4. [10 points ] Consider the Taylor expansions (assume h>0, and h small) f(x+h)=f(x)+hf(x)+2!h2f(x)+3!h3f(+)f(xh)=f(x)hf(x)+2!h2f(x)3!h3f() Substracting (2) from (1) and re-arranging, we get f(x)=2hf(x+h)f(xh)+O(h2) where the O(h2) term involves f, and is of the form 3h2(f(+)+f()). Assuming f(x)M for all x[xh,x+h], the approximation f(x)2hf(x+h)f(xh) is said to be of second order. Now consider taking a different stepsize left and right of x. Show that f(x)=hR(hR+hL)hLhR2u(xhL)+(hR2hL2)u(x)+hL2u(x+hR)+O(hRhL) and derive the form of the term O(hRhL). (The arising approximation to f(x) can also be called of second order assuming hL and hR decrease at the same rate, and f(x)M for all x[xhL,x+hR]. But it is nonuniform, while (3) is uniform.)