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研究所、轉學考(插大)-微積分
> 103年 - 103 國立暨南國際大學_轉學生入學考試試題_資工系二、土木系二、應化系二、電機系二、應光系二：微積分#123440
103年 - 103 國立暨南國際大學_轉學生入學考試試題_資工系二、土木系二、應化系二、電機系二、應光系二：微積分#123440
科目：
研究所、轉學考(插大)-微積分 |
年份：
103年 |
選擇題數：
0 |
申論題數：
12
試卷資訊
所屬科目：
研究所、轉學考(插大)-微積分
選擇題 (0)
申論題 (12)
(a) (5%) Sketch the graph of the function f(x).
(b) (5%) According to the Definition A, show that f '(x) =
.
(c) (5%) Find the tangent line that passes through the point (2, 1/2) with slope f'(2).
2. (10%) Does there exist a differentiable function f that satisfies f(0) = 2, f(2) = 5, and f'(x) ≤ 1 on (0, 2)? If not, why not?
(a) (10%) Then, the function F defined on [a, b] by F(x) =
f(t)dt is continuous on [a, b] and differentiable on (a, b). Prove that F'(x) = f(x) for all x in (a, b). Hint: F(x+h)-F(x)
f(x) × h if h is small enough. (圖示證明(pictorial proof)即可)
(b) (5%) Let F(x) =
(t-sin
2
t)dt, find F'(x).
4. (10%) Evaluate the integral
sin x dx
5. (10%) Show that the improper integral
diverges.
6. (10%) Find the Maclaurin series for
and its radius of convergence.
7. (10%) Let g(x, y) =
. Show that
g(x, y)does not exist.
8. (10%) Find the linear approximation of the function f(x, y) =
at (2, 1) and use it to approximate f(1.95,1.08).
9. (10%) Find the directional derivative of the function
f(x, y, z) = x cos y sin z at the point (1, π, π/4) in the direction of the vector 2i-j+4k.
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f(t)dt is continuous on [a, b] and differentiable on (a, b). Prove that F'(x) = f(x) for all x in (a, b). Hint: F(x+h)-F(x) 
f(x) × h if h is small enough. (圖示證明(pictorial proof)即可)
(t-sin2t)dt, find F'(x).
sin x dx
diverges.
and its radius of convergence.
. Show that
g(x, y)does not exist.
at (2, 1) and use it to approximate f(1.95,1.08).