Another way to ​​approximate derivatives is to use the centered difference quotient: f' (a) ≈ f(a+h) - f(a- h) / 2h. Again, consider f(x) = √x.
c. Explain why it is not necessary to use negative values of h in the table of part (b).

Verified step by step guidance

1. Step 1: Understand the centered difference quotient formula: The centered difference quotient is given by \( f'(a) \approx \frac{f(a+h) - f(a-h)}{2h} \). This formula is used to approximate the derivative of a function at a point \( a \).
2. Step 2: Recognize the role of \( h \): In the formula, \( h \) is a small positive number that represents a small change in \( x \). The choice of \( h \) affects the accuracy of the approximation.
3. Step 3: Consider symmetry in the formula: The centered difference quotient uses both \( f(a+h) \) and \( f(a-h) \), which means it inherently considers both positive and negative changes around \( a \).
4. Step 4: Analyze the effect of negative \( h \): Using a negative \( h \) would simply reverse the roles of \( f(a+h) \) and \( f(a-h) \), but since the formula already accounts for both directions, negative values of \( h \) are redundant.
5. Step 5: Conclude on the necessity of negative \( h \): Since the centered difference quotient already incorporates both forward and backward differences through its symmetric structure, using negative values of \( h \) does not provide additional information or improve the approximation.

Key Concepts

Centered Difference Quotient

The centered difference quotient is a method for approximating the derivative of a function at a point by averaging the slopes of secant lines on either side of that point. It is defined as f'(a) ≈ (f(a+h) - f(a-h)) / (2h), where h is a small positive value. This approach provides a more accurate estimate of the derivative compared to the forward or backward difference quotients, especially when h is small.

Symmetry in Function Values

When using the centered difference quotient, the function values f(a+h) and f(a-h) are symmetrically located around the point a. This symmetry means that the contributions of positive and negative h values effectively cancel out any asymmetries in the function's behavior at a, allowing for a more accurate approximation of the derivative without needing to consider negative h values.

Behavior of the Function Near a Point

The behavior of the function f(x) near the point a is crucial for understanding why negative values of h are unnecessary. If the function is continuous and differentiable at a, the values of f(a+h) and f(a-h) will converge to the same limit as h approaches zero. Thus, using only positive h values suffices to capture the local behavior of the function around a, ensuring a reliable approximation of the derivative.