What is the difference between an interpolated point and an extrapolated point? What assumption is interpolation and extrapolation based on? Which technique is more prone to error?

interpolated: between two known points

extrapolated: beyond the last known point

extrapolation is generally more prone to error. Both are based on the theory of differentials.

What is the theory of differentials?

Interpolation and extrapolation are two techniques used to estimate points on a curve or a line. The main difference between them lies in the region where the estimated point lies.

Interpolation is used to estimate points that fall within the known data range. It involves estimating a value between two or more data points that are already given. In other words, it fills in the gaps between the known data points. For example, if you have data points at x=1 and x=5, interpolation can help you estimate the value of y at x=3.

Extrapolation, on the other hand, is used to estimate points that lie beyond the known data range. It involves extending the trend or pattern observed in the given data to predict values outside the known range. For example, if you have data points at x=1 and x=5, extrapolation can help you estimate the value of y at x=10.

Both interpolation and extrapolation are based on the assumption that the underlying data follows a certain pattern or trend. Interpolation assumes that the relationship between the known data points is continuous, so the estimated points should reasonably fall within the same pattern. Extrapolation assumes that the trend observed in the known data will continue outside the given range.

In terms of error, extrapolation is generally more prone to error compared to interpolation. This is because when you extrapolate, you are extending the trend beyond the known data, making assumptions about its behavior that may not hold true. Factors like data variability, measurement errors, or unforeseen changes in the underlying pattern can lead to significant errors in extrapolated estimates. On the other hand, interpolation deals with estimating points within the range of the known data, which is typically more reliable since it relies on the existing data points.

In summary, interpolation estimates points within the known data range, while extrapolation estimates points outside the known data range. Both techniques are based on assumptions about the underlying pattern, but extrapolation is more prone to errors due to uncertainties in extending the trend beyond the available data.