Always rework my examples successfully on your own BEFORE attempting the homework.

Here we use Newton–Gregory interpolating polynomials to approximate derivatives from tabulated (numeric) data.

See these various difference formulas for derivatives. This is merely a sampling of various differentiation formulas.

1. Use the data given on Slide 13 of the lecture notes to approximate the velocity of the asteroid
   1. at 3:00 a.m. on Sunday using the 3-point forward difference formula.
   2. at 3:00 a.m. on Monday using the 3-point central difference formula.
   3. at 3:00 a.m. on Tuesday using the 3-point backward difference formula.
   Answers:       (a) –68,750 kph,       (b) –81,250 kph,       (c) –93,750 kph.
   Explain why the velocities are negative.

2. In class I used Formula (3P) in the lecture notes to obtain:
   • the 3 point FORWARD difference formula (3f) for f '(x₀), and
   • the 3 point CENTRAL difference formula for f '(x₁).
   Now you use Formula (3P) to obtain the 3 point BACKWARD difference formula (3b) to approximate f '(x₂).
   Then write the formula in standard form so that the interpolant is called x₀ rather than x₂.

3. Work these problems on numerical differentiation.

4. What's the fewest number of points needed in order to approximate a
   1. 3rd derivative?                   (Answer:  4 points)
   2. 5th derivative?                   (Answer:  6 points)
   3. 8th derivative?                   (Answer:  9 points)
   Explain why.

   Answers:

   1. 4 points
   2. 6 points
   3. 9 points

5. Why might we get large errors when approximating a high order derivative by Newton-Gregory polynomials?

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