96年 - 96 國立暨南國際大學_轉學生入學考試試題_資工系二:微積分#124712

1. (15%) Sketch the graph of the function f(x) =.

2. (15%) Find .
[Hint: limit of Riemann sums]

3. (10%) Calculate ∫e^ˣcosx dx. Hint: cosxdx=dsinx & integration by parts

4. (10%) Find the area enclosed by the ellipse (x²/a²)+(y²/b²)=1.
(Hint: cos²x=(1+cos2x)/2.)

5. (10%) Let
Show that g(x,y) does not exist.

(a) (5%) Find given
that x(t) = t and y(t) = t². (These functions parameterize the parabola y=x²)

(b) (5%) Find given
that x(t) =1/4(t+4) and y(t) = t. (These functions parametrize the line y=4x-4)

(c) (5%) Compute the directional derivative of f at (2,4) in the direction of i + 4j.

(d) (5%) Notice that i + 4j is a direction vector for the line y=4x-4 and this line is tangent to the parabola y=x² at (2,4). Explain why the computations in ( a ), ( b ), and( c ) yield different values.

7. (10%) Show that the series converges on [-1,1).

8. (10%) Maximize f(x,y,z) = xyz subject to the side condition
x³ + y³+ z³ = 1, with x≥0, y≥0, z≥0.