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 ;)
Final Reflection
Well, the class we all anticipated to fail is finally over and I think most of us learned a lot of calculus this year.
Some topics I caught on to right away such as tangent line and normal line since it was only taking a derivative and plugging in numbers. I also learned RRAM, LRAM, TRAM, and MRAM right away. This is partly because I did both my comments on these topics weekly for almost 2 months. It really is just plugging in formulas.
Other topics such as optimization and integration took longer to catch on too. I did not understand optimization at all until Brob gave us steps on the board one day. Those steps applied to every problem we saw. I think its easier to follow steps then to have to decipher the problem by yourself, at leas it is for me. Integration is not a hard topic, but it just took time to catch on too. A LOT of practice problems were done especially substitution problems. I still forget my +c sometimes though. I never really did get into the habit of putting it. I also did not know how to do anything in my calculator until around the last month of school. That is when John started giving me funny looks when correcting APs because I tried to do everything by hand. Who knew those a hundred something dollars would actually be worth it.
And finally, the topics that I still struggle with are graphs, angle of elevation, and related rates. Graphs I never did get from day one. I did get a little better with them since they were on EVERY AP, but I would have liked to have had a better understanding on them. I always struggled with word problems hence my problem with angle of elevation and related rates. Every problem differs so these cannot have a set of rules like optimization does. I have a problem with drawing the picture correctly and picking out the given and what we are looking for.
All in all, I feel it was a pretty successful year, and I want to thank you brob for HOPEFULLY getting me out of calculus in college; although, I cannot picture next year without it.
Also, a thanks to all my classmates for asking plenty of questions, so I did not have too =)
And a thanks to John for answering my thousands of questions even though many of them were repeated.
I will miss all of you!
Some topics I caught on to right away such as tangent line and normal line since it was only taking a derivative and plugging in numbers. I also learned RRAM, LRAM, TRAM, and MRAM right away. This is partly because I did both my comments on these topics weekly for almost 2 months. It really is just plugging in formulas.
Other topics such as optimization and integration took longer to catch on too. I did not understand optimization at all until Brob gave us steps on the board one day. Those steps applied to every problem we saw. I think its easier to follow steps then to have to decipher the problem by yourself, at leas it is for me. Integration is not a hard topic, but it just took time to catch on too. A LOT of practice problems were done especially substitution problems. I still forget my +c sometimes though. I never really did get into the habit of putting it. I also did not know how to do anything in my calculator until around the last month of school. That is when John started giving me funny looks when correcting APs because I tried to do everything by hand. Who knew those a hundred something dollars would actually be worth it.
And finally, the topics that I still struggle with are graphs, angle of elevation, and related rates. Graphs I never did get from day one. I did get a little better with them since they were on EVERY AP, but I would have liked to have had a better understanding on them. I always struggled with word problems hence my problem with angle of elevation and related rates. Every problem differs so these cannot have a set of rules like optimization does. I have a problem with drawing the picture correctly and picking out the given and what we are looking for.
All in all, I feel it was a pretty successful year, and I want to thank you brob for HOPEFULLY getting me out of calculus in college; although, I cannot picture next year without it.
Also, a thanks to all my classmates for asking plenty of questions, so I did not have too =)
And a thanks to John for answering my thousands of questions even though many of them were repeated.
I will miss all of you!
sarah's final blog part 2.
OMG I ALMOST FORGOT!
To B-ROB: you are an AMAZING teacher. I don't care what anyone says, hands-down.. best teacher at riverside academy. you have so much dedication for your students and you believe in all of us. even the annoying ones in the class... ;) you are one of the only teachers i know that will work with you in all of your extra hours UNTIL you get it. no matter how long it takes, you will make it click in our brains. which you did for most of us. i wish i could put a note on here for all incoming calc students saying that brob is willing to help as long as you're willing to put forth the effort. that is one thing that i've learned also, which i'm very glad i did. i can honestly say i think i'm speaking on behalf of the entire class when i say: WE LOVE YOU BROB. thank you for believing in us.
To B-ROB: you are an AMAZING teacher. I don't care what anyone says, hands-down.. best teacher at riverside academy. you have so much dedication for your students and you believe in all of us. even the annoying ones in the class... ;) you are one of the only teachers i know that will work with you in all of your extra hours UNTIL you get it. no matter how long it takes, you will make it click in our brains. which you did for most of us. i wish i could put a note on here for all incoming calc students saying that brob is willing to help as long as you're willing to put forth the effort. that is one thing that i've learned also, which i'm very glad i did. i can honestly say i think i'm speaking on behalf of the entire class when i say: WE LOVE YOU BROB. thank you for believing in us.
Kaitlyn's Final Reflection
Well after anticipating all year, it's finally here. My last few days of high school. First I just want to say how bittersweet this class was for me. The class itself, i loved it! The people, the messing around, the stressing out, the late night mini van trips, it was unforgettable. I really enjoyed getting to know everyone and love our ability to have fun yet still get a lot of work done. Now for the bitter part...the actually calculus. I'm pretty sure it's safe to say i suck at it. But hey, it's not the end of the world. Don't worry, i don't plan on building bridges or roller coasters in my future, so y'all should be safe ;)
Now onto the "concept" that never gave up on me and kept running into my skull until it finally got into my brain...
what else could it be..... AVERAGE VALUE!
It's just such a simple integral problem, how can anyone get this wrong. The formula is 1/b-a bSa f(x). You simply just plug in to the formula with what you were given. And half the time it's in the calculator portion so you just have to plug the integral in your calculator then times it by 1/b-a. Voila, you get a beautiful answer that you actually think is right :) AMEN to that.
Just so i don't seem like i only accomplished one thing in calculus all year, some other things i understood were average rate of change, mean value theorem, simple derivatives and integrals, HORIZONTAL TANGENTS, and i'm sure there are a couple of others.
So onto my farewells:
my seniors- We've had an amazing run. We're all so close and we worked hard to get where we are. No matter how mean i may come off sometimes, I love each and every one of y'all and wish every one luck in their future endeavors. Now let's finish this year off right and make a name.
oh let's doooo it
the juniors- i absolutely love y'all. i never really talked to y'all much before this class but i'm so glad that we had it together. Every one of y'all cracks me up, and i know y'all will be successful b/c of how determined you are. Keep it up and don't slack off, and have fun in calculs bc next year. I might miss you guys ;)
B-robbbb- thanks for putting up with me these last two years. You believed in me and kept pushing me even when i said it was impossible to stick this what i call "garbage" in my brain. I know i didn't catch on to a lot at first, but i did try and i truly believe no other teacher could put up with us as well as you did. You're an awesome person and i truly admire you and all of your accomplishments. Thanks again for everything :)
Now i think it's time to close this blog, my LAST one.
Goodbye Calculus, Goodbye Riverside, Hello Life :)
Now onto the "concept" that never gave up on me and kept running into my skull until it finally got into my brain...
what else could it be..... AVERAGE VALUE!
It's just such a simple integral problem, how can anyone get this wrong. The formula is 1/b-a bSa f(x). You simply just plug in to the formula with what you were given. And half the time it's in the calculator portion so you just have to plug the integral in your calculator then times it by 1/b-a. Voila, you get a beautiful answer that you actually think is right :) AMEN to that.
Just so i don't seem like i only accomplished one thing in calculus all year, some other things i understood were average rate of change, mean value theorem, simple derivatives and integrals, HORIZONTAL TANGENTS, and i'm sure there are a couple of others.
So onto my farewells:
my seniors- We've had an amazing run. We're all so close and we worked hard to get where we are. No matter how mean i may come off sometimes, I love each and every one of y'all and wish every one luck in their future endeavors. Now let's finish this year off right and make a name.
oh let's doooo it
the juniors- i absolutely love y'all. i never really talked to y'all much before this class but i'm so glad that we had it together. Every one of y'all cracks me up, and i know y'all will be successful b/c of how determined you are. Keep it up and don't slack off, and have fun in calculs bc next year. I might miss you guys ;)
B-robbbb- thanks for putting up with me these last two years. You believed in me and kept pushing me even when i said it was impossible to stick this what i call "garbage" in my brain. I know i didn't catch on to a lot at first, but i did try and i truly believe no other teacher could put up with us as well as you did. You're an awesome person and i truly admire you and all of your accomplishments. Thanks again for everything :)
Now i think it's time to close this blog, my LAST one.
Goodbye Calculus, Goodbye Riverside, Hello Life :)
sarah's final blog.
wow. I can't believe it's finally over. In a few days, i'll no longer by an underclassmen. I never realized how hard this was gonna be until it actually happened. I sound like i'm leaving the earth or something. ha, sorry. This is only my junior year and I'm already freaking out. I'm a big baby.
To my seniors: I love you guys more than you know. Every last one of you. Including Mher, although he always seems to not be in class. Yall had an awesome year, I can tell. I hope my senior years proves to be just as great. You guys will be missed, EXTREMELY. especially those of you who have been at RA since I first walked in the front door when I was 4. I always looked up to you guys... cuz yall were older :) and it's so weird to think that next year, we will be the role models. It still hasn't hit me yet.
To my juniors: Well, what can I say. it's been one heck of a ride.. From preschool, to middle school, to soon-to-be seniors. The five of you in the calc class are the people I am most close to out of the ENTIRE jr class. because I spend 90% of my day with you mon-fri... on weekends @ math trips, and during the summer @ practices. I love yall more than you can imagine. I can't wait for next year's calc BC class, it will be fun with just the six of us :)
So i'll stop being all mushy for a moment and talk about calc this year.
Most helpful thing: working in groups at the end of year to go over what you missed on the practice ap tests. one-on-one with another student that will actually make you do your work (thanks ellie) helped me more than you can understand. best idea yet B-ROB.
Thing i struggled with most: integration. i honestly still am not 100% with it, idk what it is. substitution, ln, tan inverse. none of it came to me easily. more practice was needed probably? idk.
Thing I picked up on the quickest: the derivative formulas. I can still say them to you probably quicker than anyone in the class. all those derivative practice sheets in the beginning were SOOOO helpful.
My epiphany of the year: the riemann sum formulas!!!!!!!!! honestly, i didn't understand all that f(a) + f(2 + a) stuff. and multiply by delta x whatever. it must easier when taught that TRAM is
b' + b'' / 2 (n) + b'' + b''' /2 (n). also the LRAM and RRAM are easier when taught that you add to x's divide by n (start all the way to left with LRAM and 2nd number with RRAM). that literally clicked in my head 2 days before the exam.
so, even though I don't want it to, this year is coming to a close. and quick too. I'm going to miss all you entirely too much. I will probably cry... a lot. just because I'm about to be a senior. i'll be sad everyday in calculus when i realize how small the class is. (& when i realize that mher IS actually not there). we had a great year, from partying it up with mr. st pierre to re-partying it up with brob. & of course getting people to sneak out of school to buy us some soft drinks because ricky never brings his stuff! i will miss it all.
GOOD LUCK TO YOU, seniors of 2010. you're gifted in more ways than you can imagine. embrace that.. you can do anything you want in your life. be proud of yourselves. i hope for the best for everyone and i'll remember you everytime i walk the halls of riverside academy.
To my seniors: I love you guys more than you know. Every last one of you. Including Mher, although he always seems to not be in class. Yall had an awesome year, I can tell. I hope my senior years proves to be just as great. You guys will be missed, EXTREMELY. especially those of you who have been at RA since I first walked in the front door when I was 4. I always looked up to you guys... cuz yall were older :) and it's so weird to think that next year, we will be the role models. It still hasn't hit me yet.
To my juniors: Well, what can I say. it's been one heck of a ride.. From preschool, to middle school, to soon-to-be seniors. The five of you in the calc class are the people I am most close to out of the ENTIRE jr class. because I spend 90% of my day with you mon-fri... on weekends @ math trips, and during the summer @ practices. I love yall more than you can imagine. I can't wait for next year's calc BC class, it will be fun with just the six of us :)
So i'll stop being all mushy for a moment and talk about calc this year.
Most helpful thing: working in groups at the end of year to go over what you missed on the practice ap tests. one-on-one with another student that will actually make you do your work (thanks ellie) helped me more than you can understand. best idea yet B-ROB.
Thing i struggled with most: integration. i honestly still am not 100% with it, idk what it is. substitution, ln, tan inverse. none of it came to me easily. more practice was needed probably? idk.
Thing I picked up on the quickest: the derivative formulas. I can still say them to you probably quicker than anyone in the class. all those derivative practice sheets in the beginning were SOOOO helpful.
My epiphany of the year: the riemann sum formulas!!!!!!!!! honestly, i didn't understand all that f(a) + f(2 + a) stuff. and multiply by delta x whatever. it must easier when taught that TRAM is
b' + b'' / 2 (n) + b'' + b''' /2 (n). also the LRAM and RRAM are easier when taught that you add to x's divide by n (start all the way to left with LRAM and 2nd number with RRAM). that literally clicked in my head 2 days before the exam.
so, even though I don't want it to, this year is coming to a close. and quick too. I'm going to miss all you entirely too much. I will probably cry... a lot. just because I'm about to be a senior. i'll be sad everyday in calculus when i realize how small the class is. (& when i realize that mher IS actually not there). we had a great year, from partying it up with mr. st pierre to re-partying it up with brob. & of course getting people to sneak out of school to buy us some soft drinks because ricky never brings his stuff! i will miss it all.
GOOD LUCK TO YOU, seniors of 2010. you're gifted in more ways than you can imagine. embrace that.. you can do anything you want in your life. be proud of yourselves. i hope for the best for everyone and i'll remember you everytime i walk the halls of riverside academy.
Stephanie's Final Blog
As this year comes to a close, I'd like to tell everyone how great our class was this year. Especially with our mad covert partying skills, our ability to goof off and still get our work done, and lastly, our great personalities. The seniors this year are great, and i'll miss every one of yall. To my juniors, LETS GET THIS PARTY STARTED. I'm ready to own the school, yall bout it?
So, something i couldn't understand then it just smoked me in the face is substitution. I think it took Ryan Breaud just yelling at me about how to substitute, then somehow making it involve either a mean girl quote, bonquiqui, or darrel, spelt like darrel, but pronounced like darrel. I can't explain how it really stuck in my head besides taht. But give me something today, and i'll substitute the numbers out of it. Watch out.
This was definately one of my favorite classes this year, besides the MASSIVE struggles i've had in this class, it definately taught me that i can achieve what I strive to do. And to the seniors leaving, i'm glad i got to take this class with yall, and yall better remember the good times we had..and if anyone gets famous remember the girl who mission impossibled it to the cafeteria to get ice for our cups.
To the juniors, well seniors now, lets make this year the greatest ever. I mean i'm already great, and yall have class with me, so that makes you offliated with greatness, which means you're semi-great..and so add that together, then you get that we're all great. because of me, of course. :)
Also, thanks to Mrs. Robinson for helping me this year and not giving up on me when i answer free response questions without thinking. I know i couldn't of done it with out you.
Lastly, i have to tell everyone thanks for making my first hour great, and you better not ever forget me. :)
So, something i couldn't understand then it just smoked me in the face is substitution. I think it took Ryan Breaud just yelling at me about how to substitute, then somehow making it involve either a mean girl quote, bonquiqui, or darrel, spelt like darrel, but pronounced like darrel. I can't explain how it really stuck in my head besides taht. But give me something today, and i'll substitute the numbers out of it. Watch out.
This was definately one of my favorite classes this year, besides the MASSIVE struggles i've had in this class, it definately taught me that i can achieve what I strive to do. And to the seniors leaving, i'm glad i got to take this class with yall, and yall better remember the good times we had..and if anyone gets famous remember the girl who mission impossibled it to the cafeteria to get ice for our cups.
To the juniors, well seniors now, lets make this year the greatest ever. I mean i'm already great, and yall have class with me, so that makes you offliated with greatness, which means you're semi-great..and so add that together, then you get that we're all great. because of me, of course. :)
Also, thanks to Mrs. Robinson for helping me this year and not giving up on me when i answer free response questions without thinking. I know i couldn't of done it with out you.
Lastly, i have to tell everyone thanks for making my first hour great, and you better not ever forget me. :)
Ellie's Final Reflection
Alright you guys!!! We've gotten through calculus and we've all made it. When this class first started, I wasn't stressed out by the least. Having a teacher like Mrs. Robinson the year before, I knew what I was getting into. The first week was oddly enough, expected! We got so many formulas to memorize and so many ways of taking a derivative a lot of us were going crazy, but not me, I was kind of like "whatever" with it. Turns out two or three weeks into it, I got really scared, my grades weren't really slipping but I wasn't getting the new, or harder, derivatives. After that, I buckled down and got to work. Here's what I've learned throughout the year...
Easy Concepts For Me:
--How to Take a Derivative:
You take the exponent and times it by the coefficient, what you get is the new coefficient and you're new exponent will be the original exponent minus one!
--The First Derivative Test:
You have to take the derivative of the function and set it equal to zero. Then solve for the critical values (x values). Set those values up into intervals between negative infinity and infinity. Plug in numbers between these intervals into the first derivative to see if there are max or mins or if the graph is increasing or decreasing.
--The Second Derivative Test:
You take the derivative of the function twice and set it equal to zero. Solve for the x values and set them up into intervals between negative infinity and infinity. Plug in numbers between those intervals into the second derivative to see where the graph is concave up, concave down, or where there is a point of inflection.
--Maximums, minimums, critical values, increasing, decreasing:
All of these are related to first derivative test. it's simple. you take the derivative, set equal to 0, solve for x. Set up some intervals using these numbers. Plug in numbers and test your intervals. pos to neg is a min. neg to pos is a max. pos = increasing, neg = decreasing. simple stuff. remember it.
--Point of inflection, concave up, concave down:
It's the second derivative test. set up intervals, if the intervals change signs, it is a point of inflection there. also, if its negative, that interval is concave down, positive is concave up.
--Finding Critical Values:
To find critical values, first take the derivative of the function and set it equal to zero, solve for x. The answers you get for x are your critical values.
--Limits:
If the degree on top is smaller than the degree on the bottom, the limit is zero.
If the degree on top is bigger than the degree on the bottom, the limit is infinity.
If the degree on top is the same as the degree on the bottom, you divide the coefficients to get the limit.
Some Concepts that...after reviewing over and over for weeks/months...it just clicked:
--Slope of a normal line:
Take derivative, plug in x. get a slope. however, make sure you use the negative reciprocal of the slope (normal means perpendicular to). use point-slope formula.
--Absolute Extrema:
If you are given a point, plug those numbers into the original function to get another number. Alos, solve for critical values and plug those into the original function. Once you get your second numbers, you set each pair into new sets of points. The highest point is the absolute max and the smallest point is the absolute min.
--Average Rate of Change:
f(b)-f(a)/(b-a)
--Average Value:
This is just an integral times by 1/(b-a).
--Equation of a Tangent Line:
Take the derivative and plug in the x value.
If you are not given a y value, plug into the original equation to get the y value.
then plug those numbers into point slope form: y − y1 = m(x − x1)
And-of course-the things that I always struggled with, and hope to never encounter again even though I know I will....:
--linearization:
Identify the equation
Use the formula f(x)+f ' (x)dx
Determine your dx in the problem
Then determine your x in the problem
Plug in everything you get
Solve the equation
--Riemanns Sums:
LRAM-Left hand approximation=delta x[f(a)+f(a+delta x)+...f(b-delta x)]
RRAM-Right hand approximation=delta x[f(a+delta x)+...f(b)]
MRAM-Middle approximation=delta x[f(mid)+f(mid)+...]
Trapezoidal-delta x/2[f(a)+2f(a+delta x)+2f(a+2 delta x)+...f(b)]
delta x=b-a/number of subintervals
--Volume by Disks:The formula is pi times the integral of the [function given] squared times dx. so just solve it by taking the integral of it and then pluging in the numbers they give you. just like before you'll have two numbers so whatever the answer is for the top one will be first and then you subtract the answer you get for the bottom one. then graph
--Volume by Washers:
The formla is pie times the integral of the [top function] squared minus the [bottom function] squared times dx. so to do this, if you don't have the in between number you have to set the functions equal, but if you do, then it's worked the same way as above. square the formula's that were given and simplify. then take the integral of it and plug in the numbers they give you or you found by setting the formulas equal to each other and then solve like any other one by subracting them. then graph.
--OPTIMIZATION...
I just can't do it =(
Just a shout out:
To the seniors in the Calculus Class of 2010 - - I'll Miss Yall, we've grown up together basically. I won't lie, I didn't like MANY of yall when I first came here because of different things that happened; however, we've all matured [thank God] and we've become friends. Although some of us aren't best friends I can honestly say that I will miss each and every one of you all because yall are the people I spent all day every day with for the past 6 years.
To the juniors in the Calculus Class of 2010 - - I'll miss yall just as much. There's many reason's why I will..some were my first friend at Riverside Academy because of the Cheerleading Squad, some have always been in my life because we grew up living next door to each other, some have been with me through the years with dancing and just being friends, some HOWEVER have become my friend in this past year and believe me I love yall just as much!
So back to the FARWELLS:
No, I'm not sad I'm graduating. My parents say that it hasn't hit me yet...but most of yall know me already and I'm so happy to finally be just about OUT of Riverside Academy that nothing else matters. Also, I'm not sad that I'm leaving any of yall. Yall have been great friends and I'm glad we've become as close as we have or, not as close as we have in some cases, because LIFE GOES ON..We cannot dwell in the past much less live it. I won't be back at Riverside once I'm out, there's no need....I'm finally getting my life started and I CAN'T WAIT!! =) Yeah, I'll miss yall when certain things come up and I realize what [so and so] would've said and they wont be there..but most of all .. I'll miss US as a class together and how we are with each other! So to everyone - - THANKS! And have a GREAT time either being a senior, or going out into the REAL WORLD and EXPERIENCING everything!
Love Always,
Ellie Kliebert
Easy Concepts For Me:
--How to Take a Derivative:
You take the exponent and times it by the coefficient, what you get is the new coefficient and you're new exponent will be the original exponent minus one!
--The First Derivative Test:
You have to take the derivative of the function and set it equal to zero. Then solve for the critical values (x values). Set those values up into intervals between negative infinity and infinity. Plug in numbers between these intervals into the first derivative to see if there are max or mins or if the graph is increasing or decreasing.
--The Second Derivative Test:
You take the derivative of the function twice and set it equal to zero. Solve for the x values and set them up into intervals between negative infinity and infinity. Plug in numbers between those intervals into the second derivative to see where the graph is concave up, concave down, or where there is a point of inflection.
--Maximums, minimums, critical values, increasing, decreasing:
All of these are related to first derivative test. it's simple. you take the derivative, set equal to 0, solve for x. Set up some intervals using these numbers. Plug in numbers and test your intervals. pos to neg is a min. neg to pos is a max. pos = increasing, neg = decreasing. simple stuff. remember it.
--Point of inflection, concave up, concave down:
It's the second derivative test. set up intervals, if the intervals change signs, it is a point of inflection there. also, if its negative, that interval is concave down, positive is concave up.
--Finding Critical Values:
To find critical values, first take the derivative of the function and set it equal to zero, solve for x. The answers you get for x are your critical values.
--Limits:
If the degree on top is smaller than the degree on the bottom, the limit is zero.
If the degree on top is bigger than the degree on the bottom, the limit is infinity.
If the degree on top is the same as the degree on the bottom, you divide the coefficients to get the limit.
Some Concepts that...after reviewing over and over for weeks/months...it just clicked:
--Slope of a normal line:
Take derivative, plug in x. get a slope. however, make sure you use the negative reciprocal of the slope (normal means perpendicular to). use point-slope formula.
--Absolute Extrema:
If you are given a point, plug those numbers into the original function to get another number. Alos, solve for critical values and plug those into the original function. Once you get your second numbers, you set each pair into new sets of points. The highest point is the absolute max and the smallest point is the absolute min.
--Average Rate of Change:
f(b)-f(a)/(b-a)
--Average Value:
This is just an integral times by 1/(b-a).
--Equation of a Tangent Line:
Take the derivative and plug in the x value.
If you are not given a y value, plug into the original equation to get the y value.
then plug those numbers into point slope form: y − y1 = m(x − x1)
And-of course-the things that I always struggled with, and hope to never encounter again even though I know I will....:
--linearization:
Identify the equation
Use the formula f(x)+f ' (x)dx
Determine your dx in the problem
Then determine your x in the problem
Plug in everything you get
Solve the equation
--Riemanns Sums:
LRAM-Left hand approximation=delta x[f(a)+f(a+delta x)+...f(b-delta x)]
RRAM-Right hand approximation=delta x[f(a+delta x)+...f(b)]
MRAM-Middle approximation=delta x[f(mid)+f(mid)+...]
Trapezoidal-delta x/2[f(a)+2f(a+delta x)+2f(a+2 delta x)+...f(b)]
delta x=b-a/number of subintervals
--Volume by Disks:The formula is pi times the integral of the [function given] squared times dx. so just solve it by taking the integral of it and then pluging in the numbers they give you. just like before you'll have two numbers so whatever the answer is for the top one will be first and then you subtract the answer you get for the bottom one. then graph
--Volume by Washers:
The formla is pie times the integral of the [top function] squared minus the [bottom function] squared times dx. so to do this, if you don't have the in between number you have to set the functions equal, but if you do, then it's worked the same way as above. square the formula's that were given and simplify. then take the integral of it and plug in the numbers they give you or you found by setting the formulas equal to each other and then solve like any other one by subracting them. then graph.
--OPTIMIZATION...
I just can't do it =(
Just a shout out:
To the seniors in the Calculus Class of 2010 - - I'll Miss Yall, we've grown up together basically. I won't lie, I didn't like MANY of yall when I first came here because of different things that happened; however, we've all matured [thank God] and we've become friends. Although some of us aren't best friends I can honestly say that I will miss each and every one of you all because yall are the people I spent all day every day with for the past 6 years.
To the juniors in the Calculus Class of 2010 - - I'll miss yall just as much. There's many reason's why I will..some were my first friend at Riverside Academy because of the Cheerleading Squad, some have always been in my life because we grew up living next door to each other, some have been with me through the years with dancing and just being friends, some HOWEVER have become my friend in this past year and believe me I love yall just as much!
So back to the FARWELLS:
No, I'm not sad I'm graduating. My parents say that it hasn't hit me yet...but most of yall know me already and I'm so happy to finally be just about OUT of Riverside Academy that nothing else matters. Also, I'm not sad that I'm leaving any of yall. Yall have been great friends and I'm glad we've become as close as we have or, not as close as we have in some cases, because LIFE GOES ON..We cannot dwell in the past much less live it. I won't be back at Riverside once I'm out, there's no need....I'm finally getting my life started and I CAN'T WAIT!! =) Yeah, I'll miss yall when certain things come up and I realize what [so and so] would've said and they wont be there..but most of all .. I'll miss US as a class together and how we are with each other! So to everyone - - THANKS! And have a GREAT time either being a senior, or going out into the REAL WORLD and EXPERIENCING everything!
Love Always,
Ellie Kliebert
Dylan's last post
Well school is finally over. The only thing that is left is graduation. Hut there are a few things I have learned this past year that will definitely stick with me.
Derivatives
The derivative of simple equations such as x^3+9x^2+7x-3. The derivative is 3x^2+18x+7.
Integrals
S 3x^2+4x-8
=x^3+2x^2-8x
Implicit derivatives
All that you do is take the derivative like normal but of x's and y's but put dy/dx right after where you take the derivative of the y then sove for dy/dx.
Limit rules
1) if the highest exponent is the same on the top and bottom then the limit is the top coefficient over the bottom coefficient of the highest exponents.
2) If the highest exponent is on the top then the limit is infinity.
3) But if the highest exponent is on the bottom then the limit is 0.
Product rule is copy the first times the derivative of the last plus copy the first times the derivative of the second.
Quotient rule is copy the bottom times derivative of the top minus copy the top times the derivative of the bottom all divided by the bottom squared.
Those are the main things that I have learned that have stuck with me all year and will probably stay with me. Calculus this year has been and interesting experience that will stick with me forever. I wish everyone good luck with everything they do.
Derivatives
The derivative of simple equations such as x^3+9x^2+7x-3. The derivative is 3x^2+18x+7.
Integrals
S 3x^2+4x-8
=x^3+2x^2-8x
Implicit derivatives
All that you do is take the derivative like normal but of x's and y's but put dy/dx right after where you take the derivative of the y then sove for dy/dx.
Limit rules
1) if the highest exponent is the same on the top and bottom then the limit is the top coefficient over the bottom coefficient of the highest exponents.
2) If the highest exponent is on the top then the limit is infinity.
3) But if the highest exponent is on the bottom then the limit is 0.
Product rule is copy the first times the derivative of the last plus copy the first times the derivative of the second.
Quotient rule is copy the bottom times derivative of the top minus copy the top times the derivative of the bottom all divided by the bottom squared.
Those are the main things that I have learned that have stuck with me all year and will probably stay with me. Calculus this year has been and interesting experience that will stick with me forever. I wish everyone good luck with everything they do.
Well here it goes.. every thing comes to an end sooner or later right?
I think over the past couple of days I’ve understood that statement more than anything and to begin I would like to say thank you to everyone who was there for me over these past couple of days because even though it doesn’t make it hurt any less, it does make it somewhat easier and for that I am forever thankful.
What can I say, calculus was calculus, we loved it and hated it at the same time. I don’t think anyone will ever be able to understand just how if feel besides all of the people who were in there with me. It was truly memorable and brought everyone closer together. One person needed help and others didn’t hesitate to lend a hand which is awesome to me. We truly excelled this year because of EACHOTHER, it was a team effort =)
To our coach, haha teacher, Brandi, words cannot express how truly thankful I am for you both as a teacher and as a person. Without you, we surely would not have been as successful as we were this year in calculus. You prepared us as much as humanly possible and even after Sara-Rachelle was born you got back to us as quickly as you could so that we would stay on track, that’s dedication. You stuck with me, you didn’t give up, and you always believed in me which means the world to me, thank you so much for an amazing couple of years and you will always be remembered.
OKAYY lets get to some calculusss!
So some things that will stay with me forever and I actually get excited when I see them:
Mmkay, well I realized I love derivatives. Of course the slope formula because it’s on AP free response and I loveeeeeeeee it. Limit rules and the basic integrals.
For derivatives
x^4 + 3x^3 + 6x
multiply the coefficient in front of your variable by it's exponent, then subtract 1 from your exponent.
4x^3 + 9x^2 + 6
Hmmm, some things that I never really caught on to, only because I didn’t focus on them, were substitution and of course how to integrate a fraction, ahhhah I never looked at the comment on my blogs =)
MY Epiphany: I FINALLY HAVE A RELATIONSHIP WITH MY CALCULATER AND IT DOES AMAZING THINGS FOR ME, I LOVE IT.
To my seniors: it’s been a great run guys, I love ALL of y’all. We’re such a close group, I consider each of you family and I don’t know where I’d be today without y’all. It’s all coming to an end, what we’ve waited for our whole high school careers, so let’s hit up my pool these next couple weeks, walk across that stage and never look back. I have complete confidence that everyone will be successful in what ever they chose to do. I LOVE Y’ALL!
To the juniors: yall have also made my senior year one of the best. Yall are an amazing bunch of kids with an extraordinary about of talent, don’t let any one tell y’all differently. I hope that y’all will also have an amazing year next year and remember not to take this time for granted, or anything for that matter. Good luck, I love every one of you.
so this is it, goodbye blogggggerrr!
I think over the past couple of days I’ve understood that statement more than anything and to begin I would like to say thank you to everyone who was there for me over these past couple of days because even though it doesn’t make it hurt any less, it does make it somewhat easier and for that I am forever thankful.
What can I say, calculus was calculus, we loved it and hated it at the same time. I don’t think anyone will ever be able to understand just how if feel besides all of the people who were in there with me. It was truly memorable and brought everyone closer together. One person needed help and others didn’t hesitate to lend a hand which is awesome to me. We truly excelled this year because of EACHOTHER, it was a team effort =)
To our coach, haha teacher, Brandi, words cannot express how truly thankful I am for you both as a teacher and as a person. Without you, we surely would not have been as successful as we were this year in calculus. You prepared us as much as humanly possible and even after Sara-Rachelle was born you got back to us as quickly as you could so that we would stay on track, that’s dedication. You stuck with me, you didn’t give up, and you always believed in me which means the world to me, thank you so much for an amazing couple of years and you will always be remembered.
OKAYY lets get to some calculusss!
So some things that will stay with me forever and I actually get excited when I see them:
Mmkay, well I realized I love derivatives. Of course the slope formula because it’s on AP free response and I loveeeeeeeee it. Limit rules and the basic integrals.
For derivatives
x^4 + 3x^3 + 6x
multiply the coefficient in front of your variable by it's exponent, then subtract 1 from your exponent.
4x^3 + 9x^2 + 6
Hmmm, some things that I never really caught on to, only because I didn’t focus on them, were substitution and of course how to integrate a fraction, ahhhah I never looked at the comment on my blogs =)
MY Epiphany: I FINALLY HAVE A RELATIONSHIP WITH MY CALCULATER AND IT DOES AMAZING THINGS FOR ME, I LOVE IT.
To my seniors: it’s been a great run guys, I love ALL of y’all. We’re such a close group, I consider each of you family and I don’t know where I’d be today without y’all. It’s all coming to an end, what we’ve waited for our whole high school careers, so let’s hit up my pool these next couple weeks, walk across that stage and never look back. I have complete confidence that everyone will be successful in what ever they chose to do. I LOVE Y’ALL!
To the juniors: yall have also made my senior year one of the best. Yall are an amazing bunch of kids with an extraordinary about of talent, don’t let any one tell y’all differently. I hope that y’all will also have an amazing year next year and remember not to take this time for granted, or anything for that matter. Good luck, I love every one of you.
so this is it, goodbye blogggggerrr!
Trina's Final Reflection
This year in calculus has definitely been a whirlwind for me. I came into the class worried that everything was not going to click for me right away and guess what.... i was right.. I struggled with grasping different concepts throughout the first half of calculus but with the help of some friends.. i caught the hang of a few things. Limit rules were the easiest to catch onto because all you had to do was follow the rules. When we learned how to do derivatives i understood the basic derivatives but when it came to e derivatives and ln derivatives, those took a little bit longer... but with a lot of practice and a lot of studying i came to understand those too.
When we were introduced to integrals i did not understand the concept at all... it took weeks of practice and help from my friends for me to finally grasp a piece of the concept. But like derivatives, once i had worked integrals over and over again, the concept became very easy.
Tangent lines were a pain at first as well. One day Mrs. Robinson explained how to do them once more and everything just clicked in my mind. I don't know why but it just did lol. Whenever i would see problems like that on the AP's i would be so happy because i knew i had at least one right :).
Finding volume and area on the free response portions were very easy for me. Mrs. Robinson explained how to do those during lunch after we had taken the tests and i caught on right away. If felt proud knowing that i could at least attempt these problems knowing what i was doing.
One thing i struggled with to the very end was Riemann sums, i don't know if if was because i couldn't remember the formulas or if i just didn't know what to do with them.. but eventually i got a little bit better at them :).
This class definitely helped me in preparation for any math courses i will have to be taking in college and i am glad i got the chance to take the class with my wonderful classmates. A big thanks to everyone who helped me study and for those who supported me throughout the year. A a huge thank you to Mrs. Robinson for helping me understand different concepts and calming my nerves throughout the year. I will miss you all very much and good luck to the juniors who will be taking Calculus BC next year.
Goodbye everyone :)
When we were introduced to integrals i did not understand the concept at all... it took weeks of practice and help from my friends for me to finally grasp a piece of the concept. But like derivatives, once i had worked integrals over and over again, the concept became very easy.
Tangent lines were a pain at first as well. One day Mrs. Robinson explained how to do them once more and everything just clicked in my mind. I don't know why but it just did lol. Whenever i would see problems like that on the AP's i would be so happy because i knew i had at least one right :).
Finding volume and area on the free response portions were very easy for me. Mrs. Robinson explained how to do those during lunch after we had taken the tests and i caught on right away. If felt proud knowing that i could at least attempt these problems knowing what i was doing.
One thing i struggled with to the very end was Riemann sums, i don't know if if was because i couldn't remember the formulas or if i just didn't know what to do with them.. but eventually i got a little bit better at them :).
This class definitely helped me in preparation for any math courses i will have to be taking in college and i am glad i got the chance to take the class with my wonderful classmates. A big thanks to everyone who helped me study and for those who supported me throughout the year. A a huge thank you to Mrs. Robinson for helping me understand different concepts and calming my nerves throughout the year. I will miss you all very much and good luck to the juniors who will be taking Calculus BC next year.
Goodbye everyone :)
Ryan's Final Reflection
So I don't really know how to start this, but I guess I'll try.
Well, I guess I'll have to say that Calculus was one of my favorite classes this year, not just because I liked the subject, but also because I loved my classmates.
To all the seniors (Aimee, Jessie, Ricky, Ryan, Ryne, Chelsea, Mamie, Trina, Kaitlyn, Mher, Steven, Ellie, John, and Dylan): I'm really going to miss you guys.
To all of my fellow juniors (Sarah, Stephanie, Abbey, Malerie, and Ashley): Let's make our senior year the best Riverside Academy has ever seen. And just think that we're the first Calculus BC class EVER.
(**hopefully I didn't forget anybody)
To our teacher, Mrs. Robinson aka B-Rob, thanks for teaching us one of the "hardest subjects" in high school this year.
Something I really believe I've grasped this year is taking derivatives, from just the basic x^2 to chain rules like cos(5x^3)sin(e^8x).
Something I had a struggle with this year was Riemann sums. It took me basically to the last couple of days before the AP to really understand them.
To finish it off
Class of 2010: have a great life in college, and don't forget about us :).
Class of 2011: Let's Get It!
Love you guys,
Ryan B.
Well, I guess I'll have to say that Calculus was one of my favorite classes this year, not just because I liked the subject, but also because I loved my classmates.
To all the seniors (Aimee, Jessie, Ricky, Ryan, Ryne, Chelsea, Mamie, Trina, Kaitlyn, Mher, Steven, Ellie, John, and Dylan): I'm really going to miss you guys.
To all of my fellow juniors (Sarah, Stephanie, Abbey, Malerie, and Ashley): Let's make our senior year the best Riverside Academy has ever seen. And just think that we're the first Calculus BC class EVER.
(**hopefully I didn't forget anybody)
To our teacher, Mrs. Robinson aka B-Rob, thanks for teaching us one of the "hardest subjects" in high school this year.
Something I really believe I've grasped this year is taking derivatives, from just the basic x^2 to chain rules like cos(5x^3)sin(e^8x).
Something I had a struggle with this year was Riemann sums. It took me basically to the last couple of days before the AP to really understand them.
To finish it off
Class of 2010: have a great life in college, and don't forget about us :).
Class of 2011: Let's Get It!
Love you guys,
Ryan B.
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