Question

Given the values `(7,1.395),(8,1.442),(9,1.482),(10,1.517),(11,1.549),(12,1.576), (13,1.602)`, without constructing interpolation polynomial, how do I estimate the value of the function when `x=8.5` ?

Answer 1

`1.463`

Explanation:

Given:

`((7,1.395),(8,1.442),(9,1.482),(10,1.517),(11,1.549),(12,1.576), (13,1.602))`

Notice the `x` values are equally spaced, so we can get some information using a method of differences:

Write down the initial sequence of `y` values:

`1.395, 1.442, 1.482, 1.517, 1.549, 1.576, 1.602`

Write down the sequence of differences between successive terms:

`0.047, 0.040, 0.035, 0.032, 0.027, 0.026`

Note that these are monotonically decreasing.

Let's split each roughly in half, but making the first half slightly larger to get:

`0.024, 0.023, 0.021, 0.019, 0.018, 0.017, 0.016, 0.016, 0.014, 0.013, 0.013, 0.013`

That looks like it is decreasing reasonably smoothly.

Now apply these half differences to the previous sequence in order to interpolate:

`1.395, 1.419, 1.442, color(red)(1.463), 1.482, 1.500, 1.517, 1.533, 1.549, 1.563, 1.576, 1.589, 1.602`

So the approximation for `f(8.5)` is `1.463`