Problem 78 - Essential Calculus

Question

Let \(f^{\prime \prime}(x)\) be continuous. Show that $$ \lim _{h \rightarrow 0} \frac{f(x+h)-2 f(x)+f(x-h)}{h^{2}}=f^{\prime \prime}(x) $$

Step-by-Step Solution

Answer

From the Taylor series expressions of \(f(x+h)\) and \(f(x-h)\), and through simplification and cancellation, the equation yields: \(\lim _{h \rightarrow 0} \frac{f(x+h)-2 f(x)+f(x-h)}{h^{2}} = f''(x)\), which is the required proof.

Step 1: Write the Taylor Series

Let's first express \(f(x+h)\) and \(f(x-h)\) using Taylor series up to the second derivative. The Taylor series of \(f(x+h)\) at point x yields:\[f(x+h) = f(x) + hf'(x) + \frac{h^2}{2}f''(x) + o(h^2).\]Similarly, the Taylor series of \(f(x-h)\) at point x is:\[f(x-h) = f(x) - hf'(x) + \frac{h^2}{2}f''(x) + o(h^2). \]

Step 2: Substitution

Next, substitute the expressions of \(f(x+h)\) and \(f(x-h)\) into the given equation: \[\lim _{h \rightarrow 0} \frac{f(x+h)-2 f(x)+f(x-h)}{h^{2}} = \lim _{h \rightarrow 0} \frac{(f(x) + hf'(x) + \frac{h^2}{2}f''(x) + o(h^2)) - 2f(x) + (f(x) - hf'(x) + \frac{h^2}{2}f''(x) + o(h^2))}{h^{2}}.\]

Step 3: Simplification & Cancellation

Simplify the numerator of the fraction by cancelling terms:This gives, \[\lim _{h \rightarrow 0} \frac{h^2 f''(x) + o(h^2)}{h^{2}}.\]

Step 4: Apply Limit

Finally, apply the limit to the equation:As \[h \rightarrow 0, o(h^2) \rightarrow 0.\]Therefore, \[\lim _{h \rightarrow 0} \frac{h^2 f''(x) + o(h^2)}{h^{2}} = f''(x).\]

Key Concepts

Limit Definition

In calculus, the concept of a limit is fundamental and helps us understand how functions behave near specific points. The limit describes the value that a function approaches as the input approaches a particular value. In the exercise, we take the limit as \(h\) approaches zero to find

• The behavior of our function's expression as \(h\) becomes practically insignificant,
• And to evaluate how small changes around \(x\) influence the quotient formed.

A limit essentially helps us find values of derivatives and crucial points for function analysis.
In this scenario, the limit\[\lim_{h \rightarrow 0} \frac{f(x+h)-2f(x)+f(x-h)}{h^2}\]is used to determine the second derivative, \(f''(x)\). This is because the limit helps remove the small error terms, \(o(h^2)\), and focus on the main components as \(h\rightarrow 0\). Adjusting the expression into a clean, simple component emphasizes the role of \(f''(x)\) in the function's change.

Second Derivative

The second derivative of a function, denoted by \(f''(x)\), represents the rate of change of its first derivative. While the first derivative \(f'(x)\) tells us how the function itself increases or decreases, the second derivative reveals how the slope of the function changes.
It impacts our understanding of the function's curvature, allowing us to analyze concavity and inflection points.

• When \(f''(x) > 0\), the function \(f(x)\) is concave up, resembling a U-shape.
• When \(f''(x) < 0\), the function \(f(x)\) is concave down, resembling an upside-down U-shape.

In the exercise, the limit evaluated leads directly to \(f''(x)\), proving that the second derivative is just the ordered change when analyzing the expression across small neighborhoods around \(x\). The consistent result obtained when taking the limit confirms the association with the second derivative.

Continuous Functions

Continuity of functions is a crucial property in calculus, ensuring that small changes in input lead to small changes in output. A function is deemed continuous at a point \(x\) if the limit of the function as it approaches \(x\) from any direction equals the function value at \(x\).
For a function to have a well-defined second derivative like \(f''(x)\), it must be at least twice differentiable, making it continuous beyond the first derivative.
Continuous functions also ensure that approximation techniques like Taylor series remain reliable and accurate for calculating derivatives and other analytical purposes. When applying Taylor series, the assumption of continuity allows the second-order terms in the expression to align with the behavior of \(f''(x)\).When analyzing our problem, the smooth and connected nature of continuous functions plays a key role. It also ensures no sudden jumps or breaks in the curve, and the reliable transition of derivatives across the specified points derived through the limit. This confirms the theoretical foundation that underpins the entire process of limiting, substituting, and simplifying in our exercise.