※ [本文轉錄自 NTU-Exam 看板]
作者: cawaiik (Dance or Die) 看板: NTU-Exam
標題: [試題] 張秀瑜 微積分乙期中考
時間: Fri Nov 19 14:23:56 2004
課程名稱︰微積分乙
課程性質︰管院必修
課程教師︰張秀瑜
開課系所︰管院
考試時間︰2004.11.9
試題 :
1. Evaluate the limit , if it exist.
(1) lim 1－√1－x*2 / x
x→0
(2) lim x*3 / tan*3 2x
x→0
2. Draw the graph of f(x)= 4－│x－2│
(1) Is f continuous at x=2 ? (Explain it by definition)
(2) Is f differentiable at x=2 ? (Explain it by definition)
3. Find y' , if y = sin ( tan √ sinx )
4. Show that x*101 + x*51 + x － 2 ＝ 0 has exactly one real
root .
5. Show that the curves 2x*2 + y*2 = 3 and x = y*2 are
orthogonal.
6. Let g(x) = f(x*2) , where f is twice differentiable for all x ,
f'(x) > 0 for all x ≠ 0 , and f is concave downward on
(－∞ , 0 ) and concave upward on ( 0 , ∞ ) .
(1) At what numbers does g have an extreme value ?
(2) Discuss the concavity of g .
7. State the Mean Value Theorem and use it to prove that
√1 + x < 1 + 1/2 x if x > 0 .
8. Sketch the graph of y = x / x*2 － 9
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