Problem 78 - Calculus Single Variable

Question

Show that if \(f(x)=x^{2},\) then the central difference quotient \(D_{0} f(c, h)\) for approximating \(f^{\prime}(c)\) is exactly equal to \(f^{\prime}(c)\)

Step-by-Step Solution

Answer

The central difference quotient equals the derivative; both are \(2c\) for \(f(x) = x^2\).

Step 1: Define the Central Difference Quotient

The central difference quotient is defined as \(D_{0} f(c, h) = \frac{f(c+h) - f(c-h)}{2h}\). Here, \(f(x) = x^2\), so we can substitute to find \(f(c+h)\) and \(f(c-h)\).

Step 2: Substitute into the Function

Calculate \(f(c+h)\) and \(f(c-h)\): \(f(c+h) = (c+h)^2 = c^2 + 2ch + h^2\) and \(f(c-h) = (c-h)^2 = c^2 - 2ch + h^2\).

Step 3: Compute the Difference

Substitute the expressions for \(f(c+h)\) and \(f(c-h)\) into the central difference quotient. This gives:\[D_{0} f(c, h) = \frac{(c^2 + 2ch + h^2) - (c^2 - 2ch + h^2)}{2h}\]. Simplify the numerator to get \(4ch\).

Step 4: Simplify the Quotient

Simplify \(D_{0} f(c, h)\) by cancelling terms:\[D_{0} f(c, h) = \frac{4ch}{2h} = 2c\].

Step 5: Compare to the Actual Derivative

The actual derivative of \(f(x) = x^2\) with respect to \(x\) is \(f^{\prime}(x) = 2x\). At \(x = c\), \(f^{\prime}(c) = 2c\). This matches the result from the central difference quotient.

Key Concepts

Central Difference Quotient

The central difference quotient is a crucial tool in the realm of numerical differentiation. It provides an effective way to approximate the derivative of a function at a particular point. Here's how it works:

• The formula for the central difference quotient is: \(D_{0} f(c, h) = \frac{f(c+h) - f(c-h)}{2h}\).
• This formula calculates the average rate of change using two points on either side of the given point \(c\), specifically the points \(c+h\) and \(c-h\).
• This helps in reducing error compared to other numerical methods.

Breaking it down further, when you substitute into the formula, you're essentially taking values of the function slightly ahead and slightly behind the point of interest. This symmetrically centered approach provides a more accurate approximation of the slope or derivative at that point, compared to methods using only forward or backward steps. In practice, the value of \(h\) should be small to get a finer approximation, but not too small to a point where numerical errors might occur due to limitations in computational precision.

Numerical Differentiation

Numerical differentiation is the process of finding an approximate derivative of a function using numerical methods. Understanding its importance requires a grasp of its application in situations where:

• The exact symbolic derivative is difficult or impossible to determine.
• The function is based on discrete data points from experiments or simulations.

Unlike analytical differentiation, which involves calculus, numerical differentiation utilizes various formulas, such as the central difference quotient, to achieve approximation. An essential part of numerical differentiation is choosing the right formula and step-size \(h\). A small \(h\) increases accuracy, but too small an \(h\) can lead to significant precision errors. Therefore, selecting \(h\) is typically a balance between accuracy and numerical stability. Computation errors due to subtractive cancellation can also creep in, requiring careful consideration when using these methods in practice.

Polynomial Functions

Polynomial functions, like \(f(x) = x^2\), are mathematical expressions involving powers of \(x\). These functions are among the simplest and most fundamental forms studied in calculus. They have a few key characteristics:

• Polynomial functions are continuous and differentiable everywhere across their domain.
• Simpler to differentiate since they follow straightforward rules of calculus.
• The derivative of \(f(x) = x^n\) is \(f'(x) = nx^{n-1}\).

For the quadratic function \(f(x) = x^2\), the derivative is \(f'(x) = 2x\). Polynomial derivatives are linear combinations of simpler functions and are powerful due to their straightforward analysis and manipulation. They're highly useful in modeling and solving practical problems because they approximate more complex functions well over small intervals.