The arc length of a curve is a fairly simple concept. We've already learned how to find the arc length of circles, but what about functions like cos(x+5) . . . Well first let's define arc length:
The length of a curve if were to be “rectified”.
"Rectified" is simply straightening out a curve from it's "curvy" shape to a linear line that can be measured. Now, you could do this with string, but you can also do it with calculus. Before we get into the calculus, here is a link to visualize a rectified curve: click here
So here is the calculus–again, nothing you don't already know how to do. Since, we're only going to go over how to find the arc length rectangularly
The first step is having a function that you would like to have the , and the formula.
For specifically rectangular form, ds is replaced with a formula showed below:
Since you already know how to find derivatives, the process is fairly simple from here on out. You can either use your graphing calculator by using "fnINT" of the rectangular formula above, or you can use Geogebra! Below is a 10-minute video I made to show how to take a real life object, and find the arc length of it. Not surprisingly, solving with calculus is much easier than measuring a curve with string.
If you would like to check out some examples of how to solve polar functions or parabolic functions, check out this website: click here. The formulas are also fairly simple, and easy to solve based on the calculus you know. Adios amigos.
For the first half of class we took the first Unit 3 quiz, some of us crossing our fingers that the one question we didn't do on the IW was not of the quiz. Just remember, you can redo the IWs, but you cannot redo a quiz, so try to keep up with the IWs, even with the behemoth of stress from applying to college. If your having trouble with stress, take this gentlemen's word of advice:
"The greatest weapon against stress is our ability to choose one thought over another."
- William James
So, if your stressed out about all of the things going on in senior year, take a breath, sit back, and just think about one thing–calculus! Don't let your mind wander to the pending applications to your first choice school, or that environmental science forestry report, just sit there and focus on math, and the stress will just melt away like hot butter on pancakes...It's really that easy!
We then went over the Unit 2 Supercorrection Test; remember if you scored lower than a 70, you can retake it for a score up to 70 points. We went over every question during class, so if you need extra help, or missed class, Mr. O'Brien will be more than happy to help you out. If you are in that situation, keep in mind that since the infamous nordic team is beginning, Mr. O'Brien will not be around after school to assist you when there is practice. Plan to get help during a study hall, or ask another teacher.
Then the new stuff begins. Today was a discussion of all of the functions we've been looking at, that are increasing. Increasing is determined by the derivative, which is always positive when a function is increasing. The concept we explored today was the difference between functions that are increasing more rapidly as the value of x increases, or in other words, the derivative gets larger and larger. Also functions that are increasing more slowly as the value of x is greater, or in other words, the derivative gets smaller as x increases, but never becomes negative. Below is a homemade diagram that I think will help you visualize some of these concepts.
You'll notice two things: one, each of the graphs is increasing (the derivative is greater than zero); two, there are two new vocabulary words. Concavity is visual for me, but incase you're more of a definition kind of guy, then here are the straight facts. Concave up is when the derivative is increasing, and the second derivative is positive. Concave down is where the derivative is decreasing, and the second derivative is negative.
If you are wondering what happens if you have a function where the derivative is increasing, but then switches to decreasing, then you are on the right track because that's called the Inflection Point. The inflection point is simply the point or coordinate where function changes concavity, or in other words when the derivative switches from increasing to decreasing, or vice versa. Here is a homemade diagram to enhance your understanding.
You can also think about this inflection point, as where the second derivative changes sign, from negative to positive, or positive to negative. Also, when the second derivative is undefined, or possibly zero. Remember, just because the second derivative is zero, does not guarantee there is an inflection point.
Lastly, we reviewed the first and second derivative tests. Instead of just showing you a diagram, let's watch a video of a wonderful lady who provides examples to apply these two tests. The video is below.
A quick tip though, whether a critical point is a min or a max can be determined by, the max is when the second derivative is negative, and the min is where the second derivative is positive. It's not what you would like to think, that the max is positive, and the min is negative, but just know that.
Remember to do the homework, it's IW #4: p. 219/7, 13, 21, 23, 39, 45-55 odd, 56-60 all. Good luck, and hope you enjoyed your holiday! UPDATE: So. It's been a long quarter–we've learned a lot. However you can remember to look back on your Unit 3 Finely Crafted Opprotunity Day, whip out those supercorrections, and take a look at problems 20, 17, 7, 10, and 11, and that should help you a lot to prepare for the Midterm exam, or even the AP exam. The relationships between the first derivative and second derivative are not so scary after taking a fourth look (IWs, test day, supercorrections, and make-up test) at those O'B test questions.
We began class with what Mr. O’Brien referred to as the “Big O’Brien Pep-talk”. This speech was an effort to motivate calculus students, particularly the period 4 class, to ask more questions for the IW. Some of his points consisted of: even though we have a lot going on senior year (often more than we can handle), math is still important; asking questions should not be a shameful task, it should be gratifying and rewarding to the class; and “Period 2 is kicking your butt.” For some reason this was directed towards myself (Eliot), because Cole Ellison is in Period 2 and also on the mountain biking team. I believe O’Brien was trying to raise competitive spirits in our class. On the subject of biking, here is an inspirational quote from an amazing cyclist about working hard:
“I have always struggled to achieve excellence. One thing that cycling has taught me is that if you can achieve something without a struggle it's not going to be satisfying.”
- Greg LeMond
The message concerning IW questions is, like painfully struggling to achieve excellence in cycling, if you struggle with the homework, you will understand concepts better and therefore achieve excellence that is satisfying. Sliding by calculus without struggling with the homework and falling into the copying routine, you will not be satisfied in a way that is meaningful. That’s enough words of wisdom.
Before we took the quiz O’Brien showed us the holy grail of math, the solutions manual to all of the problems in the book. That’s right, all of the workings and answers to everything: uncensored mathematics. Then we took the quiz on IW’s 1-6.
After the quiz O’Brien let us know that we probably made mistakes because the questions were hard, especially considering that our class didn’t ask a sufficient amount of questions on the homeworks. However, he gave us hope, as he stated that there was a reason we were having trouble finding the derivatives on the quiz, here’s why:
“We know how to take derivatives of functions like the square root and linear functions . We know how to take derivatives of their sums and differences and even of their products and quotients. But, we don’t know how to take the derivatives of compositions .” *Note: Compositions are functions within functions, or to sub a function into another.
So, now we can learn a rule that we can really appreciate. This is called the Chain Rule, which is an alternative way of finding derivatives, just like the tactic of simplifying before applying the Damion-trick. Here is the Chain Rule defined numerically speaking:
Wowzah. Let’s break it down into words so that it’s a little bit more easy to understand.
“Take the derivative of the outer function evaluated at the inner function and multiply it by the derivative of the inner function.”
So now that we have this all powerful rule, let’s try applying it to some of the problems we’ve already seen before. Refer to the Chain Rule above to follow along.
Unit 2 Quiz 2: 5.) Find the derivative of at
To clarify the rule, think of it in steps. 1. Take the derivative on the outside function, the f function. 2. Evaluate not at x, but at g(x). 3. Then multiply it by the derivative of the inside function
IW #6: problem 43 Find the derivative of
Ok, so two functions composed together. The cosine function whose derivative you know (it’s ), and the linear function whose derivative you know (it’s ). The chain rule says you take the derivative cosine, and evaluate it at . We would like to just stop there, but the Chain rule says we must multiply what we just did by the derivative of the inside function. How nice is that? Compared to the double angle stuff we had to deal with in 43 originally, this is amazingly simple! Let’s do one more.
Problem 6 Evaluate this derivative:
So, now that we’ve learned this rule it’s all over right? No, because we have yet to learn how to find the derivative of logarithmic and exponential functions. If you’re still not satisfied with this explanation of the Chain Rule, check out this video out from our old friend partickJMT, it can be helpful to look at some different examples of the Chain Rule.
Remember, don’t be afraid to ask questions on the homework! IW #7 is pg. 158 / 15, 17, 19, 23, 27, 29, 33, 39, 53, 55, 58, 72, and 73 UPDATE: There is some valuable information that I have for everyone. Mr. O'Brien has been keeping something from us, something so valuable, it will change the way you think about mathematics. In this post we explored finding a few derivatives, but the reality of it is, the second step is already the derivative. The only reason we simplify is make all the calculus look pretty. It has been leaked that the AP AB Calculus exam does not require you to simplify on free response questions, however Mr. O'Brien would like us to learn to simplify anyways. This is most likely true, since simplifying skills will be crucial in order to answer the multiple choice questions, but, no need to freak out on a quiz anymore because you don't have time or don't have the knowledge to simplify.
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