Chapter 10, Problem 56RE

Derive a 2nd order accurate finite difference expression for the first derivative at point i-1, using the values at points i and i+1, in the case of general (non-uniform) spacing. Then reduce this expression in the case of uniform spacing. Use polynomial fitting.

Let f be a function that has derivatives of all orders on the interval (5, 7). Use the values in the table below and the formula for Taylor polynomials to give the 4" degree Taylor polynomial for f centered at x = = 6: f(6) f'(6) f"(6) | f"(6) | f(4)(6) -3 4 8 9. -3 3 x³ + 4x² + 4x – 3 1 b) O-3 + 4 (x + 6) +4(x + 6)° + 2(x + 6)3. 8 3 c) O-3 + 4 (x – 6) + 4(x – 6)² + x – 6)³ - (x – 6)* d) O-3x* + 9x³ + 8x2 + 4x – 3 e) O-3 +4 (x – 6) + 8(x – 6)² + 9(x – 6)³ – 3(x – 6)* f) O-3 + 4 (x + 6) + 8(x + 6)² + 9(x + 6) – 3(x + 6)* g) None of the above.

Let f(x) = x* – 3x2 + 2 be a polynomial function. Which one the following intervals, the function f(x) is definitely concave down ? (-0, –1) A) (-2,0) B) (0, 00) D) (1, 00) E)