Finally, a whole year is gone. There have been many things that we learned in Calculus, like derivatives, integrals, limits, related rates, optimization, angle of elevation, and other mathematical processes that could help us in our near future.
Calculus, for the most part, came easy to me, whether it was memorizing the long list of differentials to applying our knowledge of graphs to find how many cars passed by within 3 hours, it was like a fun, challenging puzzle. One particular subject we were learning that through me off for a bit was integrating where the area was a base and the cross sections were squares or semi-circles. After a bit of playing around with some problems, I finally figured out how to work them correctly and moved on to something new.
One thing that Alex mentioned was by-parts integration. With things winding down in the last couple of months of school, Brandi decided to teach some of us how to integrate by parts one day during lunch. That was definitely a challenge, but John managed to create a funny saying that helped me learn it, 'uv minus vdu'. Saying that over and over again, anyone can memorize the by-parts integration formula with ease, and eventually be able to integrate fairly tricky problems with ease.
One last thing; I feel sorry for the juniors. Spending the entire year with Alex during seventh hour showed me a bit of Calculus BC, which was very annoying. There is going to be a lot to memorize and everything has little tricks that will annoy the hell out of you. He's not joking about sequences and series!!! One day, he showed me some work he was doing with Taylor and MacClaurin Series and it blew my mind. Good luck and spend your last year of high school wisely Juniors.
Thursday, May 13, 2010
Alex Tir's Final Reflection
Hello to all and farewell to all.
This is my first post on this blog and my last post on this blog.
I will say what I have to say for the end of this year and the start of another.
It was a pleasure being with all of you this year in Claculus AB as an ASO student. I usually never had many classes with my graduating class, but this gave me another class to be with you all this year. For this final reflection, I am going to reflect on my entire two years of Calculus experience.
Starting off with Calculus my first year, I had a very easy time learning limits and derivatives. I would like to say that taking derivatives are probably the easiest thing to do, but when this concept is put into a word problem-format, sometimes I still make mistakes. There are always key words to derivatives: slope, rate of change, rate, maximum, minimum, etc. For limits, one thing that I always forget is that a limit can exist at a removable.
When I got to integration, things became a bit tougher. I actually learned integration by-parts when I was in Calculus, so we had harder integration. I'm pretty sure you guys didn't learn it, but that's okay.
I was a good AP Calculus student, but I was a bad Mu Alpha Theta Calc student when it came to integration. Mrs. Robinson would bash my head all the time whenever I missed an "easy" integration problem. Psh! It was easy to her. But entering into Calculus BC, I realized it was easy. I had more practice with integration, and I learned more advanced integration topics. Trigonometric substition is probably my favorite type, and partial fractions is always a fun puzzle to solve at times.
For you juniors this year who will be in Calculus BC next year, don't worry! Calculus BC is about 75% Calculus AB, so I became better at what I learned in AB through BC. You will basically learn a little more advanced ways of doing what you've already learned in AB.
If there is one thing in Calculus BC that I have to say, it is SEQUENCES AND SERIES ARE BEAST! The basic concepts of sequences and series are not too bad, but once you get to Taylor and MacClaurin Series, your mind will be baffled. For AP, you should memorize your series, which I didn't do until right before my AP test lol, but for working out of the book, your work will look ridiculous on paper, but you will be okay.
Anyway, I hope many of you can take a lot of things out of Calculus AB and bring it into Calculus BC or into college. For those of you who may not have understood things so much, or those of you who may have spent more time complaning than learning, or those of you who were too lazy to do some of the work, I apologize. Not on your behalf, but I apologize to you that you may not have gotten the best out of one of the best AP courses in the area.
Take care all! It was a pleasure. Good luck!
This is my first post on this blog and my last post on this blog.
I will say what I have to say for the end of this year and the start of another.
It was a pleasure being with all of you this year in Claculus AB as an ASO student. I usually never had many classes with my graduating class, but this gave me another class to be with you all this year. For this final reflection, I am going to reflect on my entire two years of Calculus experience.
Starting off with Calculus my first year, I had a very easy time learning limits and derivatives. I would like to say that taking derivatives are probably the easiest thing to do, but when this concept is put into a word problem-format, sometimes I still make mistakes. There are always key words to derivatives: slope, rate of change, rate, maximum, minimum, etc. For limits, one thing that I always forget is that a limit can exist at a removable.
When I got to integration, things became a bit tougher. I actually learned integration by-parts when I was in Calculus, so we had harder integration. I'm pretty sure you guys didn't learn it, but that's okay.
I was a good AP Calculus student, but I was a bad Mu Alpha Theta Calc student when it came to integration. Mrs. Robinson would bash my head all the time whenever I missed an "easy" integration problem. Psh! It was easy to her. But entering into Calculus BC, I realized it was easy. I had more practice with integration, and I learned more advanced integration topics. Trigonometric substition is probably my favorite type, and partial fractions is always a fun puzzle to solve at times.
For you juniors this year who will be in Calculus BC next year, don't worry! Calculus BC is about 75% Calculus AB, so I became better at what I learned in AB through BC. You will basically learn a little more advanced ways of doing what you've already learned in AB.
If there is one thing in Calculus BC that I have to say, it is SEQUENCES AND SERIES ARE BEAST! The basic concepts of sequences and series are not too bad, but once you get to Taylor and MacClaurin Series, your mind will be baffled. For AP, you should memorize your series, which I didn't do until right before my AP test lol, but for working out of the book, your work will look ridiculous on paper, but you will be okay.
Anyway, I hope many of you can take a lot of things out of Calculus AB and bring it into Calculus BC or into college. For those of you who may not have understood things so much, or those of you who may have spent more time complaning than learning, or those of you who were too lazy to do some of the work, I apologize. Not on your behalf, but I apologize to you that you may not have gotten the best out of one of the best AP courses in the area.
Take care all! It was a pleasure. Good luck!
Tuesday, May 11, 2010
Final Reflection
Wow. We're done with Calculus. (well, for the sole 6 juniors, maybe not, but you get the picture) Who would've ever thought that we'd make it alive, through all the blood and glory (I know! so melodramatic...). But, seriously! This year has been, in one word, CRAZY.
So, in all things math related, we learned some concepts that might actually help us later on in life. For example, I'm pretty sure everyone will remember that rate=derivative=slope. Also, an integral is the area under a curve! PLUS, the derivative of 2 is ZEEERROOOO!!! Hopefully, you remember that.
Okay, so for the semi-sad part. All of the seniors are leaving. How depressing. The 2009-2010 Calculus class.
Tir-...do I have to say anything?
John John-the go to guy for everything and anything (moderately loud)
Mamie & Chelsea (because you can't have one without the other-John's co-conspirators/enemies in the Calculus realm; also, mamie=insne; chelsea=insane on certain days
Ryne-the loud one
Gonzales-the semi-stupid one (maybe just the stupid one)
Aimee-"I didn't do that."
Ricky-DROID (seriously, that's the one thing I will remember...the "best phone ever")
Kaitlyn-no chocolate...
Trina-Ms. Stress over calculus forever. (really and truly dedicated)
Ellie-almost the same as Trina, except she's a little...more...outspoken? ahmm...
Dylan-another loud one...when he talked
Jessie-quiet one most of the time (I think it was too early) and the sickly one.
And I almost forgot Mher---the door closer who gave me pencils. How trivial..
And then of course there's all the Juniors...Sarah, Steph, Abbey, ME, Ashley, Milky...and that's about it.
OH! the most important part...B-ROB:
who left us, unfortunately, to have Sarah-Rachelle (during which time we had Mr. St. Pierre and had a proper going away party for). She's probably the best Calculus teacher on the planet because although we hated doing the work, she really did prepare us for the AP..which is the final goal..
SO, to the Seniors: have a nice life, and don't forget to visit.
to Brandi: Thanks.
to the Juniors: we have another year left...gosh.
So, in all things math related, we learned some concepts that might actually help us later on in life. For example, I'm pretty sure everyone will remember that rate=derivative=slope. Also, an integral is the area under a curve! PLUS, the derivative of 2 is ZEEERROOOO!!! Hopefully, you remember that.
Okay, so for the semi-sad part. All of the seniors are leaving. How depressing. The 2009-2010 Calculus class.
Tir-...do I have to say anything?
John John-the go to guy for everything and anything (moderately loud)
Mamie & Chelsea (because you can't have one without the other-John's co-conspirators/enemies in the Calculus realm; also, mamie=insne; chelsea=insane on certain days
Ryne-the loud one
Gonzales-the semi-stupid one (maybe just the stupid one)
Aimee-"I didn't do that."
Ricky-DROID (seriously, that's the one thing I will remember...the "best phone ever")
Kaitlyn-no chocolate...
Trina-Ms. Stress over calculus forever. (really and truly dedicated)
Ellie-almost the same as Trina, except she's a little...more...outspoken? ahmm...
Dylan-another loud one...when he talked
Jessie-quiet one most of the time (I think it was too early) and the sickly one.
And I almost forgot Mher---the door closer who gave me pencils. How trivial..
And then of course there's all the Juniors...Sarah, Steph, Abbey, ME, Ashley, Milky...and that's about it.
OH! the most important part...B-ROB:
who left us, unfortunately, to have Sarah-Rachelle (during which time we had Mr. St. Pierre and had a proper going away party for). She's probably the best Calculus teacher on the planet because although we hated doing the work, she really did prepare us for the AP..which is the final goal..
SO, to the Seniors: have a nice life, and don't forget to visit.
to Brandi: Thanks.
to the Juniors: we have another year left...gosh.
Monday, May 10, 2010
Ryans Final Reflection
This year has been a very upside and backwards year, not just including Calculus. Graduating is a stepping stone in life that is very complicated filled with mixed feelings. Although I am ready to move on and begin a new chapter in my life, i know that i will miss high school and all my friends very much so. Looking back on the year and class as a whole, I have some regrets but who wouldn't. I also have some memories that will last a lifetime. Late night study groups where we all freaked out and crammed the night before a test, John and Tir getting aggravated and trying not to show it while they help us,and People falling flat on there face in the middle of class are all memories that stick out in my mind. (Lets not hope anybody has spitwater in there ear when i said that because i ment it) Ill remember all of my classmates, they all have something different and wonderful to contribute to the group. Although this class has been some what over barring and grueling at times I am thankful for having taken it as i know it has prepared me for the next level.
One concept that i am taking away from calculus is that an intergral is the area under the curve. This is very useful. lol
To all reading this pick up on intergrating as quick as possible it will help in the long run... and watch out for particle problems.
well I said what I had to say. Whether it was getting me ready for college or yes even finding me a girlfriend ;) calculus has helped. I wish all of my classmates, whether you have another year left or you are begging a whole new chapter, the best of luck. Turn the book one page at a time. Stay persistent in reaching your goals and I wish everybody the best of success in whatever it is they are doing.
One concept that i am taking away from calculus is that an intergral is the area under the curve. This is very useful. lol
To all reading this pick up on intergrating as quick as possible it will help in the long run... and watch out for particle problems.
well I said what I had to say. Whether it was getting me ready for college or yes even finding me a girlfriend ;) calculus has helped. I wish all of my classmates, whether you have another year left or you are begging a whole new chapter, the best of luck. Turn the book one page at a time. Stay persistent in reaching your goals and I wish everybody the best of success in whatever it is they are doing.
Abbey's Final Reflection
Well I'm late on blogs...again haha.
So, this year is coming to a close, and all I can say is yessss! Calculus this year has made me laugh and cry at the same time. I feel this year I really became close to the seniors. Seniors I like to thank you: to Kaitlyn, I now know about a new brand of goldfish; Aimee, for using all my loose leaf paper; John, for answering my 215445585 questions about the same thing; Mamie, for letting me borrow pencils. And obviously thanks to this calculus class I have Ryan G. now haha. I'm really going to miss yall and the random late night study groups trying to finish corrections. Also to the six juniors of our class...yall have become some of my closest friends. I can't wait until next year when we are seniors! :)
Now let me tell you one concept, perhaps the only concept I knew all year long...AVERAGE VALUE. formula: 1/b-a aSb equation
*just plug in and if you have a calcultor it is even easier
Also, I now love area, graph, and table free response questions because I know what I'm doing! haha
Something that took me until two days before the ap test to learn was tram. Mrs. Robinson had to teach me a completely different formula just because it wouldn't click! (b-rob: thanks for answering all my questions and helping me at sixth hour!)
So, the new formula is: [(b2-b1/2)(h) + (b2-b1/2)(h)...]
Some things I just want to point out that I always forgot...but now I know!
1. use your calculator, it is your friend not enemy
2. that calculator that I just talked about can actually find derivatives, integrals, intersection or bounds, and x-intercepts
3. when the original graph is increasing the derivative will be positve, original decreasing then derivative negative
4. if you get a graph and they want area, look to see if you can divide the graph into shapes
This year flew by. Good luck to all of the seniors and yall will be missed!
Finally I just want to say: I defintily stuggled with learing calculus, but now that we are done with the ap and I take a look back, I feel like I have learned a lot! Bring on Calculus BC HAHAHA
Abbey has left the building....
So, this year is coming to a close, and all I can say is yessss! Calculus this year has made me laugh and cry at the same time. I feel this year I really became close to the seniors. Seniors I like to thank you: to Kaitlyn, I now know about a new brand of goldfish; Aimee, for using all my loose leaf paper; John, for answering my 215445585 questions about the same thing; Mamie, for letting me borrow pencils. And obviously thanks to this calculus class I have Ryan G. now haha. I'm really going to miss yall and the random late night study groups trying to finish corrections. Also to the six juniors of our class...yall have become some of my closest friends. I can't wait until next year when we are seniors! :)
Now let me tell you one concept, perhaps the only concept I knew all year long...AVERAGE VALUE. formula: 1/b-a aSb equation
*just plug in and if you have a calcultor it is even easier
Also, I now love area, graph, and table free response questions because I know what I'm doing! haha
Something that took me until two days before the ap test to learn was tram. Mrs. Robinson had to teach me a completely different formula just because it wouldn't click! (b-rob: thanks for answering all my questions and helping me at sixth hour!)
So, the new formula is: [(b2-b1/2)(h) + (b2-b1/2)(h)...]
Some things I just want to point out that I always forgot...but now I know!
1. use your calculator, it is your friend not enemy
2. that calculator that I just talked about can actually find derivatives, integrals, intersection or bounds, and x-intercepts
3. when the original graph is increasing the derivative will be positve, original decreasing then derivative negative
4. if you get a graph and they want area, look to see if you can divide the graph into shapes
This year flew by. Good luck to all of the seniors and yall will be missed!
Finally I just want to say: I defintily stuggled with learing calculus, but now that we are done with the ap and I take a look back, I feel like I have learned a lot! Bring on Calculus BC HAHAHA
Abbey has left the building....
Mamie's final post
As you can see, it's 6:17 on Monday morning; the day after I was supposed to do my blog. Surprise, surprise.
Where should I start? This year was a challenging one to say the least. It was very fast paste at the beginning of the year. We learned most things in the matter of only a few days. Maybe even just one. Throughout the year, though, we went over the same topics again and again, but some people--like me--still never caught on to some of the things taught until the very end. Some things I eithe don't remember how to do or never really understood include optimization and angle of elevation. These two are related a good bit. I'm pretty sure I was awesome at optimization problems. I think I also understood angle of elevation a little, but now if you'd ask me to do a problem, I couldn't. I was also confused toward the end about natural log integrals. I would never recognize them. I would also never recognize the trig functions that gave us so much trouble in advanced math.
I did pretty much master derivatives, though. Some other things I'm pretty decent at include regular integrals, e integration, limit rules, area graphs and volume graphs.
Many people mess up on substitution. I do sometimes too, but I think I do well at remembering what has to be done in the process. e integration is also really easy. The exponent of e will always be u and its derivative will always be du. You set it up as e^u then solve that way. Limit rules: if the degree of the top is greater than the degree of the bottom the answer is infinity, if the degree of the top is less than the degree of the bottom the answer is 0. If the degrees are equal, you make a fraction with the two terms infront of x. Also sinax/bx will always be a/b. I remembered this at the beginning of calculus, but now I find myself using quotient rule to figure it out.
Area and volume graph problems are always the same. You just integrate. For area, it's top -bottom. For volume it's top^2-bottom^2 times pi.
This year really has been fun with all of you. It still didn't hit me yet that my last day of high school is going to be Wednesday. I'll really miss you all and I'll never forget any of you. Except maybe Mher, because none of us really remembered he was in our Calculus AB calss when he was there.
Bye everyone!!!!
Where should I start? This year was a challenging one to say the least. It was very fast paste at the beginning of the year. We learned most things in the matter of only a few days. Maybe even just one. Throughout the year, though, we went over the same topics again and again, but some people--like me--still never caught on to some of the things taught until the very end. Some things I eithe don't remember how to do or never really understood include optimization and angle of elevation. These two are related a good bit. I'm pretty sure I was awesome at optimization problems. I think I also understood angle of elevation a little, but now if you'd ask me to do a problem, I couldn't. I was also confused toward the end about natural log integrals. I would never recognize them. I would also never recognize the trig functions that gave us so much trouble in advanced math.
I did pretty much master derivatives, though. Some other things I'm pretty decent at include regular integrals, e integration, limit rules, area graphs and volume graphs.
Many people mess up on substitution. I do sometimes too, but I think I do well at remembering what has to be done in the process. e integration is also really easy. The exponent of e will always be u and its derivative will always be du. You set it up as e^u then solve that way. Limit rules: if the degree of the top is greater than the degree of the bottom the answer is infinity, if the degree of the top is less than the degree of the bottom the answer is 0. If the degrees are equal, you make a fraction with the two terms infront of x. Also sinax/bx will always be a/b. I remembered this at the beginning of calculus, but now I find myself using quotient rule to figure it out.
Area and volume graph problems are always the same. You just integrate. For area, it's top -bottom. For volume it's top^2-bottom^2 times pi.
This year really has been fun with all of you. It still didn't hit me yet that my last day of high school is going to be Wednesday. I'll really miss you all and I'll never forget any of you. Except maybe Mher, because none of us really remembered he was in our Calculus AB calss when he was there.
Bye everyone!!!!
Sunday, May 9, 2010
Ash's 38th and Final Post..
As much as I hate to admit it, this is actually kind of depressing. This school year seems to have flown by! I'm really going to miss all of the seniors and I'm so glad I had those in Calculus in my class, otherwise I wouldn't have gotten to know their cooky personalities X] Good luck to all of you guys! I know all of yall will succeed in college and then later on in life, wherever that may take you =]
Enough of the mushy-gushy stuff for now and onto the last bit of Calculus I'll have to type on here for a while.
The concept that I felt really stupid when I finally understood was Definition of Derivatives. I'm not exactly sure how it clicked, but I remember Mrs. Robinson sitting at her desk explaining it once again..when a light bulb when off in my head! =] I felt so accomplished after I worked that first problem understanding the concept..then I felt like an idiot for not getting it before X]
But how about I go over real quick what is it? ^^
f'(x) = Lim f(x+h)-f(x)
H->0 h
You merely take the derivative of f(x+h) by following these steps :D
First plug in 0 for H
Then you take the derivative
Then you simplify!
The end!
I can't believe it took me forever to grasp that X]
And a final note before I say goodbye to Calculus AB...thank you guys for such a great year! I'm never going to forget this class and I'm pretty sure I never want to. Thanks to Mrs. Robinson for having the patience to teach us, the motivation to push us and herself as hard as she could, the love and care she showed throughout the entire year. We love you! =]
Goodbye Calculus AB!
Enough of the mushy-gushy stuff for now and onto the last bit of Calculus I'll have to type on here for a while.
The concept that I felt really stupid when I finally understood was Definition of Derivatives. I'm not exactly sure how it clicked, but I remember Mrs. Robinson sitting at her desk explaining it once again..when a light bulb when off in my head! =] I felt so accomplished after I worked that first problem understanding the concept..then I felt like an idiot for not getting it before X]
But how about I go over real quick what is it? ^^
f'(x) = Lim f(x+h)-f(x)
H->0 h
You merely take the derivative of f(x+h) by following these steps :D
First plug in 0 for H
Then you take the derivative
Then you simplify!
The end!
I can't believe it took me forever to grasp that X]
And a final note before I say goodbye to Calculus AB...thank you guys for such a great year! I'm never going to forget this class and I'm pretty sure I never want to. Thanks to Mrs. Robinson for having the patience to teach us, the motivation to push us and herself as hard as she could, the love and care she showed throughout the entire year. We love you! =]
Goodbye Calculus AB!
John's Final Reflection
Wow.
It really is amazing that the year is over. It seems like it has gone by so fast...it's like I remember people in class asking how to take a derivative, and I feel like it was days ago...wait, that was days ago, like Monday when we reviewed. :-P.
Jokes aside, I can't express how awesome this year has been for me in Calculus. It feels like we flew right through it, and it was not as bad as some people made it out to be. Calculus is pretty easy once you get the concepts down (derivatives and integrals). It is of course taught by an amazing teacher, so I know that everyone after me will do just fine with very little effort.
I first want to congratulate and acknowledge Mrs. Brandi Robinson for being an amazing teacher. Not only did you have a baby and manage to keep your math classes running, but you also taught us so many things and prepared us so well for the AP exam. I can honestly say that I took the AP exam with absolutely no studying outside of the school day, and I'm positive I did well on it. I would not be able to say that without you, and I greatly thank you and appreciate everything you've done for me and the class especially.
Second, I want to say that I could not have done so well in Calculus without all of my classmates as well. The reason I was able to take the AP without studying is because of you guys, mainly. You guys pestered me and annoyed me and asked as many questions as you could possibly come up with, sometimes asking the same exact question without realizing it, and I thank you all for it. Through all of the explaining and answering of your questions, I was able to study almost every class period of my day. You guys have made my year successful, and I want to thank you for that. I also hope that I have helped make your year just as successful, because that would only be the fair thing :-).
Last, I want to thank Chelsea specifically. Of every classmate in Calculus, she has been the most useful to me. Whether it was her asking a million and one questions or whether it was discussing calculus problems over AIM, she helped me in a million ways, and I owe my deepest gratitude and thanks to her for all of her help. :-)
Overall, Calculus AB has been a complete success on so many levels. Not only will it get me credit hours in college, but it really has furthered my knowledge in mathematics and Calculus specifically, which is going to help my career (Engineering) in so many ways.
P.S. I just want to thank you again, B-Rob, for being an amazing teacher and mentor. I cannot fathom the words to describe how awesome you have been to me this year with all of your help and support. With that being said, I hope you understand how much I truly cherish the teachings and knowledge you have passed to me.
-John
It really is amazing that the year is over. It seems like it has gone by so fast...it's like I remember people in class asking how to take a derivative, and I feel like it was days ago...wait, that was days ago, like Monday when we reviewed. :-P.
Jokes aside, I can't express how awesome this year has been for me in Calculus. It feels like we flew right through it, and it was not as bad as some people made it out to be. Calculus is pretty easy once you get the concepts down (derivatives and integrals). It is of course taught by an amazing teacher, so I know that everyone after me will do just fine with very little effort.
I first want to congratulate and acknowledge Mrs. Brandi Robinson for being an amazing teacher. Not only did you have a baby and manage to keep your math classes running, but you also taught us so many things and prepared us so well for the AP exam. I can honestly say that I took the AP exam with absolutely no studying outside of the school day, and I'm positive I did well on it. I would not be able to say that without you, and I greatly thank you and appreciate everything you've done for me and the class especially.
Second, I want to say that I could not have done so well in Calculus without all of my classmates as well. The reason I was able to take the AP without studying is because of you guys, mainly. You guys pestered me and annoyed me and asked as many questions as you could possibly come up with, sometimes asking the same exact question without realizing it, and I thank you all for it. Through all of the explaining and answering of your questions, I was able to study almost every class period of my day. You guys have made my year successful, and I want to thank you for that. I also hope that I have helped make your year just as successful, because that would only be the fair thing :-).
Last, I want to thank Chelsea specifically. Of every classmate in Calculus, she has been the most useful to me. Whether it was her asking a million and one questions or whether it was discussing calculus problems over AIM, she helped me in a million ways, and I owe my deepest gratitude and thanks to her for all of her help. :-)
Overall, Calculus AB has been a complete success on so many levels. Not only will it get me credit hours in college, but it really has furthered my knowledge in mathematics and Calculus specifically, which is going to help my career (Engineering) in so many ways.
P.S. I just want to thank you again, B-Rob, for being an amazing teacher and mentor. I cannot fathom the words to describe how awesome you have been to me this year with all of your help and support. With that being said, I hope you understand how much I truly cherish the teachings and knowledge you have passed to me.
-John
final post
alright, well some of the things i've gotten comfortable doing this year in calculus is the first and second derivative test. since those were basically the first things we did this year, i learned those the best, and i got kinda lazy throughout the rest of the year, so that's why i was the best at that.
For this post i am going to go into detail on the second derivative test to find all possible points of inflection and intervals of concavity. remember, points of inflection only happen where there is a change of concavity.
Example: f'(x)= 6/(x^(2)+3)
First, you have to take the derivative of that, and you have to use the quotient rule, so the beginning of the problem will look like, [(x^(2)+3)(0)-6(2x)]/(x^(2)+3)^(2), which simplifies to -12x/(x^(2)+3)^2 remember, that was just the first derivative.
Second step is to take the derivative of the first derivative, that would make this step called taking the second derivative.
Once again u need to use the quotient rule, so f''(x)={(x2+3)^2-(12)-[(-12x)2(x^(2)+3)2x} all that over (x^(2)+3)^4 then you get a bunch of stuff, then you simplify, then you cancel, so I am just going to type the end answer of the second derivative. Which is, (3)(6)(x+1)(x-1) all over (x^2+3)^3
The possible points of inflection are found in the numerator of the finished second derivative, in this case, if you look, it would be x=1, and x=-1
so then you set up your points, (-infinity, -1) u (-1, 1) u (1, infinity)
then you plug in. f''(-2)= positive value f''(0)=negative value f''(2)=positive value
then you know that your intervals concave up at (-infinity, -1) u (1,infinity) or x<1,>1
and it is concave down at (-1,1) or -1
and you're points of inflection are x=-1, and x=1
okay, one thing i wish i would have done was do my homework at the beginning of the year, because that would have made learning all the other stuff throughout the year much easier. and my advice to anyone else taking this class is to learn derivatives, integrals, related rates, substitution, limits, and implicit derivatives really well, because that's pretty much all the main focal points of the ap test and calculus in general. and thanks ms robinson for teaching me so much and being such a cool teacher :)
For this post i am going to go into detail on the second derivative test to find all possible points of inflection and intervals of concavity. remember, points of inflection only happen where there is a change of concavity.
Example: f'(x)= 6/(x^(2)+3)
First, you have to take the derivative of that, and you have to use the quotient rule, so the beginning of the problem will look like, [(x^(2)+3)(0)-6(2x)]/(x^(2)+3)^(2), which simplifies to -12x/(x^(2)+3)^2 remember, that was just the first derivative.
Second step is to take the derivative of the first derivative, that would make this step called taking the second derivative.
Once again u need to use the quotient rule, so f''(x)={(x2+3)^2-(12)-[(-12x)2(x^(2)+3)2x} all that over (x^(2)+3)^4 then you get a bunch of stuff, then you simplify, then you cancel, so I am just going to type the end answer of the second derivative. Which is, (3)(6)(x+1)(x-1) all over (x^2+3)^3
The possible points of inflection are found in the numerator of the finished second derivative, in this case, if you look, it would be x=1, and x=-1
so then you set up your points, (-infinity, -1) u (-1, 1) u (1, infinity)
then you plug in. f''(-2)= positive value f''(0)=negative value f''(2)=positive value
then you know that your intervals concave up at (-infinity, -1) u (1,infinity) or x<1,>1
and it is concave down at (-1,1) or -1
and you're points of inflection are x=-1, and x=1
okay, one thing i wish i would have done was do my homework at the beginning of the year, because that would have made learning all the other stuff throughout the year much easier. and my advice to anyone else taking this class is to learn derivatives, integrals, related rates, substitution, limits, and implicit derivatives really well, because that's pretty much all the main focal points of the ap test and calculus in general. and thanks ms robinson for teaching me so much and being such a cool teacher :)
Ricky's Last Blog
After all the anticipation, its finally over. High school and more importantly CALCULUS. I'm not a math person as many of u know but im glad i have been in advanced math and in calculus with all of you. i love each and every one of you and im glad to call yall my classmates. From watching ryan get yelled at to making stupid math jokes. yall are amazing. i would be lost without most of you so i would like to say thanks for putting up with me sometimes.
I will have many things that stick with me from calculus but lets see what i can really remember. first and second derivative tests is something i def cant forget. WE DID IT A MILLION TIMES.
I will def remember limit rules.
1. If the degree of the top is larger than the degree of the bottom, the limit approaches infinity
2. If the degree of the bottom is larger than the degree of the tip, the limit approaches zero
3. If the degree of the bottom is equal to the degree of the top, then you make a fraction out of the coefficients in front of the largest degree.
I know the optimization steps i just suck at doing them hahaahahah i know how to take integrals and how to substitute. i know eintegration and many other useful tools that i have learned along the way of my calculus year.
One thing i have learned more than anything is to never give up and strive for something better. i didnt ahve to take calculus as a senior buti chose to and im glad i did. the hard work will help me in college just like it will help the rest of the class.
To Brandi:
Even though i wanted to kill myself during the year im glad to have had you as a teacher these past two years. From making you laugh to me almost crying cuz your tests were beast i had a great time. This year was a great one w/ alot of things that will help me in the future. thank you for being there for me and being my teacher even though im not the easiest to teach.
TO THE JUNIORS:
I LOVE YALL JUST AS IF YALL WERE SENIORS. I WISH YALLL COULD GRADUATE WITH US BUT YALL HAVE YALLS OWN THING TO DO NEXT YEAR. IMA BE AROUND NEXT YEAR BTW SO YALL CANT MISS ME TOOO MUCH :D
TO THE CLASS OF 2010:
LETS DO THIS :) WE WORKED OUR A$$E$ OFF AND ITS GONNA PAY OFF. I LOVE YOU ALL AS IF YOU WERE MY BROTHERS AND SISTERS. IN FACT YALL ARE LIKE BROTHERS AND SISTERS. IF ANY OFYOU NEED ANYTHING IN THE FUTURE U KNOW THE NUMBER.
Signing out,
Ricky Johnson ;)
I will have many things that stick with me from calculus but lets see what i can really remember. first and second derivative tests is something i def cant forget. WE DID IT A MILLION TIMES.
I will def remember limit rules.
1. If the degree of the top is larger than the degree of the bottom, the limit approaches infinity
2. If the degree of the bottom is larger than the degree of the tip, the limit approaches zero
3. If the degree of the bottom is equal to the degree of the top, then you make a fraction out of the coefficients in front of the largest degree.
I know the optimization steps i just suck at doing them hahaahahah i know how to take integrals and how to substitute. i know eintegration and many other useful tools that i have learned along the way of my calculus year.
One thing i have learned more than anything is to never give up and strive for something better. i didnt ahve to take calculus as a senior buti chose to and im glad i did. the hard work will help me in college just like it will help the rest of the class.
To Brandi:
Even though i wanted to kill myself during the year im glad to have had you as a teacher these past two years. From making you laugh to me almost crying cuz your tests were beast i had a great time. This year was a great one w/ alot of things that will help me in the future. thank you for being there for me and being my teacher even though im not the easiest to teach.
TO THE JUNIORS:
I LOVE YALL JUST AS IF YALL WERE SENIORS. I WISH YALLL COULD GRADUATE WITH US BUT YALL HAVE YALLS OWN THING TO DO NEXT YEAR. IMA BE AROUND NEXT YEAR BTW SO YALL CANT MISS ME TOOO MUCH :D
TO THE CLASS OF 2010:
LETS DO THIS :) WE WORKED OUR A$$E$ OFF AND ITS GONNA PAY OFF. I LOVE YOU ALL AS IF YOU WERE MY BROTHERS AND SISTERS. IN FACT YALL ARE LIKE BROTHERS AND SISTERS. IF ANY OFYOU NEED ANYTHING IN THE FUTURE U KNOW THE NUMBER.
Signing out,
Ricky Johnson ;)
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