課程名稱︰微積分一
課程性質︰數學系大一必修
課程教師︰齊震宇教授
開課學院：理學院
開課系所︰數學系
考試日期（年月日）︰2015/11/12
考試時限：7.5 hr
試題 :
第一部分敘述題
1. Let A,B are included in R. We say that (A,B) is a Dedekind cut of R if:
2. Let a_n be a sequence in a metric space X with metric d and L belonds to X. We say that lim_{n→∞} a_n = L if:
3. A sequence a_n in a metric space X with metric d is a Cauchy sequence if:
4. We say that a series Σ_{n=1}^∞ a_n converges if:
5. We say that a series Σ_{n=1}^∞ a_n converges conditionally if:
6. A function d: XxX→R is a metric on X if:
7. Let X be a metric space. We say that a subset K of X is compact if:
8. Let f: X→Y be a map between metric spaces X and Y(with metric d_X and d_Y respectively) and a belongs to X. Please write down and ε-δ definition of "f is continuous at a."
9. Let X be metric space, S is included in X, and a belongs to X. a is a limit point of S if:
10. A map f: X→Y between metric spaces X and Y(with metrics d_x and d_Y respectively) is uniformly continuous if:
11. Let f: S→R be a function and a belongs to S. For an integer k≧2, we say that f is k-times differentiable at a if:
12. Let f: X→Y and g: X→Y be two maps between metric spaces(with metrics d_X and d_Y respectively). For any β＞0 and a belonging to X, we say that f(x) is approximated by g(x) to order β as x→a if:
第一部分證明題
1.(6 points)
Show that Σ_{n=1}^∞ a_n = sup{a_{n_1} + ... + a_{n_k}|n_1＜...＜n_k, k belongs to N} if a_n≧0 for all n. (Here we allow any side to be ∞. More precisely, we say that the left(resp. right) hand side is ∞ if its partial sum sequence of the series Σa_n (resp. the set {...}) has no upper bond.)
2.(4 points)
Let a_n and b_n (n belongs to N) be bounded sequence of real numbers. Show that limsup_{n→∞} (a_n + b_n) ≦ limsup_{n→∞} a_n + limsup_{n→∞} b_n.
3.
Let f: X→Y be a map between metric spaces. Show that
(1)(5 points)
f is continuous iff. f^(-1)(V) is open in X for every open set V in Y.
(2)(5 points)
if K(included in X) is compact and f is continuous, then f(K) is a compact subset of Y.
4.(8 points)
Let K be a compact subset in a metric space X and O_α(α belongs to A) is an open cover of K. Show that there exists δ＞0 such that for every pair of points x, x' belonging to K, if d(x,x')＜δ, then there exists α belonging to A such that x, x' belonging to O_α
5.(10 points)
State Rolle's theorem and the generalized mean value theorem and prove them.
6.(10 points)
For any k belonging to N, functions f: S→T, g: T→R with a belonging to S included in R and T included in R, and polynomials P(x) and Q(y) with f(a) = P(a) and g(f(a)) = Q(f(a)), show that if f(x) is approximated by P(x) to order k as x→a and g(y) is approximated by Q(y) to order k as y→f(a), then g(f(x)) is approximated by Q(P(x)) to order k as x→a.
第二部分是非題
1. A uniformly continuous function on a metric space must be Lipschitz.
2. If a function f: R→R is differentiable, then f' is continuous on R.
3. If a function f: R→R is k-times differentialbe at a and is approximated by a polynomial P(x) as x→a, then f^(j)(a)=P^(j)(a) for j=0,...,k.
4. Let f: R^2→R be a function whose partial derivatives exist everywhere. If (a,b) belongs to S(included in R) and f|_S achieves its maximum at (a,b), then f_x(a,b) = f_y(a,b) = 0.
5. Let f(x,y) = x^4 + 3x^2 y^2 + y^6 and S = {(x,y) belonging to R^2|y≧x^2}. Then f|_S achieves its minimum.
6. If an alternating series Σ_{n=1}^∞ (-1)^{n-1} b_n converges with b_n＞0, then b_n↘0 as n→∞.
7. Let f: R→R be a differentiable function. Then f'(x)＞0 for every x belonging to R if and only if f is strictly increasing, i.e. if x＞x' then f(x)＞f(x').
8. If f: (-a,a)→R(where a＞0) is even(resp. odd) and f' is differentiable on (-a,a), then f' is and odd(resp. even) function. (We call such a function f an even(resp. odd) function if f(-x) = f(x)(resp. = -f(x)) for all x belonging to (-a,a).)
9. Let X be a metric space and S is included in X. Then {all limit points and isolated points of S} is the intersection of all closed sets of X containing S.
10. If both f,g: R→R are differentiableat 0, then so g(f(x)).
第二部分證明題
1.(5 points)
Given 0＜c＜d, we define two sequences c_n and d_n as follows: c_1 = c, d_1 = d, c_{n+1} = sqrt(c_n d_n), d_{n+1} = (c_n + d_n) / 2 for all n belonging to N. Show that both lim_{n→∞} c_n and lim_{n→∞} d_n exist and are equal.
2.
Let f(x) = x^x, x>0.
(1)(5 points)
Compute f'(5).
(2)(5 points)
Does lim_{x→0+} f(x) exists? If it does, compute it; if it does not, give a proof.
(3)(5 points)
Consider the function g: (0,∞)xR→R. Does lim_{(x,y)→(0,0)} g(x,y) exist? If it does, compute it; if it does not, give a proof.
3.
Consider the function f: R^3→R, f(x,y,z) = x^4 + y^4 + z^4 - 4xyz. Let K = {(x,y,z) belonging to R^3|yz≦0 and x^2 + y^2 + z^2 ≦ 3}.
(1)(3 points)
Show that K is a closed set in R^3.
(2)(7 points)
Find the maximum and minimum of f|_K and where they are achieved.
4.(10 points)
Suppose that f and g are differentiable function on R such that
(a) lim_{x→∞} 1/|g(x)| = 0,
(b) g'(x) ≠ 0 for every x belonging to R, and
(c) lim_{x→∞} f'(x)/g'(x) exists.
Show that lim_{x→∞} f(x)/g(x) = lim_{x→∞} f'(x)/g'(x).
5.
Let X be a metric space. We say that a function f: X→R is upper semicontinuous(abbr. usc) if limsup_{x→a} f(x) = f(a) foor every a belonging to X.
(1)(5 points)
Show that a function f: X→R is usc if and only if for every c belonging to R the set {x belongs to X|f(x)＜c} is open.
(2)(10 points)
Show that if f: X→R is usc and X is compact, then f achieves its maximum on X, i.e. there exists a belonging to X such that f(a)≧f(x) for all x belonging to X.
6.(10 points)
Does the following limit exist?
lim_{x→0} (sin(tan x) - tan(sin x)) / (arcsin(arctan x) - arctan(arcsin x))
If it does, compute it; if it does not, give a proof.
7.
Let f: R→R be a function such that lim_{t→a} f(x) exists for every a belonging to R.
(1)(7 points)
Show that the function g: R→R, g(x) = lim_{t→x} f(t) is continuous.
(2)(8 points)
Let g be the function defined in (1). Show that for every ε＞0 and every pair of real numbers a,b with a＜b, the set A := {x belonging to [a,b]||f(x)-g(x)|＞ε} is finite. (Hint. You may want to use Bolzano-Weierstrass theorem.)