MATH 295 PROBLEMS 12

IGOR KRIZ

1. Using the axioms for ln x and theorems proven in class, compute:

lim_{x→0} \frac{a^{x} − 1}{x} (a > 0).

2. Compute the derivative of

\sqrt[3]{2e^{x} − 2^{x} + 1} + (ln x)^{5}.

3. Prove that a is a double root of a polynomial f(x) (i.e. f(x) = (x − a)^{2}g(x) for some polynomial g(x)) if and only if a is the root of both f and its derivative f′.

4. If f is three times differentiable and f′(x) ≠ 0, the Schwarzian derivative of f at x is defined to be

Df(x) = \frac{f′′′(x)}{f′(x)} − \frac{3}{2} (\frac{f′′(x)}{f′(x)})^{2} .

(a) Show that

D(f ∘ g) = [Df ∘ g].(g′)^{2} + Dg.

(b) Show that if f(x) = \frac{ax+b}{cx+d}, with ad − bc ≠ 0, then Df = 0. Consequently, in that case, D(f ∘ g) = Dg.

5. Suppose that f(x) = xg(x) for some function g which is continuous at 0. Prove that f is differentiable at 0, and find f′(0) in terms of g.