Calculating Taylor Polynomials T2 And T3 For F(x) = 25ln(x+1) At X=0

Introduction

In the realm of mathematical analysis, Taylor polynomials stand as a powerful tool for approximating functions using polynomial expressions. These polynomials, named after mathematician Brook Taylor, provide a local approximation of a function around a specific point. The higher the degree of the Taylor polynomial, the better the approximation, especially in the vicinity of the point of expansion. This article delves into the process of calculating Taylor polynomials, specifically focusing on finding the second-degree Taylor polynomial ($T_2$) and the third-degree Taylor polynomial ($T_3$) for the function $f(x) = 25\ln(x+1)$ centered at $x=0$. This exploration will involve understanding the fundamental formula for Taylor polynomials, computing derivatives, and applying these concepts to the given function. By understanding how to calculate these approximations, readers can gain a deeper appreciation for the way complex functions can be represented and analyzed through simpler polynomial forms.

Understanding Taylor Polynomials

Before diving into the calculations, it's crucial to grasp the concept of Taylor polynomials. A Taylor polynomial of degree $n$ for a function $f(x)$ centered at $x=a$ is given by the formula:

$T_n(x) = f(a) + f'(a)(x-a) + \frac{f''(a)}{2!}(x-a)^2 + \frac{f'''(a)}{3!}(x-a)^3 + ... + \frac{f^{(n)}(a)}{n!}(x-a)^n$

Where:

• $f(a)$ is the value of the function at $x=a$.
• $f'(a)$, $f''(a)$, $f'''(a)$, ..., $f^{(n)}(a)$ are the first, second, third, and $n$-th derivatives of the function evaluated at $x=a$, respectively.
• $n!$ denotes the factorial of $n$.

This formula essentially constructs a polynomial that matches the function's value and its first $n$ derivatives at the point $x=a$. Thus, the Taylor polynomial provides a good approximation of the function near this point. For a given function, the accuracy of the approximation generally improves as the degree $n$ increases.

Key Components of the Taylor Polynomial Formula

To fully understand the formula for Taylor polynomials, it's essential to dissect its key components:

• Function Evaluation at the Center ($f(a)$): This term represents the value of the function at the point around which the Taylor polynomial is being constructed. It sets the baseline for the approximation, ensuring that the polynomial matches the function's value at the center.
• Derivatives of the Function ($f'(a), f''(a), ..., f^{(n)}(a)$): These derivatives capture the rate of change of the function and its higher-order rates of change at the center point. Each derivative term contributes to how well the polynomial mimics the function's behavior in the vicinity of the center.
• Factorials ($n!$): Factorials in the denominators serve to scale the derivative terms appropriately. They arise from the process of repeated differentiation in calculus and play a crucial role in ensuring the Taylor polynomial converges to the function as the degree increases.
• Powers of $(x-a)$: These terms determine the shape of the polynomial around the center point $a$. The higher the power, the greater the influence of the corresponding derivative on the polynomial's behavior further away from the center.

Understanding these components allows for a more intuitive grasp of how Taylor polynomials work and how they can be used to approximate functions.

Calculating the Derivatives of f(x) = 25ln(x+1)

To compute the Taylor polynomials $T_2(x)$ and $T_3(x)$, we first need to find the derivatives of the function $f(x) = 25\ln(x+1)$. This involves applying the rules of differentiation, particularly the chain rule and the derivative of the natural logarithm.

First Derivative

The first derivative, denoted as $f'(x)$, represents the rate of change of the function. Applying the chain rule, we have:

$f'(x) = \frac{d}{dx} [25\ln(x+1)] = 25 \cdot \frac{1}{x+1} \cdot \frac{d}{dx}(x+1) = \frac{25}{x+1}$

Second Derivative

The second derivative, $f''(x)$, represents the rate of change of the first derivative. Differentiating $f'(x)$ with respect to $x$, we get:

$f''(x) = \frac{d}{dx} [\frac{25}{x+1}] = 25 \cdot \frac{d}{dx} [(x+1)^{-1}] = 25 \cdot (-1)(x+1)^{-2} \cdot \frac{d}{dx}(x+1) = -\frac{25}{(x+1)^2}$

Third Derivative

The third derivative, $f'''(x)$, represents the rate of change of the second derivative. Differentiating $f''(x)$ with respect to $x$, we have:

$f'''(x) = \frac{d}{dx} [-\frac{25}{(x+1)^2}] = -25 \cdot \frac{d}{dx} [(x+1)^{-2}] = -25 \cdot (-2)(x+1)^{-3} \cdot \frac{d}{dx}(x+1) = \frac{50}{(x+1)^3}$

Summary of Derivatives

In summary, we have calculated the following derivatives:

• $f'(x) = \frac{25}{x+1}$
• $f''(x) = -\frac{25}{(x+1)^2}$
• $f'''(x) = \frac{50}{(x+1)^3}$

These derivatives are essential for constructing the Taylor polynomials $T_2(x)$ and $T_3(x)$. By evaluating these derivatives at the center point $a=0$, we can determine the coefficients of the polynomial terms.

Evaluating Derivatives at a = 0

Now that we have the derivatives of $f(x)$, we need to evaluate them at the center point $a=0$. This will give us the coefficients needed for the Taylor polynomial formula. We also need to evaluate the original function $f(x)$ at $a=0$.

Evaluating f(x) at a = 0

$f(0) = 25\ln(0+1) = 25\ln(1) = 25 \cdot 0 = 0$

Evaluating the First Derivative at a = 0

$f'(0) = \frac{25}{0+1} = 25$

Evaluating the Second Derivative at a = 0

$f''(0) = -\frac{25}{(0+1)^2} = -25$

Evaluating the Third Derivative at a = 0

$f'''(0) = \frac{50}{(0+1)^3} = 50$

Summary of Evaluations

In summary, we have the following values at $a=0$:

• $f(0) = 0$
• $f'(0) = 25$
• $f''(0) = -25$
• $f'''(0) = 50$

These values are the key ingredients for building the Taylor polynomials $T_2(x)$ and $T_3(x)$. By substituting these values into the Taylor polynomial formula, we can derive the polynomial approximations for the function $f(x)$ around the point $x=0$.

Calculating the Taylor Polynomial T2(x)

Now that we have the function and its derivatives evaluated at $a=0$, we can proceed to calculate the second-degree Taylor polynomial, $T_2(x)$. The formula for $T_2(x)$ is a specific case of the general Taylor polynomial formula, truncated at the second-degree term:

$T_2(x) = f(a) + f'(a)(x-a) + \frac{f''(a)}{2!}(x-a)^2$

In our case, $a=0$, $f(0) = 0$, $f'(0) = 25$, and $f''(0) = -25$. Substituting these values into the formula, we get:

$T_2(x) = 0 + 25(x-0) + \frac{-25}{2!}(x-0)^2$

Simplifying, we have:

$T_2(x) = 25x - \frac{25}{2}x^2$

Thus, the second-degree Taylor polynomial for $f(x) = 25\ln(x+1)$ centered at $x=0$ is $T_2(x) = 25x - \frac{25}{2}x^2$. This polynomial provides a quadratic approximation of the function near $x=0$. The closer $x$ is to 0, the better the approximation will be.

Significance of T2(x)

The Taylor polynomial $T_2(x)$ offers a valuable approximation of the original function $f(x)$ near the point $x=0$. It captures the function's value and its first two derivatives at this point, providing a quadratic representation of the function's behavior. This approximation is particularly useful in situations where the original function is complex or difficult to evaluate directly. For example, $T_2(x)$ can be used to estimate the value of $25\ln(x+1)$ for small values of $x$ without needing to calculate the logarithm.

Calculating the Taylor Polynomial T3(x)

Next, we calculate the third-degree Taylor polynomial, $T_3(x)$. This polynomial includes one more term than $T_2(x)$, providing a potentially more accurate approximation of the function. The formula for $T_3(x)$ is:

$T_3(x) = f(a) + f'(a)(x-a) + \frac{f''(a)}{2!}(x-a)^2 + \frac{f'''(a)}{3!}(x-a)^3$

We already have the values for $f(0)$, $f'(0)$, and $f''(0)$. We also calculated $f'''(0) = 50$. Plugging these values and $a=0$ into the formula, we get:

$T_3(x) = 0 + 25(x-0) + \frac{-25}{2!}(x-0)^2 + \frac{50}{3!}(x-0)^3$

Simplifying, we have:

$T_3(x) = 25x - \frac{25}{2}x^2 + \frac{50}{6}x^3$

Which can be further simplified to:

$T_3(x) = 25x - \frac{25}{2}x^2 + \frac{25}{3}x^3$

Therefore, the third-degree Taylor polynomial for $f(x) = 25\ln(x+1)$ centered at $x=0$ is $T_3(x) = 25x - \frac{25}{2}x^2 + \frac{25}{3}x^3$. This cubic polynomial provides an even better approximation of the function near $x=0$ compared to $T_2(x)$.

Significance of T3(x)

The inclusion of the third-degree term in $T_3(x)$ allows the polynomial to capture more of the function's curvature near the center point. This means that $T_3(x)$ typically provides a more accurate approximation of $f(x)$ over a wider interval around $x=0$ compared to $T_2(x)$. In practical applications, using higher-degree Taylor polynomials like $T_3(x)$ can lead to more reliable estimations and predictions.

Summary of Taylor Polynomials

In this exploration, we successfully calculated the Taylor polynomials $T_2(x)$ and $T_3(x)$ for the function $f(x) = 25\ln(x+1)$ centered at $x=0$. We found that:

• The second-degree Taylor polynomial is $T_2(x) = 25x - \frac{25}{2}x^2$.
• The third-degree Taylor polynomial is $T_3(x) = 25x - \frac{25}{2}x^2 + \frac{25}{3}x^3$.

These polynomials provide approximations of the function $f(x)$ near the point $x=0$. The higher the degree of the polynomial, the better the approximation, especially in the vicinity of the center point. Taylor polynomials are a fundamental tool in calculus and are widely used in various fields, including physics, engineering, and computer science, for approximating functions and solving complex problems.

Applications and Further Exploration

Taylor polynomials have numerous applications in various fields. They are used to:

• Approximate function values: As demonstrated in this article, Taylor polynomials can provide accurate estimations of function values near the center point.
• Solve differential equations: Taylor series, the infinite extension of Taylor polynomials, can be used to find solutions to differential equations that cannot be solved analytically.
• Analyze function behavior: Taylor polynomials can help understand the local behavior of functions, such as identifying local maxima, minima, and inflection points.
• Develop numerical methods: Many numerical methods, such as Newton's method for finding roots, rely on Taylor series approximations.

Further exploration of Taylor polynomials might involve:

• Investigating the convergence of Taylor series: Understanding the conditions under which the Taylor series converges to the original function.
• Exploring applications in different fields: Learning how Taylor polynomials are used in physics, engineering, and other disciplines.
• Using software tools to compute Taylor polynomials: Utilizing computer algebra systems to efficiently calculate Taylor polynomials for complex functions.

By delving deeper into these topics, readers can further enhance their understanding of Taylor polynomials and their significance in mathematics and beyond.