1. **Problem Statement:** Explain interpolation and extrapolation based on their definitions and provide two example applications in electrical engineering. 2. **Definitions:** - **Interpolation** is the process of estimating unknown values that fall within the range of known data points. - **Extrapolation** is the process of estimating values outside the range of known data points. 3. **Formula and Explanation:** - For interpolation, a common method is linear interpolation using two known points $(x_0, y_0)$ and $(x_1, y_1)$: $$y = y_0 + \frac{(x - x_0)(y_1 - y_0)}{x_1 - x_0}$$ This formula estimates $y$ at a point $x$ between $x_0$ and $x_1$. - Extrapolation uses the same formula but for $x$ values outside the interval $[x_0, x_1]$. 4. **Important Rules:** - Interpolation is generally more reliable because it estimates within known data. - Extrapolation is less certain as it predicts beyond known data and can lead to larger errors. 5. **Example Applications in Electrical Engineering:** **Example 1: Interpolation in Sensor Calibration** - Sensors provide discrete calibration points. - To find sensor output at intermediate values, interpolation estimates the output accurately. - For instance, if a temperature sensor is calibrated at 20°C and 30°C, interpolation estimates output at 25°C. **Example 2: Extrapolation in Circuit Behavior Prediction** - Engineers may extrapolate the behavior of a circuit beyond tested voltage ranges. - For example, predicting current at voltages higher than tested to ensure safety margins. 6. **Summary:** - Interpolation fills gaps within data. - Extrapolation predicts beyond data. - Both are essential tools in electrical engineering for analysis and design.