Approximating square roots Let p1 and q1 be the first-order...

Approximating square roots Let $p_{1}$ and $q_{1}$ be the first-order Taylor polynomials for $f(x)=\sqrt{x},$ centered at 36 and $49,$ respectively. a. Find $p_{1}$ and $q_{1}$. b. Complete the following table showing the errors when using $p_{1}$ and $q_{1}$ to approximate $f(x)$ at $x=37,39,41,43,45,$ and 47 Use a calculator to obtain an exact value of $f(x)$. $$\begin{array}{|c|c|c|} \hline x & \left|\sqrt{x}-p_{1}(x)\right| & \left|\sqrt{x}-q_{1}(x)\right| \\ \hline 37 & & \\ \hline 39 & & \\ \hline 41 & & \\ \hline 43 & & \\ \hline 45 & & \\ \hline 47 & & \\ \hline \end{array}$$ c. At which points in the table is $p_{1}$ a better approximation to $f$ than $q_{1}$ ? Explain this result.

Approximating square roots Let $p_{1}$ and $q_{1}$ be the first-order Taylor polynomials for $f(x)=\sqrt{x},$ centered at 36 and $49,$ respectively.
a. Find $p_{1}$ and $q_{1}$.
b. Complete the following table showing the errors when using $p_{1}$ and $q_{1}$ to approximate $f(x)$ at $x=37,39,41,43,45,$ and 47 Use a calculator to obtain an exact value of $f(x)$.
$$\begin{array}{|c|c|c|}
\hline x & \left|\sqrt{x}-p_{1}(x)\right| & \left|\sqrt{x}-q_{1}(x)\right| \\
\hline 37 & & \\
\hline 39 & & \\
\hline 41 & & \\
\hline 43 & & \\
\hline 45 & & \\
\hline 47 & & \\
\hline
\end{array}$$
c. At which points in the table is $p_{1}$ a better approximation to $f$ than $q_{1}$ ? Explain this result.

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Key Concepts

Effect of Center Proximity

The choice of the center point for the Taylor expansion is vital since the approximation is most accurate in a neighborhood around this point. If the evaluation point is closer to the center, the approximation generally has a smaller error, whereas it degrades as the point moves further away.

Taylor Polynomial

A Taylor polynomial approximates a function by summing terms derived from the function's derivatives at a specific center point. This technique transforms a complex function into a polynomial that closely models the function’s behavior around the point of expansion.

Linear Approximation

The first-order Taylor polynomial, also known as the linear approximation, uses the function's value and its first derivative at a particular point. This yields a linear function that serves as an estimate for the original function near the center, providing a simple yet effective approximation method.

Error Analysis

Error analysis in Taylor approximations involves assessing the difference between the actual function value and its polynomial approximation. It is crucial for determining the accuracy of the approximation and understanding how factors like the distance from the center point influence the error.

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