PHYCS 6730 Lab Exercise

Exercise 1 Numerical differentiation and polynomial approximation experiment

The derivative of tan(x) is exactly 1/cos²(x). Using Maple with 10 digit accuracy, find the best estimate of the derivative at x = 1.2 using

"3 pt" central difference estimate
```
        [tan(x+h) - tan(x-h)]/(2h)
```

"5 pt" central difference estimate

     [tan(x-2h) - 8tan(x-h) + 8tan(x+h) - tan(x+2h)]/(12h)

Do this by varying the value of h and comparing the estimate with the known exact result. You will see that as h is made smaller, the accuracy improves. But when h is too small, precision is lost. Optimize h only up to integer powers of 10, such as 1e-2 or 1e-3, etc.

In your answer file show a table of your results.

Exercise 2. Chebyshev polynomial

Use Maple to plot the Chebyshev polynomial T(8,x). What is the largest magnitude of this function on [-1,1]? Could you have anticipated this result from the general formula
```
      T(n,x) = cos(n arccos x)?
```
Find the zeros of T(8,x). These are proposed polynomial interpolation points for approximating a function on the interval [-1,1].