所屬科目：研究所、轉學考(插大)-微積分
(a) (5 points) Evaluate the limit
(b) (5 points) Find the horizontal asymptote of the graph y =for x >0 if it exists.
2. (10 points) Define f(x) = for x ≠2. Give a value of f(2) x-2such that f is continuous at 2.
3. (10 points) Let f(x) =
be the invesse function of f. A curve C satisfies the equation 2x2y + xy2 = 8. Find a point (a, b) such that
(a) (5 points) Find the Taylor expansion for f about x = 0. (In the fomm with a general formula for ak)
5. (10 points) Let u(x,y) be a differentiable function with (4,1) = 2. dy Suppose that x = st and y= and define h(s, t) = u(x(s, t),y(s, t). At the point (2,2), find a unit vector u in the st-plane such that h increases most rapidly in the direction. (4, 1) = 1 and
6. (10 points) Suppose that f(π) = 4 and [f(x) + f"(x)] sin x dx = 5. Find f(0).
7. (10 points) Find the arc length of the part of the curve r = 1 + sin θ which is inside the curve r = 3 sin θ (the solid curve in the figure).
9. (10 point) Evaluate the double integral where D is the trapezoid in the first quadrant with vertices (2,0), (4,0), (0, 4) and (0,8).
10. (10 points) Let F(x, y, z)= k. Compute the surface integral (using the outward pointing normal), when S is the surface x2 + y2 + z2 = 225.
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for
x >0 if it exists.
for x ≠2. Give a value of f(2)
x-2such that f is continuous at 2.
with a general formula for ak)
(4,1) = 2.
dy
Suppose that x = st and y=
and define h(s, t) = u(x(s, t),y(s, t). At the point
(2,2), find a unit vector u in the st-plane such that h increases most rapidly in the
direction.
(4, 1) = 1 and
[f(x) + f"(x)] sin x dx = 5. Find f(0).
where D is the trapezoid in the first quadrant with vertices (2,0), (4,0), (0, 4) and
(0,8). 
k. Compute
the surface integral
(using the outward pointing normal), when S is the surface x2 + y2 + z2 = 225.