94年 - 94 中原大學轉學生招生考試_理工、化學群組二年級：微積分#124618

1. Let x ＞ 0. =_________

2. =_________

3. Find the slope of the line tangent to the graph y² + x² y = 3x² at the point (2,2) :_________

4. Find =_________

5. Find the point on the graph of y = that is nearest the point (1,0):_________

6. Find the interval on which the function f(x) = x³ - 3x² - 9x + 12 is decreasing:_________

7. =_________

8.=_________

9.=_________

10. Find the area of the region bounded by the graph of f(x) = 2 + x - x² and the x-axis:_________ .

11. Find the value of the integral =_________

12. Find +ydA where Q={ (x, y) : -1 ≤ x ≤ 2, 1≤ y ≤3 } : _________.

13. Let f(x, y) = , x = 1 + t²  and y = 1 - t• =_________

14. Let f(x, y) = cos(x² + y), x = and  y = st =_________

15. Find :_________.

16. Let f(x, y, z) = x + 2xy - 3z². Find the direction vector along which the directional derivative at the point (1, 0, 4) reaches its maximum. =_________

(a) Plot the domain of integration of (*).

(b) Evaluate (*).

2. (a) Prove that for r ∈ R.

(b) Evaluate