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Differentiation plays a crucial role in the process of creating Taylor polynomials. The core idea behind differentiation is to find the rate at which a function is changing at any given point. For Taylor polynomials, we use these rates of change to build a polynomial that approximates a function around a specific point. In this exercise, the function given is \( f(x) = \sqrt{x} \). The process begins with finding the derivatives of \( f(x) \) up to the third order, since a third-degree Taylor polynomial is required.

To help you better grasp this, here is a quick rundown of derivatives:

• The first derivative, \( f'(x) = \frac{1}{2}x^{-1/2} \), represents the slope of the tangent line to the curve at any point \( x \).
• The second derivative, \( f''(x) = -\frac{1}{4}x^{-3/2} \), provides information about the concavity of the function, telling us how the slope is changing.
• The third derivative, \( f'''(x) = \frac{3}{8}x^{-5/2} \), helps us understand the rate of change of the concavity.

Derivatives are evaluated at \( x=1 \) because the Taylor polynomial approximates the function near this point.

Taylor polynomials are a powerful tool for function approximation. They use information about a function's derivatives to create a polynomial that closely resembles the function near a specific point. This is particularly useful when dealing with functions that are complex or difficult to work with directly.

In this exercise, we approximate \( f(x) = \sqrt{x} \) using a Taylor polynomial centered at \( x=1 \). The polynomial is constructed using the derivatives of the function evaluated at this point. The formula for the third Taylor polynomial is:\[P_3(x) = f(1) + f'(1)(x-1) + \frac{f''(1)}{2!}(x-1)^2 + \frac{f'''(1)}{3!}(x-1)^3\]Substituting the calculated values, we get:\[P_3(x) = 1 + \frac{1}{2}(x-1) - \frac{1}{8}(x-1)^2 + \frac{1}{16}(x-1)^3\]This polynomial gives us an approximate value of \( \sqrt{x} \) near the point \( x=1 \). Despite its simplicity, this approximation can be remarkably accurate.

Graphing provides a visual representation of the function and its approximation, allowing us to compare how well the Taylor polynomial mimics the original function. By plotting both \( \sqrt{x} \) and its Taylor polynomial \( P_3(x) \), over a given interval, we can observe the accuracy and limitations of the approximation.

For this task, graph both functions over the interval \([-1, 3]\). Note, however, that \( \sqrt{x} \) is only defined for \( x \geq 0 \). The graph of \( \sqrt{x} \) will appear only on the right side of the y-axis, whereas the Taylor polynomial, being a complete polynomial, will extend across the entire interval.

By comparing the two graphs, you'll notice how the Taylor polynomial closely follows \( \sqrt{x} \) near \( x=1 \), but the further away from this point, the more the approximation deviates from the actual function. This characteristic highlights the trade-off between polynomial degree and approximation accuracy.