AP Calculus » Class Discussions

Corrections to my post on 12/6/07

scribe — Mon, 10 Dec 2007 14:50:51 +0000

Okay, so I realized in class today that the product rule is written u’v +uv’ (in my blog I used capitals- sorry!)

posts from now on

scribe — Wed, 17 Jan 2007 03:38:22 +0000

the pdf. files are great but do you guys think that from now on we could write what topics we went over that class as a heading. I think it will make it easier for everyone if they are looking for notes on a specific topic and don’t exactly remember the day we went over it so that we don’t have to l

Average and Instantaneous Rate

scribe — Thu, 02 Nov 2006 23:56:54 +0000

From these labswe did in class I learned about the many differences in instantaneous rates and average rates. Ella did a very good job describing the differences and why the labs were helpful. One lab showed showed how inaccurate it is taking an average rate in relation to large interval. The average rate is focused more on the specific behavior of a specific point with smaller intervals. This then makes it easy to see that deriative is the limit of the average rate as delta x approaches zero. Theses activities also showed how derivatives on a graph are positive as the slope increases and negative as the slope decreases. Also the x intercepts are the maximum and minumum points of the function.

Reflection on the class labs

scribe — Sun, 29 Oct 2006 15:48:12 +0000

I have just completed the labs on instantaneous rate of change and tangents. I didn’t really learn much from the instantaneous rate of change lab, but it the work on the lab made it clear that the derivative of a point found by the secant line is much more accurate if c is small. (Given that c is the differance between x and a nearby point). From the tangent line lab, I learned that the derivative is equal to the slope of the tangent line at any point x. On a sine curve, the derivative has the greatest value at the x-intercepts of 0, 2Pi, 4Pi, Ect. These labs, overall, were a great review of all the ways in which derivatives can be found.

Lab, october 12

scribe — Thu, 12 Oct 2006 17:08:16 +0000

What I did learn from the exercises

• I have learnt what the average rate of change between two points on a function looks like. And when one point approaches the other, average rate approaches instantaneous rate. Even more I have learnt what the instantaneous rate look like.

• A tangent line is a line that interact the function at a point. At this point you find instantaneous rate for the function.

• About the slope: The slope can have positive slope which means it’s increasing. When the function is increasing the derivative is positive than zero. On the opposite when the slope is decreasing the derivative is less than zero.

• From looking at the points on your graph you can tell if there is a maximum or a minimum. For example if the graph is positive, zero and than negative you have got a increasing maximum. On the other hand if you got a negative, zero and then positive you must have a minimum where the graph is decreasing.

Reflection on the Two Class Labs

scribe — Thu, 12 Oct 2006 05:41:06 +0000

The Getting Off on a Tangent lab showed a number of things. It first pointed out that the average rate of change between two points on a function is the slope of the secant line between the two points. Then as one point approaches the other, the average rate approaches the instantaneous rate. The tangent’s slope = instantaneous rate at P = f’(x_p). For increasing functions the derivative is greater than zero. For decreasing functions the derivative is less than zero. At the high and low points in the graph the derivative equals zero. The derivative is the instantaneous rate of change of f(x) with respect to x at x = c. Also, when the tangent line is above, it is concave down and the slope is decreasing.

The Rate of Change Lab demonstrated visually what the differences are between average rate and instantaneous rate, in that you could see when one of the two methods would be most effective. The average rate of a long period of time does not effectively summarize the variable rates during the course of the time. The instantaneous rate is effective when trying to find what the rate was at a given time. As the intervals got smaller and smaller, the average rate became much more accurate and was also closer to the instantaneous rate. The last parts of the lab also showed how you can find the derivative by taking two points, and essentially finding the slope of the segment. The change in y over the change in x equals the average rate, and the slope of the secant line equals f(x) – f(c) over x – c.

Instantaneous rate vs. Average rate

scribe — Wed, 11 Oct 2006 23:12:44 +0000

These labs further differentiated between instantaneous rate of change (derivative) and average rate of change. The gsp lab illustrated using a tangent line how the slope of the derivative increases as the function increases and decreases as the function decreases. This is especially helpful in cases where we are not able to graph the function and need to describe the functions behavior at a particular point. It also showed how the high and low points of a function would have a derivative of 0 because it is neither going up or down. However this is also true for example for a cubic graph when the function is y=x^3 when x=0 the derivative is also 0.

The second lab further demonstrated how average velocity can be misleading as a variety of different graphs all very different had the same average velocity. It also showed me how inaccurate average velocity is sometimes. It also showed me how when we take the average rate of a function getting closer to the limit the more accurate and closer it is to the derivative (instantaneous rate of a point). It also illustrated how much more accurate a derivative is versus the average rate in discussing the behavior of a function at a particular point. It also made it much clearer of what a derivative is: the behavior of the graph (slope) of a function at a particular point and its connection with the change in x as it becomes more accurate as the change in x approaches 0.

**Also here is what the hw is for tonight:

all Q’s, page 81 1, 2, 3-19 odd, 20. Also fill out what you learned from the two activities on the Blog.

**my scribe post will be up first thing in the morning because i need to use the school computers so i can put in the equation editor thing it won’t let me on the pc**

-Laurie

Reflection on in class work Oct. 10th

scribe — Wed, 11 Oct 2006 00:13:08 +0000

Well ella summed the labs up very well. The first lab helped show me where we will be going, in describing the derivative in a graph and how the derivative is related to the the functution. its poositive when the function increasing, negative when the funtion is decresing, and zero at the appex of curve. This would make the graph of the perivative of a sighn wave just another sigh wave but offset from the origonal. The second lab simply once again explifies the difference between average and instantanous rate and how inacurate average rate is over a long period but average rates over samller and samller rates can determine the instantanous rate by finding the limit of a function relationg average rates.

Average Rates Versus Instantaneous Rates

scribe — Tue, 10 Oct 2006 18:04:12 +0000

From these labs I have learned about the differences between instantaneous rates and average rates. The second lab shows us about what can be inaccurate about taking an average rate over a large interval. As we take the average rate over a smaller and smaller interval, it becomes closer to describing the behavior of the graph at a specific point. This helps to make sense of the explanation of a deriative as the limit of the average rate as delta x approaches zero. Theses activities also helped to explain more about derivatives in that they are postive when the graph is increasing, negative when the graph is decreasing, and zero at maximums and minimums (or in some cases not), which will help to describe the behavior of a graph without a visual representation.

Discussions

cevrard3 — Thu, 20 Jul 2006 19:41:24 +0000

This is a place where I hope we can have rich discussions about problems or concepts etc. It will be what you make it. It’s a great place to discuss mathematical topics that we don’t have time to address in class because of time constraints.