Slope, Trends, Interpolation, and Extrapolation

Engineering data is rarely perfect. Often, you will have discrete data points (like test results from a lab) but need to know the value at a point exactly between your test data.

Slope and Trends

The Slope of a line on a chart represents the rate of change between two variables.

• Linear Trend: If the data forms a straight line, the rate of change is constant.
• Non-Linear Trend: If the data forms a curve, the rate of change is accelerating or decelerating. Recognizing trends allows engineers to predict how a system will behave as inputs change.

Interpolation (Filling the Gaps)

Interpolation is the process of estimating an unknown value that falls between two known values.

For example, if an engineering table tells you that water density is $999.7 \text{ kg/m}^3$ at $10^\circ\text{C}$, and $998.2 \text{ kg/m}^3$ at $20^\circ\text{C}$, but you need the density at exactly $15^\circ\text{C}$:

• Linear Interpolation: You assume the change between the two points follows a straight line. You can mathematically calculate the midpoint value. In this example, $15^\circ\text{C}$ is exactly halfway, so the density is estimated as the exact average: $998.95 \text{ kg/m}^3$.
• Safety Note: Linear interpolation is generally safe for small gaps, but can introduce errors if the true physical relationship is highly curved.

Extrapolation (Predicting the Unknown)

Extrapolation is the process of estimating an unknown value that falls outside the range of your known data.

If you have test data for a pump's performance between 1000 RPM and 3000 RPM, you might mathematically project the trend line outward to guess its performance at 4000 RPM.

• The Danger of Extrapolation: While interpolation is bounded and generally safe, extrapolation is highly dangerous in engineering. Physical systems do not scale infinitely. A material might behave elastically up to a certain stress, but then suddenly snap.
• Engineering Rule: Never extrapolate data critical to safety unless the physical laws governing the extended range are fully understood and verified.