6. a. Write a function to approximate the change of an arbitrary function f(-) with respect to a small change in X, say dx, via its Taylor's series expansion: For any infinitely differentiable function f(-), one can write df (x) 1 dn f(x) 1 (n+1(E) f(x+Ax) = f(x)+ Ax+ (4x)" + dx 2 dx2 n! dxn 1 d2 f() (Ac)2+...+ (n+1)! dxn+l(Ax)+1, I where & E (X, X + Ax). Comment on the differences between the true and the approximation values as a function of the number of terms included in the expansion. b. Using Taylor's series expansion, or otherwise, justify the following approximation: P(y) Ply+dy) -D(dy), and P(y+dy) P(y){1 - D* (dy)}, where D and D* denote the dollar duration and the modified duration for a bond P(y) with yield y. You should also argue why the bond price can be infinitely differentiable with respect to y if you make use of the result in (a) to support your claim. Illustrate the approximation via a figure with R and/or Python. 6. a. Write a function to approximate the change of an arbitrary function f(-) with respect to a small change in X, say dx, via its Taylor's series expansion: For any infinitely differentiable function f(-), one can write df (x) 1 dn f(x) 1 (n+1(E) f(x+Ax) = f(x)+ Ax+ (4x)" + dx 2 dx2 n! dxn 1 d2 f() (Ac)2+...+ (n+1)! dxn+l(Ax)+1, I where & E (X, X + Ax). Comment on the differences between the true and the approximation values as a function of the number of terms included in the expansion. b. Using Taylor's series expansion, or otherwise, justify the following approximation: P(y) Ply+dy) -D(dy), and P(y+dy) P(y){1 - D* (dy)}, where D and D* denote the dollar duration and the modified duration for a bond P(y) with yield y. You should also argue why the bond price can be infinitely differentiable with respect to y if you make use of the result in (a) to support your claim. Illustrate the approximation via a figure with R and/or Python