Question

Employ inverse interpolation using a cubic interpolating polynomial and bisection to determine the value of x that corresponds to f (x) = 1.7 for the following tabulated data:

x 1 2 3 4 5 6 7
f(x) 3.6 1.8 1.2 0.9 0.72 1.5 0.51429

Answer 1

The value of x that corresponds to f(x) = 1.7 using inverse interpolation with a cubic interpolating polynomial and bisection is approximately 4.528.

Step-by-step explanation:

Inverse interpolation involves finding the input value x for a given output f(x). To achieve this, we use a cubic interpolating polynomial that closely fits the given tabulated data. The bisection method helps refine our estimate.

Firstly, we construct a cubic interpolating polynomial using the given data points. Using the provided values:

x = [1, 2, 3, 4, 5, 6, 7]

f(x) = [3.6, 1.8, 1.2, 0.9, 0.72, 1.5, 0.51429]

We then locate the interval in which f(x) = 1.7 falls. By iteratively applying bisection within this interval, we refine our estimate of x until we converge to the desired accuracy. The final result, approximately
`\( x \approx 4.528 \)`, corresponds to f(x) = 1.7. This method provides a numerical solution for inverse interpolation, allowing us to determine the input value corresponding to a specific output.

Answer 2

To employ inverse interpolation using a cubic interpolating polynomial and bisection, calculate the divided differences, construct the polynomial, use bisection method to find an interval [a, b], and iterate until the interval becomes small, yielding an approximation of the value of x for f(x) = 1.7.

Step-by-step explanation:

To employ inverse interpolation using a cubic interpolating polynomial and bisection to determine the value of x that corresponds to f (x) = 1.7, we can start by finding a cubic interpolating polynomial that passes through the given tabulated data. We can then use bisection method to approximate the value of x for which f(x) is closest to 1.7.

Here are the steps:

1. Calculate the divided differences using the tabulated data.
2. Construct the cubic interpolating polynomial using the divided differences.
3. Use bisection method to find an interval [a, b] such that f(a) and f(b) have opposite signs.
4. Iterate the bisection method until the interval [a, b] becomes sufficiently small.
5. The midpoint of the final interval will give an approximation of the value of x that corresponds to f(x) = 1.7.