101年 - 101 國立嘉義大學_碩士班(甲組)招生考試試題_應用數學系：微積分#146307

(a) \( \lim_{x \to 0} \frac{\ln((1+x)^2)}{\tan(3x)} \). (5%)

(b)\( \lim_{x \to 0} \frac{\cos(x) - 1}{x^3 + 2x} \) . (5%)

(a)\( f(x) = \frac{1}{2^x} + x^{\frac{1}{5}} \) . (5%)

(b)\( g(x) = \sqrt{\ln(x^2 + 1)} + \cos(x^3) \) . (5%)

3. Show that the equation \( 4px^3 + 6qx^2 + 2rx = p + 2q + r \) has at leastone real root between 0 and 1,\( p, q, r \) where are realconstants. (10%)

4. Show that the limit \( \lim_{(x, y, z) \to (0, 0, 0)} \frac{xyz}{x^3 + y^3 + z^3} \)does not exist. (10%)

5. Find the local extrema of the function \( f(x, y) = y^4 + x^3 - 4y - 3x - 10 \)(10 %)

6. With the definition \( \sinh(x) = \frac{e^{x} - e^{-x}}{2} \) ,we know that exists.Prove that \( \frac{d}{dx} \sinh^{-1}(x) = \frac{1}{\sqrt{x^2 + 1}} \)(10%)

(a)\( f(x) = (\sin(x))^2 \) (5%)

(b)\( f(x) = \cos(\sqrt{x}) \) (5%)

(a)\( \int x^2 \ln(x) dx \) . (5%)

(b) \( \int \frac{1}{x^2 - 5x + 6} dx \). (5%)

(a)\( \sum_{n=1}^{\infty} \frac{n^n}{n!} \) (5%)

(b)\( \sum_{n=1}^{\infty} \frac{e^{2n}}{n^n} \) (5%)

10. Suppose that the collection of pointsis \( \{ (x_1, y_1), (x_2, y_2), \cdots, (x_n, y_n) \} \) is given. Find a and b such that they yieldthe minimum of \( \sum_{i=1}^{n} [ax_{i} + b - y_{i}]^{2} \)(10%)