※ [本文轉錄自 NTU-Exam 看板]
作者: diep (隱藏人物) 看板: NTU-Exam
標題: [試題] 94上 張秀瑜 微積分乙 期中
時間: Sun Nov 13 16:11:17 2005
課程名稱︰微積分乙上
課程性質︰共同必修
課程教師︰張秀瑜
開課系所︰管院、地理、經濟
考試時間︰94/11/10
試題 :
1. ___
(a) Find lim√x+1 -2/(x-3).
x→3
(b) Prove that lim x^2cos(1/x^2) = 0.
x→0
(c) Find all values of a such that f is continuous on R, where
{ x^2, if x > a;
f(x) = {
{ x+1, if x≦ a.
(d) Give a short explanation of why the following approximation is valid.
____
√4.02 ≒ 2 + 1/4(0.02)
2.Show that the function g(x)= x∣x∣has an inflection point at (0,0), but
g"(0) does not exist.
dy 3 _
3.Find —. (a) y = 2csc (√x) (b) tan(x/y) = x+y
dx
4.If y = f(u) and u = g(x), where f and g are third differentiable function,
d^2y d^2y (du)2 dy d^2u d^3y
— = — (－) + — — . Find a formula for — similar to
dx^2 du^2 (dx) du dx^2 dx^3
the one given above.
5.Prove that if a > 0 and n is any positive integer, then the polynomial
function
2n+1
p(x) = x + ax + b
cannot have two real roots.
6.Find an equation of the line through the point (3,5) that cuts the least
area from the first quadrant.
7.Sketch the graph of y = x^3 －1/x^3 + 1
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