91Ó°ÊÓ

In calculus, derivatives are a fundamental concept that describe how a function changes as its input changes. They provide us with the slope of the function at any given point.
Especially in the context of a Taylor series, derivatives are crucial as they determine the terms in the series expansion.
For the function \( \ln x \), we begin by finding its derivatives.

• First Derivative: \( f'(x) = \frac{1}{x} \). This shows how the natural logarithm changes with respect to \( x \).
• Second Derivative: \( f''(x) = -\frac{1}{x^2} \). This illustrates the concavity of the natural logarithm function.
• Third Derivative: \( f'''(x) = \frac{2}{x^3} \). As we move to higher derivatives, we are capturing more precise aspects of the function's behavior.

When constructing a Taylor series, evaluating these derivatives at the point of expansion, in this case \( a = 3 \), helps us form the series terms.
This step-by-step calculation makes sure we include all necessary details to approximate the function accurately.

The natural logarithm function, denoted as \( \ln x \), is a logarithmic function with the base \( e \) (approximately 2.718) and is heavily used in mathematics.
This function is defined only for positive numbers and provides a unique and continuous transformation of numbers.
The slope or rate of change of the natural logarithm function is described by its first derivative, \( \frac{1}{x} \), indicating that as \( x \) increases, the rate of change of \( \ln x \) decreases, approaching zero.
Its calculus properties, like the derivatives we calculated, help in expanding the function into a Taylor series.

• This series allows us to approximate the function around a particular point \( a \), making it easier to compute \( \ln x \) for values near \( a \).
• This method is particularly useful because the exact computation of the logarithm function might not be straightforward for all values.

Through derivatives, the complexity of the natural logarithm is broken down into simpler polynomial expressions.

A Taylor series allows us to express a complicated function as an infinite sum of terms calculated from its derivatives at a particular point.
This process is known as evaluating the series at a point, and it makes understanding and computing functions much simpler.
For \( \ln x \) around \( a = 3 \), we start by evaluating the function and its derivatives at this point.

• Original Function Value: \( f(3) = \ln 3 \). This becomes the starting point or the constant term in the series.
• First Derivative \( f'(3) = \frac{1}{3} \), multiplied by \( (x - 3) \) to estimate how \( \ln x \) changes near \( x = 3 \).
• Higher derivatives \( f''(3) = -\frac{1}{9} \) and \( f'''(3) = \frac{2}{27} \) adjust the series with more terms like \( (x-3)^2 \) and \( (x-3)^3 \).

Such terms refine the approximation by capturing more of the function's curvature or change.
The Taylor series essentially builds a polynomial expression that mimics \( \ln x \) near \( x=3 \), offering a much simpler calculation than direct evaluation.