113年 - 113 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#144161

1. \[ \int \frac{x^2+1}{(x-1)(x-2)^2} dx=? \]

2.\[ \frac{d}{dx} \left\{ \log_2 \sqrt{\frac{(x^2+1)(x^3+3)}{x^4+5}} \right\} = ? \]

3. \[ \lim_{x\to 0} \frac{\sqrt{1+2x}-\sqrt[3]{1-3x}}{x+x^2} = ? \]

4. \[ \lim_{y\to 0} \frac{\cos(y)-1}{y} = ? \]

5. \[ \int_{0}^{2} \{[x^2] - [x]^2\} dx = ? \text{ where } [ \cdot ] \text{ is the greatest integer function.} \]

6.\[ \lim_{x \to 0^+} x^x = ? \]

7. Find a power series for \( f(x) = \tan^{-1}(x) \) centered at 0.

8. Find the arc length of the graph of r = 1 + cosθ, 0 ≤ θ ≤ 2π.

9. Let \(f(x) = e^{|x-1|}\). Is \(f\) differentiable at \(x=1\)?

10. Let \(f(x) = \sqrt{x + \sqrt{x + \sqrt{x}}}\). Then \(f'(x) = ?\)