[試題] 102上 李聰成 微積分乙上 期中考
課程名稱︰微積分乙上
課程性質︰必修
課程教師︰李聰成
開課學院：理學院
開課系所︰數學系
考試日期（年月日）︰2013年11月7號
考試時限（分鐘）：160分鐘
是否需發放獎勵金：是
（如未明確表示，則不予發放）
試題 :
1. Evaluate the limit.
x^1/3 －1
lim ───────
x→1 √x －1
2.Evaluate the limit.
╴╴╴╴
lim (√x^2+x+1 +X)
x→-∞
3. Find a function f and a number a such that
sin(3+x)^2－sin9
lim ──────── ＝f'(a)
x→0 x
and evaluate the limit.
4. Sketch the graph oh the given function, making use of any suitable
information you can obtain from the function and its first and second
derivatives.
x
f(x)＝───
x^2＋1
5. Show that the equation x^3＋3x^2＋6x＋1＝0 has exactly(恰好) one root.
6. Find an equation of the tangent line(切線) to the curve
x^3＋y^3＝6xy
at the point (3,3).
7. Find the values of m and b such that f is differentiable on R, where
╭ x^2, if x<＝2 ;
f(x)=│
╰ mx＋b, if x>2 .
8. Find all critical(臨界) points and the absolute maximum and absolute
minimum values of
f(x)＝│x^2＋4x－12│
on the interval [－4,4].
9. A man walks along a straight path at a speed of 1.5 m/s. A searchlight
is located on the ground 6 m from the path and is kept focused on the man.
At waht rate is the searchlight rotating when the man is 8 m from the point
on the path closest to the searchlight?
10. Find the dimensions(長,寬) of the rectangle of largest area(面積) that can
be inscribed(內接) in a circle of radius r.
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