课程名称︰微积分乙上
课程性质︰必带
课程教师︰张树城
开课学院：生科院
开课系所︰生科系.生技系
考试日期（年月日）︰2016/11/10
考试时限（分钟）：110
试题 :
I.(25%) Compute the following:
(i)
lim (√( x + √x) - √x )
x→∞
(ii) d^2 y /dx^2, where x = t - t^2, y = t - t^3.
(iii)d^2 y /dx^2, where 2x^3 - 3y^2 = 8.
(iv)d^101 y /dx^101, where y = sin x.
(v)f'(0), where cos x <= f(x) <= x^2 +cos x, |x| <= 1/8.
II.(15%) Let x^2 sin x, x ≠ 0
f(x) =
0 , x = 0
(i) Find f'(x) for x≠0.
(ii) Find f'(0).
(iii)Is f'(x) continuous at x=0?
III.(15%) Prove or disprove the followings:
(i) If |f(x)| is continuous at c, then f(x) is continuous at c.
(ii)If f(x) is differentable at c, then f(x) is continuous at c.
(iii)There is exact one real root for x^3 + 3x + 1 = 0.
IV.(15%) Let the function f(x) = x^3 - 2x^2 + x + 1
(i)Indicate where f(x) increases and decreases.
(ii)Indicate the critical points and points of inflection and the concavity.
(iii)Sketch the graph.
V.(10%) Suppose that f(x) = x^(2/3), x ε[-2,3]. Find the maximum and
minimum values of f(x) and indicate where they are taken on.
VI.(10%) Define the function f(x) = x^(1/3).
(i)Find the tangent line to the curve y = f(x) at (1000,10).
(ii) Use the linearization method (i) to esimate 三次根号(1001).
VII.(10%) Let f(x), g(x): [a,b] →R be differentable function with
g'(x)≠0. Define G(x) by
G(x) = f(x) - f(a) - [f(b)-f(a)]/[g(b)-g(a)] (g(x)-g(a))
Show that
(i) G(a) = G(b)
(ii)there exist c ε (a,b) such that
f'(c) f(b) - f(a)
──── = ───────
g'(c) g(b) - g(a)