-
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1. Let T be the triangle with vertices (0, 0), (1, 0), and (0, 1). Find
Sfe*dA=
2. Let A=
2
e- dt. Find C=
_such that
50'e-di, B =
dt["
Liber
e-ve dydx = A -
-B+C.
-T
3. Find the maximum value of the function T(1,y) = x - 2y2 –
on the region A = {(,y)|x2 + y2 <1}.
(x² – 42+6
4. Let f(x) =
if I is rational,
the
202 – 6x2 + 12x – 6 if & is irrational
_such that f'(20) exists.
Let g be a differentiable real-valued function satisfied
9
Find Jo =
(* oe) dt
) )+m7-2+ *300
P ( e) dt
gt
sin + cos z = 0 + exp
for all 2 near 0. Find g'(0) -
Two holes of radius 6 are drilled through the center of a sphere of radius
10. Assume that their axes meet at right angles. Find the volume of
the solid remaining
7. I F(x) = [ 560 dt, where f(1) = 1 * vi er
du, find F"(2) =
1
u
乙、計算題:每題15分,須詳細寫出演算過程,否則不予計分。
1. Show that the function defined by
e if < 70,
f(x)=
if x = 0.
0
is not equal to its Maclaurin series. X
正致
2. Let a and b be positive numbers with a > b. Let aj be their arithmetic
a + 6
mean and b, their geometics mean: a1 = 5,62 = Vab. Repeat this
2
process so that, in general, an+1 = On+1 = Vamor for n 2 1.
2
Show that both {am) and {br} are convergent and lim An = lim bn.
ant bn
100
700
