MATH 355 Final: MATH 355 Amherst S13M355Final

Let b > 0 and f : [0, b] → R be continuous and f(b) = 0. Assume that f is differentiable on (0, b). Show: For every n ∈ N, there exists some c ∈ (0, b) such that cf ′(c) = −nf(c).

Hint: Apply Rolle’s theorem to the function g(x) = xⁿ f(x).

we mean that for every e > 0 there exists a 8 >0 such that 1. (a) By lim f(x) = - f(t)<- whenever I > (b) At a height of 2km the barometric pressure is changing at a rate of 1 kiloPascal per km. (c) tanh x = et-e- etter Its domain is R and its codomain is (-1,1). (d) log2 (2) - log2 (5) + log2 (20) = log2 8 = 3 (e) Observe that f(0) = 2, f(1) = 1 – 2 – 3 + 2 = -2, and f() is continuous on the interval [0,1]. Thus, by the Intermediate Value Theorem there exists a root of f between 0 and 1.