Scribe Post 10/2

     We started class with a quick warmup that focused mostly on using our calculators to find derivatives numerically and graphically(to do it numerically math 8, function, x, value of x, precision(optional)). It is really hard to explain without a visual so here is a youtube video from DrPhilClark

   
     Just for review, in order to find a symmetric difference quotient use the equation   .  Mr. O'Brien told us that the calculator does not have mystical elves working inside to find the derivative for us, but instead it just uses this equation.  In order to graph a derivative on your calculator use the syntax illustrated by the picture below:

     The function we looked at in the warmup is the Y1 function in the picture above, and the picture below shows the graph of the function  :

     When we looked at the graph we found that when the slope was positive, the door was opening, and when the slope was negative, the door was closing.

      At this point Mr. O'Brien made a graph FURRY and I noticed some people wanted to know how to make their own furry graphs. For those of you who dont know how to do that here is a little video I made.  Go to the y= window and press the left key a bunch of times. Once you have the line next to Y1 alternating from / to _ just press enter: 

      Mr. O'Brien showed us something strange: when we graph the original and derivative together, the vertical scale on the graph is used for both y and y'.  On the original function, the points were in the form (time, º), but the derivative showed (time, º/second). In other words, the original shows (x,y), and the derivative shows (x, slope). The original is in blue and the derivative is in red

     After further investigation we found that there was a zero for the graph of the derivative.  We knew this because the graph started above the x axis with a y intercept of 200, and dropped below with a minimum value at about (2.885, -27.067).  According to the Intermediate Value Theorem* because 0 is between 200 and -27.067, at some point with an x value between 0 and 2.885, the function will cross the x axis.  The x value of zero for the derivative was the x value of the Maximum of the original function!  At that point, the door is neither opening or closing.
*for review of the Intermediate Value theorem check Sarah's blog post

    In order to use solver on the calculator press the MATH key then scroll down to SOLVER. Enter your equation, then press ALPHA, SOLVE(ENTER). For a more in depth explanation watch this very basic video on Solver by learning4mastery.  It involves a quadratic, but the basic concepts are applicable to derivatives.  You may not even need to watch much if you're just looking for keystrokes.

     If you have already entered a function into the y= window and you are feeling a little lazy, you can copy that function by pressing the VARS key then the right arrow to get to Y-VARS, press ENTER on the "Function" option and then select the equation you want.

     For more calculator help check out this website.

    While working on the warmup we learned new vocabulary: a Point of Inflection is the point of a graph at which the slopes curvature changes signs, or in our example below, the point at d(2.885). Flecto in Latin means to bend.  Inflection is the point at which the function changes from bending down to bending up! It goes from concave to convex! Check out wolframs definition here.

(side note: O'Brien reminded us that sinø/ø is not 1, but the lim of sin/ as x-> =1)
     A question was raised about #2 on the test, which related to #3 of the practice test(IW#8). On #3, it looked like the lines only crossed 2 times, but we have to remember that an exponential equation does shoot up eventually.

Question 3

At how many points do the graphs of the functions   (in red) and

(in blue) intersect?

At first it appears that there are only 2 intersections at points C and D, however, if we zoom WAY out we find that there is a third intersection at point E below. 

    In the same way #2 on the test at first looked like this:

How many zeroes does the function g(x)=sin(ln x) have for 0< x ≥1?

when we graph this function on a simple window we get this:

Okay, so now you might say "easy 2 intersections." Instead when you zoom in you can see 3 below, and if you zoom in more, and more, you will find more and more zeroes. The sin made the ln(x) oscillate as it got close to 0.  We usually wont be able to see these types of things on graphing calculators, but instead we need to see things like the fact that as we plug numbers close to 0 into ln(x) we get infinitely small numbers.  Then when you put all of those numbers into sine, you get numbers oscillating between -1 and 1 infinitely.  Finally by using the intermediate value theorem we see that if there are infinite oscillations over the x axis, then there is an infinite amount of 0s :

Now we checked out the Power Rule, Mr. O'Brien showed us how the rule worked with the following example:

This table shows a function of a certain power and it's derivatives.  Notice it works for negative and fractional exponents too!

The Power Rule and the Damian Trick are the same thing! when the function is , the derivative is .  What a POWERful function!

We went to the thatquiz links posted by O'Brien to work on using this rule and class ended.

     If there is a coefficient on the X, ignore the coefficient at first: k*f '(x). It is the same thing just more mathematical than the explanation offered by physics. You can prove the mathematical way by using limits to find the derivative, you take the k out to find the limit, then multiply by k.


FORESHADOWING:
Sum/Difference Rule:
[f(x)+g(x)]=f'(x)+g'(x)
[f(x)-g(x)]=f'(x)-g'(x)

The ln(3x)= ln(3)+ln(x)
       ln(20)=ln(4)+ln(5)
       ln(20)=ln(2)+ln(10)
       ln(20)=ln(1)+ln(20)
So: horizontal dilations of a derivative would be simply a translation for all log functions!

UPDATE:

I mentioned the sum/difference rule above, but what about the product and quotient rules?

Product rule:

So with this rule, you multiply f(x) with the derivative of g(x) and vise-versa, then simply add the products and you have the derivative of a product!

The Quotient rule is similar, but with a few crazy differences:

Notice that in the numerator it is the same as the product rule with one key difference: SUBTRACTION.  Also, notice the funky denominator: g(x) squared.


Cal

 NEXT SCRIBE: jk, side deals going down, Will is the next scribe again…