115年 - 115 國立嘉義大學_轉學生招生考試試題_理工學院：微積分#143894

1. Find the limit \( \lim_{x\to 0} \frac{\sin 4x}{x} \) .

2. Find the extrema of f(x) = 2x − \(3x^{\frac{2}{3}}\) on the interval [−1, 3].

3. Find the tangent line to the graph of x²(x² + x²)=y² at the point\( \left( \frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2} \right) \).

4. Evaluate the integral \( \int_{0}^{\frac{\pi}{4}} \sqrt{1+\cos 4x} dx \) .

5. Find the interval of convergence of \(\sum_{n=0}^{\infty} \frac{(-1)^n (x+1)^n}{2^n} \) .

6. Find \( \int \frac{1}{x \ln 3x} dx \).

7. Find the derivative of y = \( \ln \sqrt{\frac{x+1}{x-1}} \).

8. Let f(x) = \( \frac{[x]}{2} \) + x, where [x] denotes the greatest integer lessthan or equal to x. To determine where the function isdiscontinuous.

9. Find \( \int_{-\infty}^{0} \frac{1}{(1-2x)^{3/2}} dx. \) .