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

Here we will cover Section 5.1 with focus on using Newton–Gregory interpolating polynomials (with evenly spaced data) to approximate derivatives from tabulated 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 Tuesday using the 3-point forward difference formula.
   2. at 3:00 a.m. on Wednesday using the 3-point central difference formula.
   3. at 3:00 a.m. on Thursday 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. Work these problems on numerical differentiation.

3. In class I used Formula (3P) in the lecture notes to obtain the 3 point forward difference formula to approximate f '(x₀)
   and to obtain the 3 point central difference formula to approximate f '(x₁).

   Now you use Formula (3P) to obtain the 3 point backward difference formula to approximate f '(x₂).
   Then write the formula in standard form so that the interpolant is called x₀ rather than x₂.

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)
   Explain why.

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