| 申論題 | 1. |
| 申論題 | 2. |
| 申論題 | 3. |
| 申論題 | 4. |
| 申論題 | 5. |
| 申論題 | 6. |
| 申論題 | 7. |
| 申論題 | 8. |
| 申論題 | 9. |
| 申論題 | 10.有一個半徑1米之半個球,該半球在距球心 |
| 申論題 | 已知在地表附近之重力加速度為9.8 【題組】11.若向半米外之牆壁丟擲,打到牆上之彈著點最高可到高? |
| 申論題 | 【題組】12.若站在大操場之定點向各方向丢擲,石頭可能飛經處(不考慮石落地後之彈跳)所構成的立體之體積何? |
| 申論題 | 1. |
| 申論題 | 2. |
| 申論題 | 3.求tan-1x2對x=0的泰勒展開式=____3____(必須寫出一般式) |
| 申論題 | 4. |
| 申論題 | 5. |
| 申論題 | 6.x3十2y2=3xyz在(1,1,1)點的切面方程式為_______ |
| 申論題 | 7.(1,0)點到 |
| 申論題 | 8. |
| 申論題 | 9. |
| 申論題 | 10. |
| 申論題 | 11. |
| 申論題 | 1. |
| 申論題 | 2. |
| 申論題 | 3. |
| 申論題 | 4. |
| 申論題 | 5.設x,y平面中一曲線Γ,可用極座標方程r=sin2θ來定義,則限制在Γ上x2+y2的最大值=___⑤____ |
| 申論題 | 6.設f(x,y)=y2+λx3+x，其中λ為參數,若已知其函數圖形中有唯一的鞍點(a,b,f(a,b)),則所有可能的λ構成的集合為____⑥____, 且a+6=____⑦____(寫成λ的表式) 【題組】6. |
| 申論題 | 【題組】7. |
| 申論題 | 8. |
| 申論題 | 8. 【題組】 9. |
| 申論題 | 10. |
| 申論題 | 1. Evaluate the following limits : 【題組】 (a) |
| 申論題 | 【題組】 (b) |
| 申論題 | 2. Let C be the plane curve of the equation :
x2+xy-2y3=0. 【題組】 (a) Find the equation of the tangent line to C at the point (1,1). |
| 申論題 | 【題組】 (b) Find |
| 申論題 | 3. Let 【題組】(a) Is f continuous at x = 0? Prove your answer. |
| 申論題 | 【題組】(b) Is f differentiable at x = 0 ? Prove your answer. |
| 申論題 | 4. Let a and b be real numbers with a ＜ b. Let f be a function continuous on the closed interval [a,b] and differentiable on the open interval (a, b). If f'(x) =0 for all a ＜ x＜ b, prove that f is a constant function on [a,b]. |
| 申論題 | 5. Evaluate the following integral : |
| 申論題 | 6. Find the length of the curve y = |
| 申論題 | 7. Let f(x) = |
| 申論題 | 8. Use the definition to prove : |
| 申論題 | 9. Let 【題組】 (a) Prove that f is one-to-one. |
| 申論題 | 【題組】(b) Let g be the inverse of f. Find g'(-3). |
| 申論題 | 【題組】 (c) Does the limit of f at ∞ exist ? Prove your answer. |
| 申論題 | 1. Let f(x,y) = sin(xy). 【題組】 (a) Find the gradient |
| 申論題 | 【題組】 (b) Find the equation of the tangent plane to the graph of f at the point
|
| 申論題 | 【題組】 (c) Let |
| 申論題 | 2. Let f(x, y) = -x3 + 4xy - 2y2 + 1. Find the relative extrema of f, and find saddle points of the graph of f. |
| 申論題 | 3. Let Ω = |
| 申論題 | 4. Evaluate the iterated integral: |
| 申論題 | 5. Let C be the plane curve of the equation x4 + 4y4 = 1. Find the points on C which are closest to the origin. |
| 申論題 | 6. Let f(x, y) = 【題組】 (a) Is f continuous at (0,0) ? Prove your answer. |
| 申論題 | 【題組】 (b) Evaluate |
| 申論題 | 【題組】 (c) Evaluate |
| 申論題 | 7. Determine the convergence or divergence of the series 【題組】 (a) |
| 申論題 | 【題組】 (b) |
| 申論題 | 8. Find the interval of convergence of the power series : |
米處之密度為
,則該半球之質量中心與球心之距離為___(10)____。
,某甲丟擲石頭,脫手時石頭的初速為
，問
上函數f(x,y)=sinx siny 的所有鞍點(saddle point)為________
的最短距離=_________
at the point (1,1).
for 0 ≤ x ≤ 1.
sinx, 0 ≤ x ≤ 
, and let Ω be the region bounded
by the graph of f, the x-axis and the vertical line x =
. Find the volume of
the solid generated by revolving Ω about the x-axis.
for all real numbers x.
of f at the point 
.
.
= (-3,4) be a vector. Find the directional derivative of f at the
point 
in the direction of 
. 
. Evaluate the double
integral:
for all (x,y).
.
