111年 - 111 國立臺北大學_碩士班一般入學考試試題_財政學系：微積分#146658

(a) \( \lim_{x \to \infty} \frac{3x^5}{e^{7x}} \)

(b)\( \lim_{x \to \infty} \frac{(\ln x)^{12}}{x^6} \)

(a) log₁₀x,

(b) \( \frac{x}{\ln x} \)

(a) x = y² + 1, x = 0, y = -1, y = 2

(b)\( y = \frac{1}{2}x^{3} + 2 \), \( y = x + 1 \), \( x = 0 \), \( x = 2 \)

4. (10%) Find the Taylor Series for f(x) = 1/x² about x =-1.

5. (10%) Find dy/dx by implicit differentiation: x sin y = y.cosx.

(a) Compute the function d(m).

(b) Find a value of m at which d(m) is NOT differentiable.

7.(10%) Suppose that n is an arbitrary positive integer. Use the integration by parts formula to show that y(n) = (n - 1)!, where
\( \gamma(n) = \int_{0}^{\infty} x^{n-1} e^{-x} dx \)

8.(10%) Find dy/dx where \( y = \int_{x}^{x+3} z(5 - z) dz \)

9.(10%) Evaluate the integral: \( \int_{0}^{2} \int_{x}^{2} x \sqrt{1 + y^{3}} \, dy \, dx \)