Problem 2 - Calculus: An Integrated Approach to Functions and Their Rates of Change

Question

Do the following. (a) Compute the fourth degree Taylor polynomial for \(f(x)\) at \(x=0 .\) (b) On the same set of axes, graph \(f(x), P_{1}(x), P_{2}(x), P_{3}(x)\), and \(P_{4}(x)\). (c) Use \(P_{1}(x), P_{2}(x), P_{3}(x)\), and \(P_{4}(x)\) to approximate \(f(0.1)\) and \(f(0.3) .\) Compare these approximations to those given by a calculator. $$ f(x)=\ln (1+x) $$

Step-by-Step Solution

Answer

The 4th degree Taylor polynomial for \(f(x) = ln(1+x)\) at \(x=0\) is \(-x^4/4 + x^3/3 - x^2/2 + x\). The Taylor polynomials \(P_{1}(x), P_{2}(x), P_{3}(x)\) and \(P_{4}(x)\) are increasingly accurate approximations of \(f(x)\) at nearby points.

Step 1 Title

Differentiate the function \(f(x) = ln(1+x)\) repeatedly to get the function's derivatives

Step 2 Title

Use these derivatives at x=0 to compute the coefficients for the Taylor polynomial.

Step 3 Title

Put the coefficients into the Taylor polynomial formula to obtain the Taylor polynomials of degree 1, 2, 3 and 4.

Step 4 Title

Plot the function and the Taylor polynomials on the same graph.

Step 5 Title

Use the Taylor polynomials to approximate the value of the function at x=0.1 and x=0.3.

Step 6 Title

Compare these approximations to the precise values computed using a calculator.

Key Concepts

Function Approximation

Function approximation is a powerful mathematical technique used to estimate complex functions with simpler ones. One way to achieve this is through Taylor polynomials. These polynomials allow us to approximate a function near a specific point by using the function's value and its derivatives at that point.
In the original exercise, the function to be approximated is \( f(x) = \ln(1+x) \). The goal is to determine the Taylor polynomials of degrees 1 through 4 at \( x=0 \).
The basic idea is to sum up the series of derivatives, each divided by the factorial of the derivative’s order, and multiplied by \( x^n \). This forms the Taylor polynomial:

• \( P_n(x) = f(0) + f'(0)x + \frac{f''(0)}{2!}x^2 + \frac{f'''(0)}{3!}x^3 + \, ... \, + \frac{f^{(n)}(0)}{n!}x^n \)

Function approximation helps us understand the behavior of \( f(x) \) near \( x=0 \) without needing to rely on the actual logarithmic function entirely, making complex calculations easier and faster.

Derivatives

Derivatives play a crucial role in creating Taylor polynomials. They represent the rate at which a function changes at any given point and are pivotal in determining the polynomial's coefficients.
For the function \( f(x) = \ln(1+x) \), we start by finding the derivatives:

• First derivative: \( f'(x) = \frac{1}{1+x} \)
• Second derivative: \( f''(x) = -\frac{1}{(1+x)^2} \)
• Third derivative: \( f'''(x) = \frac{2}{(1+x)^3} \)
• Fourth derivative: \( f^{(4)}(x) = -\frac{6}{(1+x)^4} \)

Evaluating these derivatives at \( x=0 \) yields the coefficients for the Taylor polynomial:

• \( f'(0) = 1 \)
• \( f''(0) = -1 \)
• \( f'''(0) = 2 \)
• \( f^{(4)}(0) = -6 \)

These are systematically plugged into the Taylor polynomial formula to provide a series that approximates the original function effectively.

Graphing Functions

Graphing functions enables us to visually compare the original function and its approximations made by Taylor polynomials. By plotting both \( f(x) = \ln(1+x) \) and its Taylor polynomials \( P_{1}(x), P_{2}(x), P_{3}(x), \text{and} \, P_{4}(x) \), we gain insights into the accuracy of our approximations.
The graph highlights how each polynomial approximation progressively gets closer to the curve of the logarithmic function. By observing:

• Localized approximation: Lower-degree polynomials, like \( P_1(x) \), closely follow the curve at \( x=0 \), but diverge as \( x \) moves away.
• Wide-range match: Higher-degree polynomials, such as \( P_4(x) \), provide a better match over a broader range of \( x \).

Graphing is an immensely helpful tool in understanding how well a Taylor polynomial works as an approximation tool, enhancing our conceptual and practical understanding of the function's behavior.

Logarithmic Functions

The function in our exercise, \( f(x) = \ln(1+x) \), is a classic example of a logarithmic function. Logarithmic functions are the inverse of exponential functions and have applications in a variety of fields, from biology to finance.
Understanding these functions is essential for accurately applying derivatives and computing approximations with Taylor polynomials. The natural logarithm, \( \ln(x) \), specifically refers to logarithms with the base \( e \), where \( e \) is approximately 2.718.
Logarithmic functions have unique properties that make them smooth, continuous, and ever-increasing for \( x > -1 \). When graphed, they curve gently upwards, starting at \( f(x) = 0 \) when \( x = 0 \). This nature makes them a great candidate for applying Taylor series, which requires the function to be infinitely differentiable over the interval of approximation.

• Interestingly, the function is defined only for \( x > -1 \), which ensures all results remain within the domain for meaningful calculation.

With these insights, approaching logarithmic functions with Taylor polynomials becomes intuitive, thus simplifying complex calculations significantly.