1) Define the polynomial function f by f (x) = (x - ri )•• •(x — rn). Show that •h(x) 1 1 = + + f(x) x — r1 x — rn in the following two ways. a) Use (without justification) the generalized product...

1) Define the polynomial function f by f (x) = (x - ri )•• •(x — rn). Show that •h(x) 1 1 = + + f(x) x — r1 x — rn
in the following two ways. a) Use (without justification) the generalized product rule: (fif2• • fn)' = fl f2 fnfl "in + • • • + fl f2• • • f7: With sum (E) and product (H) notation this is:
b) Use logarithmic differentiation, that is, compute separately the two sides of: (Inlf(x)IY = Onl.fi(x)1 + ••• + lnifn(x)1)'
Let f be a function defined on a neighborhood of x = a. Assume that f is differentiable at x = a. Recall that the linearization L(x) := f (a)+ f'(a)(x — a) is a linear function such that L(a) = f (a) and L' (a) = f'(a). a) Assume that f is twice differentiable at x = a. Find A. B. and C so that the quadratic function Q(x) := A+ B(x — a) + C(x — a)2 satisfies: Q(a) = f(a) Q'(a) = f'(a) Q"(a) = r"(a)
b) Leta be a nonnegative integer, and assume that f is n times differentiable at x = a. Find if0 A„ so that the nth degree polynomial T (x):= Ao + Ai(x — a) —•• + A.(x — a)" satisfies T,P) (a) =. f()(a) for i = 0. I n
3) Consider a disk of radius r. Roll the disk on a flat surface at a constant angular speed of w. (Equivalently, the center of the disk moves at a constant speed of rw.) 'What are the minimum and maximum horizontal speeds of a point on the surface of the disk?'