User:Egm6321.f12.team6.bang/report7/problem 7.1

=Report 7=

Problem Statement

 * Use matlab to plot $$ \displaystyle F(5,-10;1;x) $$ near $$ X=0 $$ to display the local maximum (or maxima)in this region.
 * show that $$ \displaystyle F(5,-10;1;x)=(1-x)^{6}(1001x^{4}-1144x^{3}+396x^{2}-44x+1) $$

Given

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$$ \displaystyle F(5,-10;1;x) $$
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 * (7.1)
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$$ \displaystyle F(5,-10;1;x)=(1-x)^{6}(1001x^{4}-1144x^{3}+396x^{2}-44x+1) $$
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 * (7.2)
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Nomenclature

 * $$ \displaystyle F(a,b;c;x):=\sum_{k=0}^\infty\frac{(a)_k\,(b)_k}{(c)_k}\,\frac{x^k}{k!} $$

Solution

 * This problem is also presented in Report 5 Problem 5.10. Many of groups presented the plotting of positive zone of the graph. As professor emphasized in the class lecture, the plotting of F(5,-10;1;x) is focused on the negative x value this time. In order to generate a graph, following MATLAB code were typed in.
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 * This will generate the following outcome. The first outcome is the function F that has been solved by MATLAB.
 * Result.jpeg
 * As you can see from above, we can conclude that F=(5,-10;1;x) can be expressed in the relationship as Equation 7.2.
 * Also, the second outcome will be the plot.
 * Hypergeometric Plot.jpg
 * The zoomed in plot is not as smooth. However, if the line is smooth, then we can anticipate that when x is 0, then it will go through 1. However, when x is negative values, you can see that the zoomed-out plot goes infinitively large.
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 * The zoomed in plot is not as smooth. However, if the line is smooth, then we can anticipate that when x is 0, then it will go through 1. However, when x is negative values, you can see that the zoomed-out plot goes infinitively large.
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Therefore, we can conclude that $$ \displaystyle F(5,-10;1;x)=(1-x)^{6}(1001x^{4}-1144x^{3}+396x^{2}-44x+1) $$.
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Also within the domain of -1 to 0, The local minimum will be at 1. However, there will not be any local maximum since the graph extends to infinity.
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