Question

The technique of estimating the value of a function for any intermediate value of the given independent variable is

• A. Leibnitz rule
• B. Extrapolation
• C. Interpolation
• D. Trapezoidal rule

Hint:

Think of the prefixes: "Inter-" means within (estimating inside the data range), while "Extra-" means outside (estimating outside the data range).

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:
In numerical analysis, we often need to estimate unknown functional values based on a given set of discrete data points.

Step 2: Detailed Explanation:
Let us define the primary estimation techniques:
- Interpolation: The process of finding the value of a dependent variable $y = f(x)$ for any intermediate value of the independent variable $x$ that lies within the range of the given data points $[x_0, x_n]$. This is the correct definition for the question.
- Extrapolation: The process of estimating the value of the function for an independent variable value that lies outside the range of the given data points (either $x < x_0$ or $x > x_n$).
- Trapezoidal Rule: A method used for numerical integration rather than value estimation.
- Leibnitz Rule: A formula used for differentiating an integral or finding the $n$-th derivative of a product of two functions.
Thus, the correct term is Interpolation.

Step 3: Final Answer:
The technique described is Interpolation.