Question

In the following exercises, find the Taylor polynomials of degree two approximating the given function centered at the given point. $$ f(x)=\sqrt{x} \text { at } a=4 $$

Short answer

The Taylor polynomial is \( P_2(x) = 2 + \frac{1}{4}(x-4) - \frac{1}{64}(x-4)^2 \).

Step-by-step solution

Identify the function and point

The function given is \( f(x) = \sqrt{x} \) and we are asked to find its Taylor polynomial of degree 2 centered at \( a = 4 \).

Calculate the function's derivatives

First, we need to compute the first and second derivatives of \( f(x) \). The first derivative is \( f'(x) = \frac{1}{2\sqrt{x}} \). The second derivative is \( f''(x) = -\frac{1}{4x^{3/2}} \).

Evaluate the derivatives at the center point

Evaluate the function and its derivatives at \( x = 4 \):\[ f(4) = \sqrt{4} = 2 \] \[ f'(4) = \frac{1}{2\sqrt{4}} = \frac{1}{4} \] \[ f''(4) = -\frac{1}{4(4)^{3/2}} = -\frac{1}{32} \]

Construct the Taylor polynomial

Using the formula for a Taylor polynomial of degree 2, \[ P_2(x) = f(a) + f'(a)(x-a) + \frac{f''(a)}{2}(x-a)^2 \], substitute the values obtained:\[ P_2(x) = 2 + \frac{1}{4}(x-4) - \frac{1}{32} \cdot \frac{(x-4)^2}{2} \]. Simplifying yields \[ P_2(x) = 2 + \frac{1}{4}(x-4) - \frac{1}{64}(x-4)^2 \].

Key concepts

Derivative Calculation

Calculating the derivative is a crucial step in finding a Taylor polynomial. You need to know how a function changes to build its approximation. In mathematics, the derivative of a function represents an instantaneous rate of change. For the given function, the derivative calculations start by identifying the original function: \( f(x) = \sqrt{x} \). This tells us how one variable relates to another.

• **First Derivative:** This tells us the slope of the tangent line to the curve at any point. For \( f(x) = \sqrt{x} \), the first derivative is: \( f'(x) = \frac{1}{2\sqrt{x}} \). This fraction comes from applying the power rule, which involves decreasing the exponent and multiplying by the original exponent.


• **Second Derivative:** The second derivative shows how the rate of change of the curve itself changes. It's like finding the acceleration if the first derivative were velocity. For this function, the second derivative is: \( f''(x) = -\frac{1}{4x^{3/2}} \). Higher order derivatives can highlight more nuances over different ranges.

By understanding these steps, you can tackle various problems that involve using derivatives in polynomial approximations.

Polynomial Approximation

Polynomial approximation, often achieved using Taylor series, helps to represent complex functions with simple polynomials. This approach uses derivatives to estimate the function around a specific point. For the function \( f(x) = \sqrt{x} \) at \( a = 4 \), Taylor's formula can simplify the function's behavior near \( x = 4 \).

• **Purpose of Polynomial Approximation:** This process converts an often difficult function into a simpler polynomial form, which is easier to calculate and understand, especially near the given point \( a \).


• **Building the Polynomial:** The general formula for a Taylor polynomial of degree 2 is \( P_2(x) = f(a) + f'(a)(x-a) + \frac{f''(a)}{2}(x-a)^2 \). This formula accounts for the value of the function, its first derivative, and its second derivative calculated at the center point.


• **Resulting Polynomial:** In our example, substituting the calculated values gives us \( P_2(x) = 2 + \frac{1}{4}(x-4) - \frac{1}{64}(x-4)^2 \). Each term of this polynomial contributes to the approximation accuracy at \( x = 4 \).

This is a valuable technique in mathematics, allowing insights into the behavior of functions.

Function Evaluation

Evaluating a function and its derivatives allows us to construct the polynomial approximation effectively. This involves substituting specific values to get precise results.

• **Evaluating the Function at a Specific Point:** Begin by evaluating the function itself at the center point \( x = 4 \). For \( f(x) = \sqrt{x} \), we find \( f(4) = 2 \). This value is the foundation of your Taylor polynomial.


• **Evaluating Derivatives at the Center:** Evaluate the first and second derivatives at \( x = 4 \). The results, \( f'(4) = \frac{1}{4} \) and \( f''(4) = -\frac{1}{32} \), then integrate into the polynomial to consider the function's rate of change and curvature at that point.


• **Importance of Accurate Evaluation:** This step is crucial because any miscalculation could lead to an incorrect polynomial that poorly approximates the function. The results help you construct the most accurate Taylor polynomial possible.

By carefully evaluating each component, you create a reliable approximation that reflects the function's characteristics around the chosen point.