Scribe Post 10/18

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.