Mr. O'Brien started off class by showing us a fun new game derivative game to play. So, if are looking to have some great fun and learn some cool math at the same time, try this new derivative matching game!
http://mathdl.maa.org/mathDL/47/?pa=content&sa=viewDocument&nodeId=2651&pf=1
But, the real fun began when O'B brought us back to the dark days of Pre-AP Calc and our 4th quarter project. Here we looked at the relationship between position (x), velocity (v), and acceleration (a).
With your knowledge of derivatives, looking at these equations now, you can see that position (x), is the function and velocity (v) is the derivative. Even further than that, acceleration (a) is the derivative of velocity (v) and the second derivative of position. Therefore acceleration (a) is the anti derivative of velocity (v) and velocity (v) is the anti derivative of position. NOTE: The concept of anti derivatives may not make sense to those of us who are not in AP physics but that is OK. Don't let the AP Physics kids bully you for not knowing.
This can also be looked at when we define a derivative is a rate of change. Velocity is the rate of change of position, and acceleration is the rate of change of velocity.
When finishing up my blog post I read over our good friend Gabe's blog post from period 2. His post included a very helpful visual that may help you to more fully understand what I have just talked about:
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This brings us further into the notation of derivatives. Many times in class we have seen O'B use the notations y', y", and y"' for the first second and third derivatives. But now, using what we have just learned, we can see where notations are derived from.
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We then moved on to look at problem 28 on page 138, a revenue problem:
We learned that the derivative can be used as a rate of change in economics as well. According to Mr. O'Brien economists think they are smart because they have fancy graphs, but really marginal cost/revenue/profit is just the derivative of cost/revenue/profit. So, finding the rate of change in this revenue problem just became a lot easier.
We decided to graph this problem, and use the margin of 5 chairs to 6 chairs as an example.
A is the profit at 5 chairs.
B is the profit at 6 chairs.
a: equation for the tangent line
b: profit at 6 chairs, according to the tangent line
m: slope of the tangent line
When graphing the derivative at 5 chairs we found that the difference between the two profits is $55.56. You can see this by looking at the slope of the tangent line at 5 chairs. These two differences in profits, $7.94, are close enough to each other to hold the derivative rule as true and give economists a good idea of the margin of profits by just looking at the derivative at each point.
Many people took note of the curve to the graph and wondered why revenue begins to decrease at a slower rate the more chairs that are produced. Ideas were thrown out such as labor and material costs, or possibly supply and demand.
Finally the bell has rung and the weekend has begun!! woooooohooooooooooooo
Please note that the IW due next class is pg 135 / 9, 13, 15, 19, 23, 27, 40, 42, 43
UPDATE: On the last multiple quizzes we have had to match the graph to the graph of its derivative. So my update is going to be about how to more easily do that. So if you look at the f(x) graph, the position graph, you know that the section of the graph where the slope of the f(x) graph is positive, the f'(x) graph will be above the x-axis. This also means that when the position is moving forwards, the velocity is positive. So when the position is moving backwards, and the slope is negative, the f'(x) graph is below the x-axis. So in between these change of direction on the f(x) graph, the f'(x) graph crosses the x axis. This makes sense, because right where the f(x) graph changes direction the derivative is 0, so on the graph of the derivative, f'(x), crosses the x-axis.
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