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Linear predictive analysis is a simple form of first-order extrapolation: if it has been changing at this rate then it will probably continue to change at approximately the same rate, at least in the short term. This is equivalent to fitting a tangent to the graph and extending the line.
One use of this is in Linear predictive coding which can be used as a method of reducing the amount of data needed to approximately encode a series. Suppose it is desired to store or transmit a series of values representing voice. The value at each sampling point could be transmitted. If it is known that the value rarely changes more than +/- 15 values between successive samples then we could encode the change in 5 bits. As long as the change is less than +/- 15 values in successive steps the value will exactly reproduce the desired sequence. When the rate of change exceeds +/-15 then the reconstructed values will temporarily differ from the desired value; provided fast changes that exceed the limit are rare it may be acceptable to use the approximation in order to attain the improved coding density.