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Tuesday 10/18
The purpose of these problems of the day was toillustrate a couple of concepts.

1) A function with a domain of all real numbers can have a derivative whose domain is not all real numbers. In both of these examples the domain of f ' (x) is all real numbers except zero.

2) Even though both functions have a domain all reals, and the domain of both f 's is all reals except zero, the graphs of these functions have different features at x = 0, the x value at which f' is undefined. Function f(x)= x ^(1/3) has a vertical tangent line at x = 0 whereas f(x) = x ^ (2/3) has a cusp, and no tangent line, at x = 0. To determine why this happens, and to distinguish between these two features in future problems, limits of f '(x), either side of the domain restriction, need to be considered. Following these two problems, we watched a video of a baseball pitcher pitching the baseball to a batter! media type="custom" key="10940646" We were asked to sketch two graphs (i) Distance from home plate versus time since the ball left the pitcher's hand Most graphs had all positive values, decreased for the first interval of x values, forming a cusp on the x axis, then increased for the remaining part of the flight. We talked a little about how the function curved but didn't really come to a solid conclusion. (ii) Velocity of the ball once it left the pitcher's hand. Most velocity graphs had both negative and positive values since the ball changes direction and speed during its flight towards the bat and away from the bat. We had a discussion about the moment of impact when the velocity instantaneously changes form positive to negative. The question here is whether the velocity is ever zero. Again no solid conclusions were reached. We also discussed the relationship between the two graphs since the graph of the velocity should be the derivative of the graph of the position of the ball.

The last activity was a problem in the book which just so happened to model the situation described above!

We were pleased to see that the graph of the function given in the problem was pretty much the same as the graphs we came up with.

Maybe some kind soul will post the solutions to this problem soon!



Wednesday, 10/19
We defined the terms 'explicitly defined relations' and 'implicitly defined relations' and discovered how to solve for the derivative of an implicitly defined function.







Thursday, 10/20






Friday, 10/21