Question

Can anybody help me with this, it has to be on maple or PDF

Four points on the graph of (x)=ln(x) are given: (1, 0), (3, 1.099), (6, 1.792), and (10, 2.303).
a) Find a cubic polynomial g(x) which approximates the inverse function of f(x) by fitting the cubic g(x) to the points (0, 1), (1.099, 3), (1.792, 6), (2.303, 10).
b) As you may know, the inverse function of ln(x) is exp(x). Compose the values of g(x) and In(x) at the given values of x = 0.1, 1, 1.1, 2, 2.3, 3, and 4.
c) Plot the graphs of g(x) and exp(x) in the same window for −1≤x≤3. Why are the graphs close to each other for some values of x but not for others?
d) Plot g(x) along with the following points in the same Cartesian Plane: (0, 1), (1.099, 3), (1.792, 6), (2.303, 10).

Answer 1

To find a cubic polynomial g(x) that approximates the inverse function of f(x), use Newton's divided difference formula to find the coefficients and then evaluate g(x) and In(x) at given values of x.

Step-by-step explanation:

To find a cubic polynomial g(x) that approximates the inverse function of f(x), we need to find the cubic polynomial that fits the points (0, 1), (1.099, 3), (1.792, 6), and (2.303, 10).

We can use the Newton's divided difference formula to find the coefficients of the cubic polynomial.

Once we have the cubic polynomial g(x), we can evaluate g(x) and In(x) at the given values of x and compose them.

The graphs of g(x) and exp(x) are close to each other for some values of x because g(x) is an approximation of the inverse function of ln(x), which is exp(x).

We can plot g(x) and the given points in the same Cartesian Plane to visualize the polynomial and the points.