Problem 1 - A Primer of Real Analysis

Question

Suppose \(D \subset \mathbb{R}, a\) is an interior point of \(D, f: D \rightarrow \mathbb{R},\) and \(f^{\prime \prime}(a)\) exists. Show that $$ \lim _{h \rightarrow 0} \frac{f(a+h)+f(a-h)-2 f(a)}{h^{2}}=f^{\prime \prime}(a) $$ Find an example to illustrate that this limit may exist even if \(f^{\prime \prime}(a)\) does not exist.

Step-by-Step Solution

Answer

The limit equals the second derivative if it exists; use non-smooth functions to show it might not exist generally.

Step 1: Understand the Limit Expression

The limit expression given is \( \lim _{h \rightarrow 0} \frac{f(a+h)+f(a-h)-2 f(a)}{h^{2}} \). It uses the function values nearby \(a\), which is related to the second derivative of the function \(f\) at the point \(a\). This formula is used to approximate or determine the second derivative.

Step 2: Recall the Definition of Second Derivative

The second derivative \( f''(a) \) is defined as \( f''(a) = \lim_{h \to 0} \frac{f'(a+h) - f'(a)}{h} \), where \( f'(x) \) is the first derivative of the function \( f(x) \). It shows how the rate of change of the function is changing.

Step 3: Investigate the Symmetric Difference Quotient

The given expression \( \frac{f(a+h)+f(a-h)-2 f(a)}{h^{2}} \) is a symmetric difference quotient that estimates the curvature at \( a \) by using values from both sides of \( a \). This symmetry is key to why it resembles the second derivative.

Step 4: Taylor Expansion

Use Taylor expansion: \( f(a+h) = f(a) + f'(a)h + \frac{f''(a)}{2}h^2 + o(h^2) \) and \( f(a-h) = f(a) - f'(a)h + \frac{f''(a)}{2}h^2 + o(h^2) \). This expansion to second order provides approximations of the function around the point \(a\).

Step 5: Substitute and Simplify

Substitute the expansions into the limit expression: \( \frac{ \left( f(a) + f'(a)h + \frac{f''(a)}{2}h^2 + o(h^2) \right) + \left( f(a) - f'(a)h + \frac{f''(a)}{2}h^2 + o(h^2) \right) - 2f(a) }{h^2} \). Simplify to get \( \frac{ h^2 f''(a) + 2o(h^2)}{h^2} \).

Step 6: Take the Limit

Taking the limit as \( h \rightarrow 0 \), we note the last terms simplifies because \( \frac{o(h^2)}{h^2} \rightarrow 0 \). Thus, the result is \( f''(a) \). Therefore, \( \lim _{h \rightarrow 0} \frac{f(a+h)+f(a-h)-2 f(a)}{h^{2}} = f^{ ext{''}}(a) \).

Step 7: Provide an Example for Nonexistence of \( f''(a) \)

Consider \( f(x) = |x| \). The expression \( \frac{|a+h| + |a-h| - 2|a|}{h^2} = \frac{|h| + |-h| - 0}{h^2}\) leads to \( \frac{2|h|}{h^2} = \frac{2}{|h|} \), which diverges as \( h \rightarrow 0 \), although the limit of the expression might still exist at points other than \(0\) if defined differently, indicating where \( f''(a) \) doesn’t exist.

Key Concepts

Limit Expression

A limit expression is a fundamental concept in calculus. It involves calculating the value that a function approaches as the variable in the expression gets closer to a particular point. In this exercise, we specifically look at the limit expression \( \lim _{h \rightarrow 0} \frac{f(a+h)+f(a-h)-2 f(a)}{h^{2}} \). This expression is essential in determining the existence and value of the second derivative of a function at a point \(a\). It uses values of the function from both sides near \(a\) to approximate the behavior of the second derivative.

• First Derivative Review: The first derivative \( f'(x) \) represents the rate of change of the function. It's the slope of the tangent line to the curve at any given point.
• Second Derivative Insight: The second derivative \( f''(a) \) tells us how the rate of change itself is varying. It relates to concepts like acceleration in physics or curvature in geometry.

By evaluating this expression, mathematicians can analyze the curvature of a function at a given point. This analysis is crucial for understanding more complex behaviors of functions.

Symmetric Difference Quotient

Symmetric difference quotients build upon the idea of approximating derivatives. The expression \( \frac{f(a+h)+f(a-h)-2 f(a)}{h^{2}} \) is a special case. It is a symmetric difference quotient aimed at approximating the second derivative at a point \(a\).

• Symmetry in Calculation: By using both \(f(a+h)\) and \(f(a-h)\), the formula takes advantage of symmetry, which helps reduce errors or biases that could arise if only one side were considered.
• Curvature Estimation: This symmetric approach helps in estimating the curvature of a function, similar to estimating how sharp or flat a curve is at a certain point.

Understanding this concept makes it clear why the symmetric difference quotient is a reliable method for calculating higher-order derivatives, especially when functions aren't easily differentiable.

Taylor Expansion

Taylor expansion is a powerful tool in mathematics for approximating functions. It allows a function to be expressed as an infinite sum of terms calculated from the values of its derivatives at a single point. In the context of this exercise, Taylor expansion is vital for analyzing and simplifying our limit expression.

• Using Expansion: We use Taylor expansion to express \( f(a+h)\) and \( f(a-h) \) as \( f(a) + f'(a)h + \frac{f''(a)}{2}h^2 + o(h^2) \) and \( f(a) - f'(a)h + \frac{f''(a)}{2}h^2 + o(h^2) \), respectively.
• Simplification: Substituting these expansions into our limit expression and simplifying reveals how the second derivative is extracted and confirmed through the limit.

This technique not only supports the calculation of the second derivative but also enhances the understanding of how functions behave around given points. Mastering Taylor expansions is essential for students aiming to dive deeper into calculus.