In the following, assume f: R → R has 3 continuous derivatives. A common usage of Taylor's theorem is to construct finite difference schemes for approximating the derivatives of f. Let's take a look at some basic problems in this vein: (a) Let x ER and h> 0. Show that there exists constants M₁, M₂, possibly depend- ing on x, h, and f and its derivatives, such that M₂h ≤ (1₁ (x) — 1(x + h) - S(x)) : h This shows the "first-order accuracy" of the forward-difference approximation. (Hint: Use Taylor's theorem for x,x + h) 0. Show that there exists constants M3, M4, possibly depend- ing on x, h, and f and its derivatives, such that Mgh² ≤ (f'(x) — f(x + h) — ƒ (x − h) 2h - h |ƒ′(z) – f(x + h) − f (x) | |f² (2) _ f(x + h) = f(x − h) - | 2h This shows the "second-order accuracy" of the centered-difference approximation. (Hint: Use Taylor's theorem for x,x+h and x, x-h) (c) Letk € R, and let f(x) = sin(kx). Show that for all x ER and h> 0, 3/2. This might happen if your function f has "high-frequency noise".

The function f(x) is continuous and differentiable on the interval [2,10]. It is known that f(2)=8 and the derivative on the given interval satisfies the condition f′(x)≤4 for all x∈(2,10). Determine the maximum possible value of the function at x=10.

For the function P(A, B), where both A and B are functions of t, r and w, use the chain rule to state the partial derivative of P with respect to w. That is, write out the chain rule for

1) Estimate the derivative at each point using a right hand approximation where possible 0 f(x) 5 1 2 3 11 14 22