課程名稱︰微積分甲上
課程性質︰數學微積分
課程教師︰周青松
開課學院：
開課系所︰限生機、生工、地質、地理、工管等系學生修習
考試日期（年月日）︰1月12日
考試時限（分鐘）：120分鐘 8:10~10:10
是否需發放獎勵金：是
（如未明確表示，則不予發放）
試題 :
Ⅰ
A) Let f be a function such that f' is continuous on [a,b]
Prove that b
∫ f'(t)dt = f(b) - f(a)
a
B) Calculate ∫ [f(x)g''(x) - f''(x)g(x)]
Ⅱ
A) Find f from the information given :
f''(x) = sinx f'(0)= - 2 f(0) = 1
B) Calculate the derivative :
d/dx 2x 2
( ∫ t √1+t dt )
tanx
Ⅲ
A) The base of a solid is the region between the parabolas
2 2
x = y and x = 3 - 2y
Find the volume of the solid given that the cross section perpendicular
to the x - axis are squares .
2/3
B) Let Ω be the region bounded below by the curve y = x + 1
bounded to the left by the y-axis , and bounded above by the line
y = 5 Find the volume of the solid generated by revolving Ω about the
y-xis .
Ⅳ
A) If a is positive and p/q is rational , prove that
p/q
a a
∫ 1/t dt = p/q∫ 1/t dt
1 1
-2x -2x
B) Calculate the following indefinite integration ∫sin e / e dx
Ⅴ
2 2 2 -1
A) Show that F(x) = x/2√a - x + a /2 sin (x/a) a > 0 is an
2 2 a 2 2
antiderivative of f(x) = √a - x amd to calculate ∫ √a - x dx
-a
-1 2
B) Prove that d/dx tanh x = 1/1-x -1 < x < 1
(每大題均20分)
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