这是一篇来自美国的关于数学期中考试的数学代写
1(a) Show that
(1 +1/n)n = 2 +nXk=2(1 − 1n )(1 − 2n)···(1 − k−1n )/k!.(1)
(10 points)
(b) By (a), show that the sequence an = (1 +1/n)n , n = 1, 2, 3, · · · , is increasing.
(10 points)
(c) Show that
(1 +1n)n < 2 +nXk=21/k!.
(5 points)
(d) Use the mathematical induction to show that k! ≥ 2k−1.
(5 points)
(e) Conclude by (a) through (d) that
2 < (1 +1/n)n < 3.
(5 points)
Note: The increasing sequence (1+
1
n
)n , n = 1, 2, 3, · · · , bounded from above, then attains a limit, which is the celebrated Euler number e.
- (a) Defifine
f(x, y) =x2y2/x2 + xy + y2 .
Verify that the domain of f is R2 \ {(0, 0)}.
(5 points)
(b) Prove that
lim(x,y)→(0,0) f(x, y) = 0.
(10 points)
- Prove that the function
f(x, y) :=x3y /x6 + y2
has no limit at (0, 0).
(5 points)
- (a) Give an example f : R −→ R and g : R −→ R such that f is continuous and g is not continuous while g ◦ f is continuous.(5 points)
(b) Use the − δ defifinition to show that the function
f : x ≥ 0 7−→ √x
is continuous at x = 10.
Hint:
f(x) − f(a) = √x − √a = x − √a/√x +√a.
(5 points)
- Give an example of a continuous function f : R subset S of R such that f(S) is not closed in R. −→ R and a closed (15 points)
Hint: A compact set in R is bounded and closed. f sends a compact set to a compact set, which is closed in R. Thus, the set S you construct can never be bounded to begin with. So, you may want to try S := N
6(a). Let f : [a, b] −→ R be continuous with the property f(a) = f(b).Prove that f is not one-to-one. (Hint: Explore the intermediate value theorem.) (10 points)
(b) Prove that a one-to-one continuous function g : [a, b]−→ R is either strictly increasing, i.e., f(x) < f(y) for any x < y, or strictly decreasing, i.e., f(x) > f(y) for any x < y. (10 points)