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Quiz 3.1-3.4

I have just finished grading the indicated quiz. The average was an 80.8. This is very low. I know the material was difficult, but we need to clear this up after harvest because we will really start running when we get back. Derivatives is a large concept that has many pieces. If you miss one, you could be lost for awhile or never come back. I want all of you to stay in this class, so I want to work on getting your skills up to scratch when we get back. I think you should really pick up and study the first three sections a little over harvest. You are getting ready to head into the real world so you are in charge of assigning yourself some work to do so that you will be ready to continue on when we return.

Some ideas that bothered most of you.
- Limit definition of the derivative, how to use it and when.
- Graphing the derivative
- Product Rule
- Quotient Rule
- Derivative notation
- Basic algebra skills (I saw a lot of illegal cancellations)

Last year, I started mandatory tutoring sessions about this time. I hope that you will come in on your own accord. Mr. Sprague and other tutors are available in the Transitions center or after school, but they are not the same as me. They are usually very good, but sometimes I teach it differently then they do. In other classes, this is usually not a problem, but in this one it can be. I hope many of you will come visit me the first week back to pick up the missing concepts. If you don't visit me, and you should, I will probably come visit you.

Also, don't hesitate to place a comment on this blog or send me an email. I am pretty good at responding. I still have students from last year writing to me to help them solve some calculus problems. I am here to help. I will stay after school till about 5pm, if I am needed. Just let me know so we can solve this issue ahead of time and not try to catch up when it is too late.

Have a nice harvest vacation, but practice some calculus.

Motion

Motion is a large topic of calculus and physics. In calculus, motion questions are used to evaluate a student's understanding of the difference quotient, derivatives, and integrals. For now we will focus on the first two, but we will look at integrals in time.

The whole show begins with a position function, s(t). If you want to find something's instantaneous velocity [v(t)], you just take the derivative of the position function. If you want to find something's instantaneous acceleration [a(t)], you just take the derivative of the velocity function. I equate these ideas to finding the tangent line, so I always remember to take the derivative.

We also talked about displacement. Displacement is your change in position. I always think about it as the distance from my starting point after I'm all done with the movement. "How far is it from where I began?"

We also talked about average values. Average velocity is the change in position divided by your change in time. Average acceleration is the change in velocity divided by your change in time. I just equate these ideas to the secant line, so I always remember what it is that I am doing.

I also asked that you look at example 6 on page 129. This talks about how derivatives are used in economics. In general, profit = revenue - cost or money in - money out. If you want to find the marginal revenue, the amount of money gained by producing one more unit or how the revenue is changing at that instant, you simply take the derivative or the revenue function. If you want to find the marginal cost, the extra cost of producing one more unit or how the cost is changing at that instant, you simply take the derivative or the cost function. If you want to find the marginal profit, you subtract the marginal cost from the marginal revenue.

Derivative Rules

There are many rules that can be used to find the derivative of a function. These are quick rules that can be proved using the limit definition of the derivative, but once you know the rules you can use them to save time when finding the derivative. The six basic rules we looked at are shown below. These rules assume that c is a constant, u and v are functions in terms of x, and that the derivative exists for all x's in the domain of the function.



You must also be able to read the notation in the book. Below is a brief list of the notation that your book and many other mathematical texts use to identify the first-order derivative and its successors.



In the next sections, we will introduce more rules that will help us find the derivatives of trigonometric, logarithmic, and exponential functions.

GeoGebra

Today, we finished our study of the derivative using GeoGebra. We found the derivative of many functions using a graphical method. The graphical method involved creating a point B that had the coordinates (x,m), where x is the x-coordinate of the point on the function that the tangent line goes through and m is the slope of the tangent line. Point B traced out our derivative and then we were able to guess the derivative from the function.

We found the following derivatives:


The next step will be to develop rules so that we can find the derivative without using the limit definition of the derivative or having to rely solely on graphs.

GeoGebra

It is very unfortunate that our technology did not work so that we can get a visual of the derivative. The lesson was to connect the function with its derivative. We are going to start learning rules that will enable us to find the derivative much quicker than using the limit definition of the derivative. Remember, the derivative is a function that tells us the slope of our original function, everywhere. By the end of the week, we will have many skills in our repertoire that will allow us to find the derivative of many complex functions (more than just polynomials).

It is important that you keep GeoGebra at the top of the pile of useful tools (right next to the unit circle). The address for GeoGebra is http://www.geogebra.org/cms/ or you can click the word GeoGebra in the title of the post. At your home, you should be able to Webstart the program and create the worksheet like we tried to in class. This program can help you graph functions, trace the values, and copy the graphs into a document very easily. I suggest that you just play with the program a little.

Since our technology will not work individually, we will work on this tomorrow as a large group to see if we can figure out the derivatives from the tracing of the tangent lines to every point on the function. Hopefully, my computer will be able to complete this task, unlike yours.

Differentiability

The derivative of a function exists if the left-hand derivative is equal to the right-hand derivative, just like with limits. There are only a few times that the derivative does not exist, and it is usually narrowed down to a specific point, or set of points.

The derivative fails to exist at a ...

1. Corner, like the one found in y=abs(x). This function is not differentiable at x=0. If you look at the left part of the function (x<0), y="x.">0), the function is described by the equation y= -x. The slope of the tangent line (the derivative) is -1. The two derivatives do not match, so the derivative does not exist at x = 0.

2. Cusp, like the one found in y = x^(2/3). If you look at this graphically you can see that the slope at x = 0 changes very quickly from positive to negative. We will look at the slope of this equation later when we have some more rules for derivatives.

3. Vertical tangent, like the one found in y = cuberoot(x). If you look at the graph of this function you will see that at x = 0, you will have a vertical tangent line. This is where I said you may want to use a small piece of paper to represent the tangent line. When you get around x = 0, your piece of paper will be pointing straight up, indicating a vertical tangent. Again, we will look at calculating this derivative (to the left and right of x =0) later on.

4. Discontinuity. We already studied discontinuities back in chapter 2. As a refresher, they are point (removable), infinite, oscillating, and jump. This means that if we find any discontinuities, our function will also not be differentiable for that particular x-value.

Now I gave you some instances, but it was merely a coincidence that our points where the functions were not differentiable were at x = 0, so don't fall into the trap of automatically thinking that is. Have a great long weekend.