Math205a Fall Term 2002

Final Examination

Print your name: Print your ID #:

You have 1 hour and 50 minutes to solve the problems. Good luck!

1. Let a sequence (x_{n})_{n∈ℕ} be defined by

\begin{cases}x_{n+1} = \sqrt{1 + x_{n}} , & n ≥ 0 \\ x_{0} = 1 . \end{cases}

Prove that it converges and compute its limit.

2. Let (x_{n})_{n∈ℕ} be a positive decreasing sequence which converges to zero. Show that the sequence (y_{n})_{n∈ℕ} defined by

y_{n} = ∑_{k=0}^{n}(−1)^{k} x_{k} , n ∈ ℕ

is convergent.

3. Let f ∈ C(ℝ, ℝ) and A ⊂ ℝ be compact. Prove or disprove: (i) f^{−1}(A) is closed. (ii) f^{−1}(A) is compact.

4. Assume that f ∈ C(ℝ, ℝ) is such that

lim_{|x|→∞} f(x) = 0 .

Prove that f attains at least its maximum or its minimum.

5. Let f ∈ C(ℝ, ℝ) satisfy

f(x + y) = f(x)f(y) , x, y ∈ ℚ and f(0) = 1 .

What is f and why?

6. Let f ∈ C(ℝ, ℝ) be periodic, that is,

f(x + T) = f(x) for x ∈ ℝ and some T > 0 .

Show that f is uniformly continuous.

7. Show that the equation

e^{e^{sin(x)}} = y

is uniquely solvable in a neighborhood U_{0} of x = 0 for y in a neighborhood V_{e} of e. Can you compute an approximation to the exact solution for y ≈ 0?

8. Let f ∈ C^{1}(ℝ, ℝ), g ∈ C(ℝ, ℝ) and assume that f(x_{0}) = 0 for some x_{0} ∈ ℝ. Prove that fg : ℝ → ℝ , x ↦ f(x)g(x) is differentiable at x_{0} and compute (fg)′(x_{0}). What can you say if g is only assumed to be bounded?

9. Assume that a, b, c ∈ ℝ with a < c < b, let f ∈ C((a, b), ℝ) and define

g(x) = \begin{cases}\frac{f(x)−f(c)}{x−c} , & x ≠ c , \\ d , & x = c . \end{cases}

What conditions on f and d ∈ ℝ make sure that g is continuous on the interval (a, b)?