Section+3.5+Beginnings

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What is a tangent line? \begin{itemize} \item A tangent line is the best linear approximation of $f(x)$ at $x=a.$

\item The secant line from $x=a$ to $x=b.$ becomes a good approximation of the tangent line when "b" approaches "a."

\item The tangent line is also the linear parts of the Taylor Expansion of $f(x)$ at $x=a,$ as well as the remainder when $f(x)$ is divided by $(x-a)^2.$ \end{itemize}

It is important to remember that while a tangent line to a circle may only cross the circle in one point, the tangent line to a polynomial may cross in more than one place and may also cross over the graph of that polynomial function. Mrs. Tyson also pointed out that every polynomial that the class had previously experienced was the Taylor Expansion of that polynomial at $x=0!$

The class proceeded to work on Check Your Understanding \#1,5

\#1 involved finding the tangent line to $y=x^2$ at the point (5,25). Synthetic division was applied twice to find the Taylor Expansion at x=5, and then was applied again, putting zero in the box (as shown below). math

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 * 1) 5 involved doing synthetic division of the polynomial $f(a)= a^2 + a$, in order to determine a general formula for finding the slope of the tangent line to the polynomial. It was determined that the general formula for the slope of the tangent line was 1 +2a. The class used this information to fill out the table associated with problem #5.

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