課程名稱︰微積分甲上
課程性質︰必修
課程教師︰薛克民
開課學院：電資學院、工學院、管理學院
開課系所︰電機系、資工系、材料系、資管系
考試日期（年月日）︰99/11/01
考試時限（分鐘）：25分鐘
是否需發放獎勵金：是
（如未明確表示，則不予發放）
試題 :
˙每題十分
˙請詳述計算過程，無計算過程的答案不予計分
1. Peter moves east from his home with velocity 10 km/min. There is a library
lies north of Peter's home with distance 8 km. Peter, the library and his
home form an angle θ. How fast is the angle θ changing when the distance
between Peter and his home is x ？
Library
▕╲
▕θ╲
8 km▕ ╲
▕ ╲
▕ x ╲
Home ￣￣￣￣￣ Peter
┐ ┐ ┐ ┐
2. Prove that: for all real numbers x and y,│cos(x) - cos(y)│≦│x - y│
┘ ┘ ┘ ┘
3. Find all local extreme o the function:
╭
│ x^4 + 4x^3 + 8x^2 + 8x + 5 if x ≦ 0
f(x) = ┤
│ 2x^3 - 9x^2 + 12x - 3 if x ＞ 0
╰
What is the absolute minimum of f in [-2,2] ？
dF
4. Suppose that: ── = 3 - F(x) and F(0) = 1
dx
(a) Find F(x)
(b) Use linear approximation to approximate the value F(0.01) and F(-0.01)
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