MATH 104, Fall 2023 Final Practice Problems
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Additional Final Practice Problems
MATH 104, Fall 2023
1. Let f : R → R be a continuous function, and deine
S = {x ∈ R : x = n(l) f (xn ) for some sequence xn -!- ∞}.
Suppose that 0, 1 ∈ S. Then show that we in fact have [0, 1] S.
2. Let f : R → R be a continuous function, and suppose that 1pq f (x) dx = 0 whenever p, q ∈ Q. Prove that f = 0.
3. Determine if the statement is true or false, with justiication: Suppose that f : R → R is a continuously diferentiable function which satisies
x(l) f (x) = a, x(l) f\ (x) = b.
Then we must have b = 0.
4. Suppose that f : R → R is a continuous function which satisies f (x) = f (x + 1) for all x ∈ R. Show that f is bounded, achieves its maximum and minimum, and is uniformly continuous.
5. Suppose that (an )n and (bn )n are two sequences of nonnegative num- bers, with limn!1 bn = 0. Also suppose we have k ∈ (0, 1) such that
an+1 kan + bn ,
for all n. Then show that limn!1 an = 0.
.
7. Suppose that f : R → R isa twice-diferentiable function which satisies
0 = f (3) = f (0) = f地 (0).
Show that f地地 (x) = 0 for some x ∈ [0, 3].
8. Suppose that f : R → R is a function which satisies |f (x)| |x| 2 , and g : R → R is a bounded function. Show that f (x)g(x) is diferentiable at x = 0.
9. Suppose that f : [0, ∞) → R isa continuous function with limx!1 f (x) =
a ∈ R. If the indeinite integral '01 f (x) dx exists, show that a = 0.
10. Let K be a compact subset of a metric space S, and let {Uj } be a family of open sets covering K. Prove there exists an E > 0 such that for every x ∈ K we have B(x, E) Uj for some j.
11. Suppose that X R is a nonempty connected set of real numbers, and X Q. Describe all possibilities for the set X .
2023-12-15