104年 - 104 國立臺北大學_學士班曁進修學士班轉學生招生考試試題_資訊工程學士、通訊工程學系學士班2年級：微積分#147542

(a) with \( y \) as the variable of the integration, i.e write \( S = \int_{1}^{1} \dots dy \) (5%)

(b) with \( x \) as the variable of the integration, i.e write \( S = \int_{1}^{1} \dots dx \) (5%)

2.(20%) The Cobb-Douglas Model for the production function is $P = bL^q K^{1-q}$, where b and $\alpha$ are positive constants and $\alpha < 1$. We want to maximize the production P with the constraint that mL + nK = q, where m, n and q are positive constants. Show that the maximum production occurs when $$L = \frac{\alpha q}{m}, \quad K = \frac{(1 - \alpha)q}{n}$$

(a) Find the power series for \( f(x) \). (10%)

(b) Use part (a) to derive \( f^{(10)}(0) \). (10%)

(a) \( \int_0^9\frac{1}{\sqrt{x - 1}} dx \)

(b) \( \int \frac{\sqrt{x^2 - 9}}{x^2} dx \)

(c) \( \int \frac{x^2 - x + 6}{x^3 + 3x} dx \)

(d) \( \int_1^2 (\ln x)^2 dx \)

5. (20%)Please sketch the curve \( y = \frac{x^3 + 4}{x^2} \). (You must describe the domain, intercepts, symmetry, asymptotes, interval of increasing or decreasing, local maximum and minimum values, and concavity and points of inflection.)

6. (5%) Prove \( \lim_{x \to 2} \frac{x^2 + x - 6}{x - 2} = 5 \) using the \( \varepsilon, \delta \) definition of a limit.

7. (5%) Find \( \frac{dy}{dx} \) if \( y\cos x = 1 + \sin (xy) \).