推 hsnuyi :differentials拼錯了喔 11/05 00:14
課程名稱︰微積分甲上
課程性質︰必修
課程教師︰蔡雅如
開課學院：電資學院 工學院
開課系所︰電機系 材料工程系 資工系 資管系
考試日期（年月日）︰2010/11/1
考試時限（分鐘）：45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 -1
1.(a) Find the tangent line of y = sinx + sin x + cosx + cos x + tanx +
-1 π
tan x at the point of (0,1+---)。
2
dy
(b) Find -- if e^y = ln(x^2+y^2) 。
dx
2.(a) A number c is called a "fixed point" of a funtion f if f(c) = c. Prove
that if f'(x)>1 for all real number x, then f has at most one fixed point.
(b) The Extreme Value Theorem states that if f is continuous on a closed
interval [a,b], then f attains an absolute maximum values f(c) for some
c in [a,b]. Show that the condition [a,b] cam not be replaced by (a,b).
That is, give a funtion f that is continuous on (a,b) but has no absolute
maximum value.
(Write down the formula for f.)
3.(a) A particle moves along the curve y = √(1+x^3) . As it reaches the point
(2,3), the y-coordinate is increasing at a rate of 4 cm/s. How fast is the
x-coordinate of the point changing at that instant?
(b) Use a linear approximation (or differentails) to estimate ln1.02 and
-1
tan 1.02, respectively.
--
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