a good deal of time studying for the SAT as well this year. In the end I got a perfect score on the math section and only missed 30 points on the English and reading for a total of 1570.

• I ran High School cross country and I ran while refereeing for the town games both of which were a lot of fun.

• For English I read a number of classics including:
• Paradigm Shift
• The Abolition of Man
• Bambi
• Steelheart
• Euclid's Elements
• A Connecticut Yankee in King Arthur's Court
• The Sword in the Stone
• The Tempest
• The 5 S's of Money
• Emma
• Anthem
• The 5,000 Year Leap
• The Present Crisis'
• Alas Babylon

• For Math along with SAT review, I began studying Multi-variable Calculus and got about half way through an Online course which I plan to continue this fall. 

• For Chemistry I did some lab work with Mr. Mcbee, making dye sensitized solar cells and I began studying Organic Chemistry on my own. This summer I am interning at MIT in the chemical engineering lab making synthetic cells.

• I Traveled with my family to Spain and England. I especially loved walking among the old buildings of Oxford and all the history there. I also went to BYU with my father to sit in on some labs and lectures and to meet the faculty in the chemical engineering department there. This fall I plan on beginning my college applications for 2020.

Year in Review

This year I learned a lot and had plenty of fun doing so. Sometimes some materials were hard to get into, but I pushed forward and ended up liking them a lot more. This year I did coursework on TJEd High, an online program for deep learning and leadership education for youth. In it I read assigned books, and posted thoughts about them on the discussion forums, and discussed interesting things about the books and our ideas on the forums. The course also had videos where instructors discussed books and principles, and gave inspiring "mid-week mentoring". The course put a heavy emphasis on reading classics, thinking deeply about them, and discussing them with others. It attempts to help us better our thinking, enabling us to have the deep, creative, analytical, and innovative thinking that can help us solve problems in the real world.  It was really fun thinking, and discussing the books. You can learn more through these links: learning.tjed.org, tjed.org . The books chosen included math classics, government classics, and other classics.  Some of the books I read include:
Mr. Bowditch, the Giver, Turn the Page, TJEd for Teens, Sword in the Stone, Alas Babylon, We Hold These Truths, Animal Farm, The Final Summit, The Lion, the Witch and the Wardrobe, The Travelers Gift and The Tools of Money; we also read short works like: If, A Mathematicians lament, Invitation to the pain of learning, A world Split Apart, the Inner Ring and the Road not Taken. Some books I read but did not finish include: the Prince, Anne of Green Gables, 1913, The coming aristocracy, Bendigo Shafter, Great Speeches by Native Americans and some essays by Emerson. You can read my posts and discussion here ( you may need to change your browser for it to load). You can see some of the discussion's I have had in my last blog post.

TJEd high helps me learn how to think, and write, and also has helped me learn more about math.
The books measuerment and Arithmetic by Lockhart are interesting ways to learn math. I have read a bit of them , and have thought up fun problems like they encourage. Here is one of them which I have solved:

Take a circle. Put an equilateral triangle in the center of it, with all three points on the circumference. Then put a square inside the triangle, connecting it to one of the sides of it. What fraction of the circle's area does the square take up? What is Square Area/Circle Area ?

You need to use the 30-60-90 triangle law to solve it. 

I have also taken two AP exams, AP Computer Science A and AP English Language, and earned a 5 and 4 on them respectively. They gave me 6 hrs of college credit and Eng fulfilled first-year writing GE requirement for BYU. 

I have been running in the morning usually between 1-3 miles, and drawing with my grandpa. I learned Bulgarian with him earlier in the year. I entered the Youthful Expressions art competition and the NAA Young and Budding Artists show, won a prize in the Youthful Expressions, and a honorable mention in NAA. I have also been playing the piano a bunch, and refereeing at ECYSA. 

Deep Discussion in TJEd High

Here are some things I have discussed on TJEd High. 

Discussion on self-defense 

Me• I know we have the right to use force in self defense, but is it our duty under natural law? Would it be illegal under this law, to just submit to any cruel harassment of ourselves by others, or destruction of our health,life,liberty and the pursuit of happiness? Would it be legal in some cases, and illegal in others?

I think the answer to this corresponds with what they said in We Hold These Truths about setting a precedent. When you refuse to defend yourself or others, you set a precedent for how you or other will be treated in the future. I've learned this the hard way, especially as a babysitter-- if you don't make it clear to the kid that punching you is not okay, they will continue to punch you because they don't know any better.

In that light, I don't think it's ever illegal under natural law to defend yourself, or legal to let others injure/hurt you. Using force may not be necessary, but you should never submit to something that decreases your freedom, because it will lower the freedoms of everyone else no matter what-- and like I said above, it sets a precedent for other people.

Thoughts?

Me•I think it is important to forgive others, but that you can also use force in self-defense ( not revenge ) when someone is harming you. Do you think that you can forgive others for wrongs they have/are committed/committing against you, while at the same time using force within the bounds of natural law to defend yourself from their harm?

I think that it is possible, but also that it's much easier once you're not being harmed. There is a difference between forced used unforgivingly( I think I would call that 'revenge') and force used in self-defense.

A basic example is when you've been physically attacked. Part of forgiving your attacker is to use force only as long as it's necessary to defend yourself. This may get drawn out if they are stubborn about it, but if they choose to retreat, part of forgiveness is not going after them unless you're sure they're going to intrude on others' freedoms. Call the police, subdue the attacker-- but don't then sit on his chest raining blows on his face just because he attacked you. Once you're no longer in danger, it becomes a matter of resisting the urge to give 'an eye for an eye'.

What do you think?

Me•Since using force in self-defense is not against natural law, it cannot possibly break the law of forgiveness on it's own. I agree that as long as we only use force in self-defense, truly forgiving our attacker in our hearts is possible. This forgiveness then leads to no act of revenge, but just a desire to help others.

I'll push this a little deeper. What if the attacker keeps attacking you? You subdue him/her and turn over to the police, then he/she does it again and you subdue again. Then the attacker promises to keep attacking as long as he/she can, as many times as possible. At some point, does it become self defense to go after the attacker and stop the continued attacks? (I ask mainly because this is at times a reality in international geopolitics.)

Me•This is a very hard and interesting question, and I don't know if I have the right answer. In my mind self-defense has always been acting after or as an offense is being made, never assuming a person is going to attack, but always defending when they do. But in this case there is the perfect knowledge that the offense will be made, so I think that you are justified in trying to stop the continued attacks. Although this may seem like an offense, it really is a defense against further attacks. It is a much more efficient way of self-defense, and I think that when self-defense is done it should be done efficiently/ the law of economy.

Although what exactly are you constituting as stopping the continued attacks? I think that if this goes further than the cumulative efforts of offense by the attacker, then it would be more of a revenge and not self-defense. Only do the minimum necessary to protect oneself. So you can only harm the attacker as much as the attacker has harmed or would harm you*. Although this might be wrong, as it sounds more like an eye for an eye, tooth for a tooth approach. This does not sound very forgiving, and I think that to forgive someone, like I said, you would need to truly forgive them in your heart, and then act in the way your forgiveness leads you too. Giving an eye for an eye, as Sam Carrel said above, is a revengeful act, not a forgiving one.

But in the heat of battle do you not need to at least use as much force as your attacker is to defend yourself? So when you are going after someone to stop them from attacking you again, is it not like a hit in, say, a boxing match, where your opponent's guard is down, but you know he is going to try to hit you later on? Does not attacking him in this moment, not just blocking his blow, constitute self-defense? Or is self-defense only the blockage of attacks, never the offensive to win the fight and stop the attacks? To have subdued an attacker, I assume you would need to use offensive blows of defense? So why are they not allowed in a larger scale of time?

*But if you think that attacker would fatally harm you, I do not think that fatally harming them is right, for you might not actually know what would happen. Also it would be better to let them live, truly caring about them, loving them as yourself, and at most physically disable them, but again this would depend.

I like the reasoning here. I agree that going beyond the level of force used moves it into retaliation rather than defense. Good point.

This is a hard question, but you stated it correctly that sometimes a preemptive action can be defensive, because past attacks show that future attacks will continue. Still, this will always be a fine line. Leaders who take such action will always be criticized by some. And leaders who don't take such action will be roundly criticized if another attack does come. One reason leadership is lonely.

Deep discussion.

Me• Imagination is a key part in fulfilling our missions, in truly thinking and creating, inventing and growing. I also have been thinking about the power of remembrance, which I think is part of imagination. Remember the end of the Giver, where the memories of sun and warmth helped give Jonas and Gabriel the strength to carry on? I think that is so applicable in our lives. In TJEd for Teens it says "there are times when the sacrifices seem great and the rewards are either a memory from the past or a hope for the future. But selfless service to the right person ... or cause brings about a sense of purpose and intrinsic reward that transcends the...immediate gratification" ( pg 60 ). Imagination through remembrance of the past, and envisioning a hope for the future can help get us through hard times by helping us focus on what really matters, to continue pressing on in our journey to fulfill our mission, and not give up or give in.

There are many wretched things that ruthlessly attempt to drown our imagination, creativity, and love for learning. There was a discussion on that last week, with Sam Carrel and others. They include things like the "conveyor belt" school system, public opinion, misguided parenting, addictive and/or leisurely things like social media, wasting time on smart phones, TV, pornography, alcohol, drugs etc. The school system, parents, public opinion come in and tell kids this is the way it should be, with forced, boring, school, that art will never get you a good job, that there is not much hope for entrepreneurial, innovative endeavors and discoveries, so stop using your imagination and get on your homework! This turns the students off to learning, and drives away their inherit curiosity, and pretty effectively blocks them from their potential to love learning, and discovering, thinking and creating. Instead they painfully endure the gruel of school and then turn to their devices, and to mindless addictive patterns, attempting, yet failing miserably to fill a void in their lives. This is super sad! So my question is how can we imagine more, be more creative and truly learn? I think that first of all we need to have our allegiance to God and His perfect goodness, and His perfect love firmly and truly planted in our hearts. Only through this can we overcome our challenges and fulfill our missions. I also think that just pondering, wondering, and thinking with sincere, deep curiosity about things can really get us going on the path of true education.

I think electronics like computers, tablets, and smartphones can take from us more than almost anything else if we let them. We spend so much time on them texting people, browsing the internet, viewing social media, or "learning" about stuff that we actually know less about because we don't have experience trying things for ourselves or talking to people face to face. Lets imagine someone who spends almost all their time on their devices. They might see hundreds of photos, videos, and stories about amazing experiences, but those all pale when compared to actually going to a waterfall, camping on a verdant mountain, going down a zip-line over a river, cliff jumping into a lake, snorkeling in a coral reef, or so many more thrilling, vibrant activities. They might have seen a bunch of cooking videos, but what good does that do them if they never cook? They would lose their communication skills, ability to read body language and the pleasure of talking face to face and doing things together with people, instead substituting bland texts to communicate. They might see beautiful pieces of art and hear great songs online, but they might never learn to play an instrument or try to write a song. They might read inspiring articles or about amazing technologies and scientific discoveries, but they have never published an inspiring article, tried to invent a machine, or investigated something new. What have they learned or done after all? I know this is an extreme example, but I think it gets the point across. If you don't build off what you learn, you lose it or it just becomes like a piece of meaningless trivia, useless facts. You lose the essence of an activity in losing the experience. It really worries me because our devices are becoming such a huge part of our every day life. This situation I've described might not seem so extreme pretty soon. I think that we as a society are in danger of losing our imagination, creativity, innovativeness and our intellects because we don't produce our own things or experience things for ourselves often enough. I think electronics can be a powerful and beneficial tool, but if we substitute our devices as our window to learn and see the world instead of experiencing and producing things for ourselves, we won't be learning that much after all.

Me•In the Giver Jonas is playing catch with Asher( pg.30) and "it was effortless and even boring" for Jonas. But then he notices the shocking phenomenon of the apple changing in mid-air, and stares at it in disbelief. He goes on to bring it home and study it, trying to figure it out, to discover what had happened and why? But then a brutal law, and culture stopped him from continuing in his search for knowledge, his journey of discovery.

Regarding Jonas's attitude towards playing catch I was wondering are all effortless things boring at first lazy glance? Or can there be some excitement in good things that are seemingly effortless for us? I say good things because there are some effortless addictive things that cannot exactly be termed boring, and I am not talking about those here.

I think that these seemingly effortless and boring things can take on a new wonder and burning curiosity in our minds and hearts if we keenly observe them. If we look deeply at these things we can discover mysteries and wonders, for which we passionately hunger to understand. Engaging in and completing this path to knowledge requires great effort, and what seemed to be effortless and boring becomes interesting, engaging, hard ,demanding and fulfilling. We just need to get a glimpse of the enigmatic wonder and then let our passion and curiosity drive us to truly learn and discover.

But how do we gain this momentous revelation? Do we, as Jonas did, just happen upon it by chance, as without foresight or seeking we just happen to be paying attention to a seemingly ordinary phenomenon? Or do we earnestly, steadfastly seek these wonders in seemingly boring things everyday of our lives, never taking things for granted, filled with optimism and the desire to learn and discover? I think that the latter method is much more effective, fulfilling and enabling. When we earnestly seek and work day after day for learning and discovery, we are all the more fulfilled and joyful when we do discover things, all the more interested to learn more. as TJEd for teens tells us, we can come to love learning by spending time on it. When Jonas just stumbles upon the "change" of the apple, he has trouble believing the Phenomenon. Do you think that if we discover wonders while actively seeking them, we will have an easier time believing and not ask others if they observed them, with a shaky confusion of the truth and attempt to ignore it?

Another way I think to better discover wonders is to focus on serving others. Someone once said "love is time". Jonas was spending time with Asher doing something that as boring to him, but which Asher enjoyed. As we strive to discover and learn because we want to serve and uplift, we will better be able to learn, and have more incentive for it. Also as we engage in service,even if it means doing "boring" things with someone which they like, we can discover and experience not only mysterious phenomena, but also the wondrous joy of service and friendship. This experience of charity can lead us to serve more, thus continuing the glorious round of kindness.

To continue in the journey of passionate, charitable, earnest learning and discovery, we need to be able to continue seeking knowledge, and applying what we learn, to better serve others. But what if there is a terrible law or custom, even culture, that blocks are path? The announcement in the Giver to eleven's who were "hoarding [snacks]", and the cultural and societal system stopped Jonas from continuing his journey in seeking to understand. Are there things in our society that do this, for example the "Fashion of thinking" described in Solzhenitsyn in "A world split apart"? Are there societal blockers of learning and discovery present today? Later on in the book Jonas is able to continue to learn about the "changing" phenomenon, and so in a sense he is following a voyage of discovery. But the society forces him to use this passion, interest, and talent for an evil, dictatorial, and degrading cause. Are there things in our society that also do this? Jonas seeks to use his discoveries for good, and overcome the societal blockers. What are some ways we can overcome ourselves, and help others overcome, our societal blockers?

Whew! I got a lot out of this! I ended up with 3+ pages of notes, so I'll focus on your ending questions. You make a great point!

(The lists included are not according to priority or strength-- I just felt that numbers would make it easier to read.)

'Are There Societal Blocks To Learning Today?'

Yes.

1) Popular Opinion

2) Fear

3) Shame

4) Mediocrity

5) Negative, Demeaning, and Stress-inducing expectations

6) The Belief that there are 'good' and 'bad' jobs (without taking into account the job's effect on someone's life mission)

7) Misguided Parental Pressure

8) Lack of Leadership Education

9) People following the wrong allegiance.

'Are there things in our society that force us to use our passion, interest, or talent for evil, dictatorial, or degrading causes?'

No. There are certainly those who try, but no one can force you to use your talent for evil.

'What are some ways we can overcome (and help others overcome) our societal blocks?'

1) Get a Leadership Education

2) Become a Mentor

3) Fulfill your Mission

4) Be an open advocate for Truth, Greatness, and Love (as well as other virtues)

5) Treat Everyone with Love and Respect, even if you don't understand them.

What things would you add to this list? I'd love to hear them!

Thank you for making this post, it really got my mind working!

Me•Awesome! True thinking is wonderfully exhilarating! I totally agree with all of those blocks. I think that along with lack of leadership education, there is a terrible mess of "conveyor belt" education. This, along with "Misguided Parental Pressure",really turn students off to true learning and discovery, and points them towards the mediocrity , fear and other negative emotions and thoughts you pointed out. This leads them to follow the wrong allegiance, and become suffocated in the sinister mass of addiction and leisure which I think is another block. They become slaves to social media, TV, browsing the Internet, pornography, and possibly, although less socially acceptable, alcohol, drugs, tobacco etc.. They really need the leadership education and right allegiance to rise above mediocrity and misery, and truly be happy and fulfill their mission.

With regards to the question about forced evil channeling of talents: You totally caught me off guard there! I guess I should have used better "precision of language" ha ha!

Do you think that Jonas was actually completely forced to use his talents/passion for evil? What is complete force? I think that in real life there is never complete force, like you said we can still choose. So then what is government, if like in the Law and We Hold These Truths it is force? Does government only give consequences for choice and make choice in some ways really hard, but not actually make it impossible?

Say in Nazi Germany there were Nuclear research Scientists who were threatened with death to themselves and their families if they would not work for the government. But if all the scientists refused to work, the Nazis could kill them all they wanted to ( although this would be excruciatingly terrible), they still would never HAVE to work for them.

Is this force? Is force the execution of consequences of choice (making a choice one way very negative-in some ways, usually physical-and another opposite choice very positive-again in some ways usually physical/temporal-) , and the austere complicating of the ability to make certain choices?

If so is there anything in our society that uses this "force" to use people's passion, interest, or talent for evil, dictatorial, or degrading causes?

Is all real-life force physical, or can there be emotional and mental force like in the Giver?

Take the average college education major student, who has a passion and talent for teaching. Say that they have never head of, or never thought much of any other system of education than the modern "conveyor belt" school system. Are they being "forced" ,in the above defined sense, to use their talents and passion for uneducated schooling, for a very block to true learning and thinking? Or are they rather being taught wrongly, misguided by ( those blockages you identified ) their parents and public opinion? I think that the school system and higher ed are using,training, even mentally misdirecting, these them to use their passion and talent to keep society in this terrible system of schooling that blocks true learning. But is this force? Or just ignorance? Are they getting forced by societal blockers to do the wrong thing, or are they choosing to follow them? Can they be forced to be ignorant of the better way, leadership education? Or is there always a choice? Is ignorance force? I think that ignorance forces you to not be able to do what you are ignorant of, until you learn it. If say they have a chance to accept leadership education, but don't because of the influence of blockages, are they actually choosing to not do it, or are the blockages influencing them in such a way in which they are being "forced" ( in a way in which they are mentally influenced so to think that there are negative consequences in leadership education, and positive ones in "conveyor belt" schooling, and/or think that wither way it would be to hard to actually do leadership education) to not do it.

This is kinda off a tangent, but it goes along with emotional and mental force:

Is the choice to get addicted like the choice to go to Sameness in the Giver? If they ( whoever that is ) chose to go to Sameness, is it actually forcing them, or only their descendants? Is addiction forcing you to continue in it, or is there still choice?

One possible example of being forced to do something in our modern day is tax. I'm not saying taxes are bad, in fact, I think that they are essential. They are just being misused. Right now the government uses taxes to fund public schools, welfare programs, and other institutions that do not fall within the proper role of government. Some would say that because of taxes, I'm being forced to support a public school math program that I do not agree with. However, right now I could just leave the country and not have to worry about the tax. I could also marshal all my time and resources to abolish the improper uses of taxes I perceive through voting and legislation changes, which might not work. Using your definition of force I am not being forced to pay taxes, but just like the technical definition of a line in math, you can't always use the technical definition of force and still be practical. Absolute force is a thing only attained by Gods.

What you added actually reflects a few of the thoughts I had last night when I was trying to argue it out in my notebook (I was tired enough that I didn't want to try to put the argument up for fear it wasn't really thought through). Generally it does seem that you can force someone to do one thing by making them completely ignorant of any other options(especially 'better' ones). And while you make the first choice to participate in something addictive, the changes that happen in your brain because of an addiction do seem to force you to perpetuate that behavior. I just don't agree with the idea that you can accuse anyone of forcing you to do something. So at this point, whatever gut disagreement I have with this is probably just stubbornness, but it still bugs me. I've always liked the idea of having the ability to choose my actions and reactions and hearing the possibility that I can be forced into something gives me a feeling not unlike catching a whiff of my dog's puke. Like I said, it could be that I'm just being stubborn.

I do think there is emotional and mental as well as physical force. I would add 'spiritual' to that list as well, because some things just move you beyond an emotional, physical, or mental place. Let me try to list examples (this list is NOT exhaustive...):

PHYSICAL:

pushing

shoving

using a weapon to threaten (gun, knife, club, spear, etc.)

grabbing

pulling

violence or threat of violence

EMOTIONAL:

Guilt

Shame

Anger

MENTAL:

Guilt-tripping

Extreme choices

Propaganda

Convoluted and Over-complicated Arguments

Mockery

Demeaning of Opposing Ideas or Arguments

Dismissiveness

Ignorance vs. Knowledge

SPIRITUAL:

Church Mandates

The Spirit, God, Supernatural Beings (if you believe in them)

Gut Sensations

Perceived Superiority or Authority to Command

Visions

It's hard to sort them into categories, because if you can come up with something that seems to have a root cause in one force, you're bound to see reactions coming out of the other forces as well. I picked these because(in my mind) they tend to lead to an action.

Again, I'm being very vague here. Part of it is that I'm out of my comfort zone. Most of it is that I haven't done much research into other arguments about it, so I'm building and guessing on a pretty weak foundation. I don't want to act as if anything is definite because I have no idea whether it is or not!

What are your thoughts on this? All I know is that I need to think about it more!

Year in Review

MATH

This year I finished Pre-calculus and Calculus BC and today I got my AP score (a 5!!! Hooray!!!)

SOCIAL STUDIES

I took a U.S. History Class up to 1877 which was interesting from BYU Independent Study this year and got an A-.
But even more fun was going to historic places like Plymouth and D.C. where I got to see The Libray of Congress and the Lincoln Memorial and all the different historical sites.

I also was assigned to speak in Church and I got to speak about genealogy and family history which is something I have done a lot of this year.
I traced my family line back to many famous people and then back to Adam.

As scouts we did backpacking and overnight camping in freezing weather in Ipswich, but it was a fun adventure.

LANGUAGE

This year I read the Chosen which I liked reading about the Jewish faith and how it affected the lives of two teenage boys.
I also read the New Rangers Apprentice Series, The Five Kingdoms Book 4 and DragonWatch.
Along with my AP studies I got to read some fun books about Math and Physics history which made me think maybe I could discover something like these famous scientists and mathematicians some day.

As a family we studied classics which my sister posted on her blog and I studied Bulgarian language and writing with my grandpa and I finished the New Testament.

SCIENCE

I started tutoring my brother in Chemistry this year and plan to continue next year.
I also studied how to make cheese from goat milk and it turned out pretty good.
I helped my grandpa build a new chicken coop and I studied AP physics C Mechanics for college Credit.

I got a 4 on that exam.

ART

This year I made a couple of 3D animations and even won an award in an online art contest for one of them.

My drawings this year were mostly portraits and religious subjects like Jesus and the apostles and the Nativity.
As a family we became members of the MFA and I got to visit there and study Monet and later we went to the Isabella Stewart Garnder museum and I learned about the stolen Rembrandt paintings.
We read chasing Vermeer after that which was also about an art heist.

MUSIC

I've done a lot of self teaching on the piano and learned like 20 new hymns plus I just recently started composing music now that I understand the chords and the way music fits together.
I play the piano in church a lot and sing while I play.

EXERCISE

This year I played 8th grade soccer with Newburyport, but next year I will be joining the Triton HS team.
I also got me US Referee certification and officiated a lot of games here in Essex County

COLLEGE PROGRESS

As of now I am entering 9th grade and I have 30 college credit hours from AP classes and Independent study.
I hope to take at least 4 more AP courses and to finish my general education requirements by the time I am 16.

Differentiation

A Derivative of a function is the instantaneous rate of change of that function at a point. It's the limit as the change in x approaches zero of the average slope of a graph. The operation of taking a derivative of a function can be written in different notation, like f"(x) or d(f(x))/dx . The rules for taking derivatives of some basic functions are outlined below:

if n is a constant and u and v v are functions of x
                                                 
when f(x)=                                             |                                                    f '(x)=
un                                                           |                                             u'(x)ᐧ (n)un-1 
nu                                                           |                                             u'(x)ᐧln(n)ᐧnu
n                                                             |                                                 0
uᐧv                                                          |                                          (u'(x)ᐧv)+(v'(x)·u)
u                                                             |                                        (u'(x)ᐧv)-(v'(x)·u)
v                                                             |                                                   v2 
u(v(x))                                                    |                                          u'(v(x))ᐧv'(x) (d(u(v(x)))/(v(x)) )(v'(x))
sin(x)                                                      |                                                   cos(x)
cos(x)                                                      |                                                   -sin(x)
tan(x)                                                     |                                                   (sec(x))2
cot(x)                                                     |                                                    -(csc(x))2
arcsin(x)                                                 |                                                 u'(x)ᐧ    1    
                                                               |                                                        √1-u2
arccos(x)                                                 |                                                     -u'(x)ᐧ    1    
                                                               |                                                               √1-u2
arctan(x)                                                 |                                                     u'(x)ᐧ    1    

                                                               |                                                               1+u2
ln(x)                                                       |                                                       1/x
log base a (x)                                         |                                                     1/x(ln(a))

to differentiate a function like xx,you need to use the natural log rule: ln(xu) =uᐧln(x).
When given a table of values you can estimate the derivative at a point at or in between input values given.

If a function is differentiable on an interval then the function is continuous on the interval but if the  function is continuous on the interval then the function is not necessarily differntiable ( i.e. functions can be non-differentiable and continuous ( with sharp cusps and the like ).

The MVT states that if a function is continuous and differentiable on an interval [a,b] then the is some point c where f '(c) is equal to the slope between a and b or the average rate of change over the interval. Rolles theorem is a special case of the mean value theorem when the average rate of change is 0.


Do differentiate parametric functions you take the derivative of y with respect to t and then divide id by the derivative of x with respect to t.

To differentiate polar functions you use the definitions y=rsinθ and x=rcosθ and then use parametric differentiation.

Two take second derivatives ( f "(x) ) you take the derivative of the derivative, to take third derivatives you take the derivative of the derivative of the derivative or the derivative of the second derivative and so on.

To take derivatives of inverse functions at a point, you take the reciprocal of the derivative of the corresponding point.

When a derivative of a function is undefined or equals zero you have got a critical point of that function. You can use derivatives to determine if a function is increasing( i.e. the derivative is positive ), or decreasing ( negative f '(x) ), concave up or concave down ( the second derivative is positive for concave up and negative for concave down. ). These determinations can be used to find maxima or minima of functions. They can also be used to find where the concavity of the graph changes ( a inflection point). There are multiple methods to do this including finding critical points and evaluating the second derivative at the points ( second derivative test, if it is positive it is a local minimum , if it is negative it is a local maximum ), or evaluating f ' before and after the critical point.
Finding local maxima and minima and points of inflection are all part of derivative applications, like useful thing you can use differential Calculus for. Approximating the value of a function using slope tangent line approximation is also part of Derivative applications. Basically you use the equation
y-y1= f '(x1)ᐧ(x-x1). Where you can approximate y values ( the y ) of x values ( the x ) near the  x1 value. The y1  is the y value at the x1.

Another application of derivatives is related rates. Which is when you have a object, and some measurement of it is changing at a given constant rate, while some other property or properties is/are changing at a non-constant rate. You use an equation that relates the two properties and then solve it for the rate of change of the non-constant one at some point in time.

One application of differential calculus has a bunch to do with Physics, namely the analyzation and study of the movement of a particle given its function of either , position, velocity, or acceleration with time. The derivative of position is velocity, the derivative of velocity is acceleration. Which means that the slope of a position versus time graph is the velocity and the slope of a velocity versus time graph is the acceleration. These definitions can help us analyze and determine when a particle is moving in a certain direction , when it is not moving, when it is slowing down or speeding up ( when the acceleration and velocity are in the same direction then the particle is speeding and when the velocity and acceleration are in different directions the particle  is slowing ), and how fast or how much a particle is accelerating at a point.

Differential calculus can also be used to solve optimization problems, in which you write a function for a thing to me maximized or minimized and then take the derivative(s) of that function to find the local an global maxima or minima.

Limits and Continuity

This is the first big idea in AP Calculus BC, and it lays the basis for most of Calculus. So Calculus is interested in finding the values of things that are infinitesimally small ( like the slope of a point or the area under a curve) and to find these values limits are used. Take for instance the area under the curve, to approximate it you can divide the curve into rectangles, but the area will only be that an approximation. The smaller the width and the more rectangles there were the better the approximation would be. So what calculus does is take the limit of the sum of the rectangles as the width of the rectangles approaches zero ( or the number of rectangles approaches infinity ) which is the exact area. I'll write more on finding areas under curves when I get to big idea 3 Integration.

So a limit of a function as the input approaches some value is really the value the output or the function approaches. So the function doesn't have to be defined at a point to have a limit there.
There are many ways to find limits of function, including using graphs and just plain algebra. Limits can come from two different directions , the negative direction ( or from the left side of the graph , i.e. the value the function approaches from input values less then the value you are taking the limit at ) and the Positive direction ( from the right side of the graph with values larger than the desired input value ). The notation for the two different types of limits looks like this:

lim f(x) = L                                                                            lim f(x) = L     
x→c -                                                                                      x→c +

From the negative direction                                         Positive Direction

So the → means approaches and the x→ c means as x approaches c . The Limits from each side do not have to be the same, and when they are not there is a weird jump in the function.

Some times to approximate limits algebraically you just have to plug in the c value for x in the function and it gives you the limit. But if this limit is undefined ( ∞/∞  or 0/0 ) then you are going to have to try some different methods. If the function has a quotient , then divide the top and bottom by the highest power of x , and then take the limit. There is also a way that you can factor a function out and then cancel expressions leaving a defined limit. A very useful rule for evaluating limits with indeterminate form  ( +∞/∞  or 0/0 ) is L'Hopitals rule which states that if you have a limit in indeterminate form , then you can take the derivative of the top and bottom ( separately  ) and then take the limit of it.

lim    f(x) =+∞/∞  or 0/0                then           lim    f(x) =lim  f '(x)           
x→c g(x)                                                           x→c g(x)  x→c g'(x)     

A function is continuous if ( all of the following conditions are met ):
            lim f(x) = lim f(x) = L                                                        
            x→c -         x→c +

( so the lim f(x)  is defined and equals L )                                                                             
             x→c 

lim f(x) = L =f(c)                                                                               
x→c 

( or in other words the function must be defined at x=c, and the limit of the function as x approaches c must be defined and equal the function's value at x=c )

A point that needs to be covered here is that if the limits of the function from the right hand and the left hand are not the same than the limit of the function at that point is not defined ( the limits from either side have to equal each other for the whole limit to be defined. )

Asymptotes  are places where either the limit as x approaches infinity equals that value ( for y value asymptotes or horizontal asymptotes ) or the limit as x approaches that value equals infinity ( for x values , or vertical asymptotes )

Continuity allows us to have some theorems, like the intermediate value theorem and the extreme value theorem ,and later on the Mean value and Rolles theorem. The Intermediate value theorem states that if a function is continuous on the interval [a,b] then there is some value c that if between the values f(a) and f(b) for the function. The extreme value theorem states that if a function is continuous on an interval it will hit a highest value on an interval. The Means value theorem ( MVT ) and Rolles theorem which is a special case of MVT have to do with derivatives.

There are different types of discontinuity , jump, removable and infinite discontinuities:

jump discontinuities happen when the right hand and the left hand limits don't equal each other ( the function " jumps " at the discontinuity )

removable discontinuous are when the limits equal each other but not the functions defined value.

Infinite discontinuities happen when functions approach positive or negative infinity at a point ( which actually means that their limit and value are undefined )

Why Calculus ?

So you may have heard that I took the AP Calculus BC exam a little while back. Your reaction could have been ugggh ! that stuff is so boring and complicated , why would you ever do that?
( or it could have been Awesome that stuff is so cool! If it was something like that then you do not have to read this but you still can ) I mean when would you EVER have to use that in regular life ?
My article today strives to answer at least part of this question " why calculus " and how it is awesome!

Calculus , like any topic, can be made boring and dissatisfying through mental prejudice and unenthusiasm. When you put it into your brain ( quite unfairly ) that this is not going to be fun and it will never be fun , in a nutshell that you dislike it. This approach must not be the way you look at calculus if you ever want to come to love it and have fun doing it, but rather you at least have to have a open approach to the subject. Preferably you should have a positive, optimistic approach to the subject having a hope that it will be fun and exciting.

What makes Calculus fun is that with knowledge you can solve problems, thereby thereby using your brainpower to get answers. Calculus is really not fun when you do not know how to do it, so learning it is an essential part of the process. Learning can be hard at times and so you need to press forward continue trying and rely on the infinite grace and mercy of our Heavenly Father to learn and understand calculus. Once you learn the material , answering problems becomes easy ( or at least doable) and the excitement of being able to find the answer emerges.

Learning Calculus can be fun to though , once you finally understand a new definition , or theorem it enables you to do something new in math that you did not know how to do before. It opens new doors in your ability to solve problems , and things that seemed impossible now become second nature.
For example , say you knew basic algebra and how to find the slope of a graph, but then you think of a curved graph ( say ln(x) ) and wonder how to find out its slope at a defined point. You can increasingly approximate the slope at the point but not quite reach  it using algebra, but with differential calculus you can literally find the slop of a point of a curved graph. For example at x=2 dy/dx of y=ln(x) or the infinemestial slope at the point x=2 of the natural log of x is equal to the 1/2.

While I was learning how to take the volume of irregular solids, I wondered if you could somehow use related rates to find the rate of change of the volume of an irregular solid where the upper integrand  is changing. So what this means is that say you have a graph of a function ( for example y=x2) and then you rotate the graph around an axis ( say the x ) so what you get is a solid with a volume that can be determined using integration and the property that each slice of the solid is a circle. The area of each of the circles is 𝛑r² and the one thing that changes between the each of the circles is the radius. The radius at each point or for each circle is equal to the value of the function at that point ( since the diameter is twice the function value ) or x²    
 If you sum the areas of every circle in the range that you want ( say from 0 to 2 , which numbers are called the lower and upper bounds of integration, or the integrands ) then you get the volume. To sum the areas of every circle you have to use an integral which is and the limit as the number of circles approaches infinity or the limit as the width of the cylinders approaches zero of the sum of each of the cylinders. It would be 𝛑∫(x²)²dx  since 𝛑(x²)² is the area of one circle and the sum of all of them is the integral. Using integration rules ( ∫xn=
xn+1

___)  Then the volume would be  𝛑(2)5

^{n+1                                                                     5}   

from 0 to 2. The rate of change of this volume was a bit hard for me to determine how to find when given that the upper integrand was changing at some rate, but I eventually got it , with Heavenly Father's help. So the fundamental theorem of Calculus state that the derivative of an integral of a function with respect to the changing upper integrand is equal to the function of the integrand.
By the chain rule the derivative of a composite function f(g(x)) with respect to x is equal to the derivative of f(g(x)) with respect to g(x) multiplied by the derivative of g(x) with respect to x. So say that you were given the rate of change of the upper integrand and asked for the rate of change of an irregular volume produced by rotating f(x) around the x axis. The way to do this is multiply f(x) by dx/dt or the function of x multiplied by the rate of change in x. This is a simpler way than what I did. I used a differential equation to solve for the upper integrand in terms of t and then plugged that into the integral as the integrand and than applied the fundamental theorem and the chain rule.

Once you have mastered Calculus answering problems gives you a "zing" to the brain and it can get mildly addictive, so do not do it to much.