109年 - 109 台灣綜合大學系統_學士班轉學考聯合招生考試：微積分C#124121

(a) (5 points)

(b) (5 points)

2. (10 points) Find the tangent line to the curve defined by the equation 4^x/y = x - y when x = 0.

3. (10 points) Compute the definite integral
dydx.

4. (10 points) Find the arc length of the polar curver =θwith 0 ≤ 0 ≤ π.

5. (10 points) Find the volume of the solid of revolution about the x-axis bounded by the curve
f(x) =with x ∈ [1,0∞).

6. (10 points) Consider the power series Find its maximal domain of convergence.

7. (10 points) Recall that the hyperbolic tangent function
tanh x=,x∈R
is one to one and has the inverse function f(x) = tanh^-1x. Find the Maclaurin series off(x) = tanh^-1x.

8. (10 points) Find the absolute maximum of f(x, y) = e^-xy on the region x² + 4y² ≤ 1.

9. (10 points) Evaluate the line integral

y³dx – x³dy, where C is the circle x² + y² = 8 with positive orientation.

10. (10 points) Let F(x, y, z) = xi + yj – 2zk. Suppose that the closed surface S is defined by the paraboloid y = x² + z² with 0 ≤ y ≤ 1. Find the flux of F outward across S.