Problem 50 - Calculus

Question

give an example of: A function \(f(x)\) that does not have a Taylor series around 0 even though \(f(0)\) is defined.

Step-by-Step Solution

Answer

An example is \( f(x) = \begin{cases} e^{-1/x^2}, & x \neq 0 \\ 0, & x = 0 \end{cases} \).

Step 1: Identify the Function

Consider the function \( f(x) = \begin{cases} e^{-1/x^2}, & \text{if} \ x eq 0 \ 0, & \text{if} \ x = 0 \end{cases} \). This function is defined for all real numbers, including at \( x = 0 \), where \( f(0) = 0 \).

Step 2: Determine Derivatives at Origin

For \( x eq 0 \), the derivatives of \( f(x) = e^{-1/x^2} \) exist, and at \( x = 0 \), all derivatives are zero. This is because they depend on the powers of \( x \), and when evaluated at zero, they vanish.

Step 3: Check for Taylor Series

The Taylor series of \( f(x) \) at \( x = 0 \) would be \( 0 + 0x + 0x^2 + \cdots = 0 \). However, this series does not represent \( f(x) \) except at the point \( x = 0 \), as \( f(x) \) is non-zero around any other value near zero, then is not equal to the Taylor series.

Step 4: Conclusion about Taylor Series

Since the Taylor series of \( f(x) \) is a constant zero polynomial while \( f(x) \) itself is not zero around the origin, \( f(x) \) does not have a valid Taylor series expansion at \( x = 0 \).

Key Concepts

Non-Differentiable Functions

In mathematics, a function is non-differentiable at a point if it doesn't have a derivative at that point, meaning the function's slope doesn't exist there. Consider it like a sharp corner or a vertical cliff on a graph, where you cannot draw a tangent line.
Non-differentiability can occur in several ways, such as discontinuities or sharp turns. For example, the simple function \( f(x) = |x| \) is non-differentiable at \( x = 0 \) because the slope of the function changes abruptly.
Sometimes, a function can be smooth and continuous everywhere and still lack a differentiation at a point. The given example of the function \( f(x) = \begin{cases} e^{-1/x^2}, & x eq 0 \ 0, & x = 0 \end{cases} \) in the exercise is smooth everywhere but behaves uniquely at zero. Here, derived values and slopes do not align to form a proper Taylor series around zero, signifying a more subtle form of non-differentiability.

Derivatives

Derivatives provide a way to understand how a function changes when its input changes. They are represented by the slope of the tangent line at any point on a function and measure the rate of change or the sensitivity of that function.
In mathematical terms, the derivative of a function \( f(x) \) is given by \( f'(x) \), which describes how \( f(x) \) behaves as \( x \) approaches a particular value. It is calculated as the limit of the average rate of change as the interval approaches zero.

• The first derivative \( f'(x) \) represents the slope or velocity.
• The second derivative \( f''(x) \) measures the curvature or acceleration.

In the original exercise, \( f(x) = e^{-1/x^2} \) has derivatives for all values of \( x eq 0 \), yet at \( x = 0 \), these derivatives calculated from power terms become zero. Consequently, the Taylor series remains a constant zero, not reflecting the behavior of \( f(x) \) near zero.

Real Analysis

Real analysis is a branch of mathematics that deals with the properties of real numbers and real-valued functions. It sets the foundation for exploring continuous functions, limits, series, and integrals.
Real analysis is crucial for understanding why some functions cannot be represented by their Taylor series. It includes examining how limits behave and how convergence works.
In the particular case of the given function, real analysis reveals that although all derivatives at zero are zero, the original function \( f(x) \) has different behavior around zero than implied by these derivatives. This discrepancy showcases a key aspect of real analysis, concerning convergence and precision in function representation.

• It helps in understanding the intricate details of function behavior.
• It highlights the importance of considering context and not relying solely on numerical values.

Hence, knowing real analysis helps explain why a function appears to have derivatives everywhere yet cannot form a valid Taylor series.