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Today, after discussing the Wiki, we started working with polynomials, using the in class experiment on page 170, which is: math

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Suppose $f(x)=x^{3}-2x^{2}+7$, so that $f(5)=82$. How close must x be to 5 to meet the following conditions?

We started solving the first of the set of problems: 1. \left | f(x)-f(5) \right |< 0.1

But what does this mean? It means that $f(x)-f(5)$ is in between $-.1$ and $.1$, and in this example, the graph of the polynomial $f(x)$ (which looks like a line because we've zoomed in so far) is between 81.9 and 82.1.

But how should we find the x-value where this is true for f(x)? We started off with a x range from 4 to 6 {}$ math



math {}$ In this graph, the polynomial looks straight, and it clearly does not fit within the boundaries of 81.9 and 82.1. To try and fix this, we shrank the x range down to $4.9< x< 5.1$. {}$ math



math {}$ Though this graph looks better, the polynomial is still crossing the lines $f(x)=81.9$ and $f(x)=82.1$. After a bit more shrinking of the x range, we found that if x is between 4.999 and 5.001, then $\left | f(x)-f(5) \right |< 0.1$. Or, in the geeky math way, $\left | x-5 \right |< .001$. {}$ math



math {}$ We then did questions 2 and 3: {}$ math math {}$ Question 2: $\left | f(x)-f(5) \right |< 0.01$ {}$ math math {}$ Answer 2: $\left | x-5 \right |< .0001$ then $\left | f(x)-82 \right |< .01$ {}$ math



math {}$ Question 3: $\left | f(x)-f(5) \right |< 0.001$ {}$ math math {}$ Answer 3: $\left | x-5 \right |< .00001$ then $\left | f(x)-82 \right |< .001$ {}$ math



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To do this on your calculator, without a super specific window you can simply find the intersections between the lines, and those will be the limits for your x range. There is also more then one right graph for each problem. Something to think about: Why does this work? Why does it work for some polynomials but not others?

\begin{itemize} \item {\bf Mrs Tyson:} Actually I think I asked why does this work for polynomials? Would it work for all functions? \item Really nice explanation. I like how you showed the graphs and spoke about what we wanted to see in the graphs. Notice that while you can use your graphing calculator to solve for a point of intersection, you can also just use your graphing calculator to find a window like the ones you showed with Wolfram alpha. \end{itemize}

\begin{itemize} \item {\bf Kaila Simpson:} Wow the graphs on this page are spectacular and they are really helpful! Good job Katie! {}$ math