課程名稱︰微積分乙下
課程性質︰必修
課程教師︰李聰成
開課學院：
開課系所︰
考試日期（年月日）︰2016年6月23日
考試時限（分鐘）：160分鐘
試題 :
請將每一步驟表達清楚，不可以只寫答案。
1. Find the local maximum and minimum values and saddle point(s) (鞍點)
of the function
f(x,y)=2x^4-xy^2+2y^2
2. Evaluate the limit or show that it does not exist.
x^2-y^2
lim ───────
(x,y)→(0,0) x^2+y^2
3. The derivative (導數) of f(x,y) at (1,2) in the direction (方向) of i+j is
2√2 and in the direction of -2j is -3. What is the derivative of f in the
direction -i-2j. Give reasons for your answer.
4. Find the volume (體積) of the solid that lies under the paraboloid (拋物面)
z=x^2+y^2, above the xy-plane, and inside the cylinder x^2+y^2=2x.
5. Evaluate the double integral.
1 1
∫ ∫ cos(x^3) dxdy
0 √y
6. The plane x+y+z=12 intersects the paraboloid (拋物面) z=x^2+y^2 in an
ellipse. Find the points on the ellipse that are
(a) closest to (最接近) and
(b) farthest from (離最遠) the origin.
7. Find an equation (方程式) for the tangent plane (切平面) and parametric (參
數) equations for the normal line (法線) to the surface with the equation
x^2+y^2+z^2=6
at the point (-1,2,1).
8. Let R be the region (區域) bounded by the graphs of y=√x, x=0, and y=3.
Evaluate
∫ ∫ (2xy^2+2ycosx) dA.
R
9. Use polar coordinates to evaluate the integral.
2 √(8-x^2) x 2
∫ ∫ (──) dydx
0 x y
10. Evaluate the integral
e^(y-4x)
∫ ∫ ──── dA,
R y+2x
where R is the region (區域) bounded by
y=4x+2, y=4x+5, y=3-2x, and y=1-2x.