課程名稱︰微積分甲
課程性質︰系定必修
課程教師︰林紹雄
開課學院：理學院
開課系所︰數學系
考試時間︰11/11/2006
是否需發放獎勵金：是
（如未明確表示，則不予發放）
試題 :
There are Problems A to I with a total of 130 points. Please write down your
computational steps, or proof procedure clearly on the answer sheets.
An answer without any reasoning will not be counted.
＿＿
A.(5 points) Find the limit lim √x+1 , and use the "definition" of the limit
x→3
to prove the answer.
B. Find the following limits. Each has 5 points.
＿＿＿＿
(a) lim (x+√x^2+x+3 ); (b) lim [tan(2x)tan((π/4)-x)].
x→∞ x→(π/4)
C.(10 points) Define the function f(x) as
f(x)={(x^2)(1-x), if x is an irrational number
{x^3, if x is a rational number.
Find all points x at which f is continuous. Is f(x) differentiable at some
points? If yes, find all such points.
D.Solve the following problems.
(a) Find the derivatives. Each has 5 points.
d sinx-xcosx d^3 x
(a) ─(─────) ; (b) ──(──────) .
dx cosx+xsinx dx^3 (1+x)^(1/3)
(b) (6 points) Find all points on the plane curves (x^2+y^2)^2=4(x^2-y^2)
where the normal lineis either horizontal or vertical.
E. Find the following definite or indefinite integrals. Each has 7 points.
3 ＿＿＿_
(a) ∫(√(x^2)-1)dx/x^4 ; (b) ∫tanx(cosx)^(1/3)dx.
1
F.(15 points) Graph the function f(x)=x/[(x^2-1)^(1/3)] . Be sure to write
down the intervals of monotonicity and concavity. Also write down its
positions of local extreme points and inflection points. Please indicate
clearly the positions of the asymptotic lines.
G. Let f(x) be a continuous function defined on x≧0 , and
x
(f(x))^2=2∫f(t) dt for all x>0 .
0
(a) (10 points) If f(x)≠0 for all x>0 , prove that f(x)=x for all x≧0 .
Is this function the only nonzero continuous function which satisfies
the conditions?
(b) (5 points) Can you find a nonzero f(x) which satisfies the conditions,
and f(x)=0 for x>100 ?
H. Solve the following problems.
(a) (10 points) Two hallways, one 8 feet wide and the other 6 feet wide,
meet at right angles. Determine the length of the longest ladder that
can be carried horizontally from one hallway into the other.
(b) (5 points) A person 6 feet tall is watching a streetlight 18 feet high
while walking toward it at a speed of 5 feet per second. At what rate
is the angle of elevation of the person's line of sight changing with
respect time when the person is 9 feet from the base of the light?
(註:b小題有圖,但我沒辦法在這裡畫,故簡單文字說明)
人身高六呎,距離路燈底x呎,路燈高18呎,人看路燈頂的仰角為θ
I. Determine which of the following statements is true. If it is true, prove
the answer. If it is false, give a counterexample. Each has 6 points.
(a) If lim f(x)=l and lim g(y)=m, then lim g(f(x))=m.
x→a y→l x→a
(b) There exists two numbers a and b such that the equation x^7+ax+b=0
has 4 distinct real roots.
(c) The function f(x) and g(x) can be nowhere continuous, but f(x)g(x) is
everywhere differentiable.
(d) There exists a continuous function f(x) defined on [0,1] , and the range
of f(x) is (0,1].
(e) If lim f(x)≠l, then lim (f(x))^3≠l^3 .
x→a x→a
試題結束.
--
Wenn der Altenberg nicht im Kaffeehaus ist, ist er am Weg dorthin.
--
※ 發信站: 批踢踢實業坊(ptt.cc)
◆ From: 203.67.108.196
※ 編輯: cisfeather 來自: 203.67.108.196 (11/13 21:55)
※ 編輯: cisfeather 來自: 203.67.108.196 (11/13 22:02)