Write down the first three terms of the Taylor series expansions of the functions dx (t-h) and dx dt (t-2h) dt about x(t). Use these two equations to eliminate d²x dt (1) and (1) drº from the Taylor series expansion of the function x(t + h) about x(t). Show that the resulting formula for x(t + h) is the third member of the Adams- Bashforth family, and hence confirm that this Adams-Bashforth method is a third-order method.

Write the Taylor series for f(x) = x³ about x = -2 as Find the first five coefficients. CO= C1= C2= C3= C4= Σen(x + 2)". n=0

Q7.4 Find the taylor series for fcx) = centered at a= 2 %3D (A) 1y"-1 (n+ 1) 22 00 (-1y+1 -(x-2)" 2 n=0 (x-2)" (B) n=0 (D) E 2-1 (x- 2) n=0 (-1)*-1 (-1)" (n + 1) (x – 2y" (F) 00 (E) (-1у (x- 2y? 2n-1 n=0 n=0 (G) E (-1)"- (n+ 1) (x - 2y 2-1 n=0 Chose 1 of the given options as your answer!

2.3.9 Write down the first four terms of the Taylor series expansion of the function x(t-h) about x(t), and the first three terms of the expansion of the function dx dt (t-h) about x(t). Use these two equations to eliminate d²x (t) and d³x dt³ (t) dt² from the Taylor series expansion of the function x(t + h) about x(t). Show that the resulting formula is Xn+1 = -4X + 5X-1 + h(4F + 2F-1) + 0(ht) Show that this method is a linear combination of the second-order Adams-Bashforth method and the central difference method (that is, the scheme based on (2.9)). What do you think, in view of this, might be its disadvantages?