114年 - 114 國立臺北大學_日間學士班轉學生招生考試試題_經濟學系日間學士班2、3年級：微積分#147894

(a) Find \( \lim_{x\to 0}\frac{\sin 3x}{x} \).

(b) Find \( \lim_{x\to -\infty}\frac{5^x + 7^x}{2\cdot 5^x - 3\cdot 7^x} \).

(c) Find the range of \( f(x) = \sqrt{x} - \ln x \).

(d) Given \( z = \ln (1 + x^2 y^3) \), find \( \frac{dz}{dx} \).

(e) Given \( y = x^3 + \tan x \), find \( \frac{dx}{dy} \) in terms of \( x \).

(f) Given \( y = \frac{e^x}{x^2 + 1} \), find \( \frac{dy}{dx} \).

(g) Given \( f(x) = x^3 + x \), find \( (f^{-1})'(2) \).

(h) Evaluate \( \int_2^8\frac{1}{x + 1} dx. \)

(i) Evaluate \( \int_0^1 x^2 e^x dx \)

(j) Evaluate \( \int0^a\int0^b (x - a)(x - b)dxdy. \)

(a) Given \( y = \frac{x1^2 + x2^2}{x1 + x2} \), find the total differential \( dy \).

(b) Given \( F(x, y, z) = xye^z + xz^2 - y = 0 \), find \( \frac{\partial z}{\partial x} \) and \( \frac{\partial z}{\partial y} \) at the point \( (1, 1, 0) \).

(c) Approximate \( \ln x \) around \( x = 1 \) using a 3-degree Taylor polynomial.

(d) Find the point \( (x,y) \) on the line \( x + 2y = 6 \) where the function \( f(x,y) = x^{2} + y^{2} \) reaches its maximum value.

(e) Given the function \( f(x) = x^4 - 8x^2 + 16 \), identify the critical points and determine their nature (local minimum, maximum, or saddle point).