Scribe Post 11/20

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.