Nyxie said:As mathman said, plug the numbers into a fourth degree polynomial. The coefficients are your unknowns. Now take the first and second derivatives of the fourth degree polynomial. Now you have 3 equations, and that's all you're going to get. But the fourth degree polynomial has 5 unknowns. The leftovers could of course be anything; they represent the coordinates of where the point that you took the derivative of was. This point could of course be anywhere. So you can't interpolate.
Interpolation with knowledge of derivatives is a mathematical technique used to estimate values between known data points using the derivative information of the function. It involves using the slope of the function at a given point to determine the value of the function at a nearby point.
Interpolation with knowledge of derivatives works by using the derivative information of a function to construct a polynomial that passes through the known data points. This polynomial can then be used to estimate the value of the function at any point between the known data points.
One of the main benefits of using interpolation with knowledge of derivatives is that it can provide more accurate estimates of values between data points compared to other interpolation methods. It also allows for the estimation of values at points outside the range of the known data.
Interpolation with knowledge of derivatives is commonly used in fields such as mathematics, engineering, and physics. It is particularly useful in situations where the data is known to have a smooth and continuous nature, and the derivative information is readily available.
One limitation of interpolation with knowledge of derivatives is that it requires the function to be differentiable at all points. If the function is not differentiable, this technique cannot be used. Additionally, it may not accurately estimate values if the data points are too far apart or if there are large variations in the derivative values.