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107 嘉義高中

10.
設\(a,b\)是實數,使得\(\displaystyle \lim_{x \to \infty}\frac{\sqrt{ax+b}-2}{x}=1\),則\((a,b)=\)   

填充10... 題目可能有打錯,應該不是 \(x \to \infty\),而是 \(x\to 0\)。

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13.
心臟線\(r=1+cos\theta\)的長度是   
(提示:\(r=f(\theta)\))所表示的曲線長度為\(\displaystyle L=\int_{\alpha}^{\beta}\sqrt{(f(\theta))^2+(f'(\theta))^2}d \theta\)

填充13. (題目下方有附極坐標的曲線長度積分公式,直接使用即可。)

所求=\(\displaystyle \int_0^{2\pi} \sqrt{\left(1+\cos\theta\right)^2+\left(-\sin\theta\right)^2}d\theta\)

  =\(\displaystyle \int_0^{2\pi} \sqrt{1+2\cos\theta+\cos^2\theta+\sin^2\theta}d\theta\)

  =\(\displaystyle \int_0^{2\pi} \sqrt{2+2\cos\theta}d\theta\)

  =\(\displaystyle \sqrt{2} \int_0^{2\pi} \sqrt{1+\cos\theta}d\theta\)

  =\(\displaystyle \sqrt{2} \int_0^{2\pi} \sqrt{1+\left(2\cos^2\frac{\theta}{2}-1\right)}d\theta\)

  =\(\displaystyle \sqrt{2} \int_0^{2\pi} \sqrt{2\cos^2\frac{\theta}{2}}d\theta\)

  =\(\displaystyle 2 \int_0^{2\pi} \left|\cos\frac{\theta}{2}\right|d\theta\)

  =\(\displaystyle 2\cdot2 \int_0^{\pi} \cos\frac{\theta}{2} d\theta\)

  =\(\displaystyle 2\cdot2\cdot2 \int_0^{\pi} \cos\frac{\theta}{2} d\frac{\theta}{2}\)

  =\(\displaystyle 2\cdot2\cdot2 \sin\frac{\theta}{2}\Bigg|_0^{\pi}\)

  =\(\displaystyle 2\cdot2\cdot2 \left(\sin\frac{\pi}{2}-\sin0\right)\)

  =\(\displaystyle 2\cdot2\cdot2 \left(1-0\right)\)

  =\(\displaystyle 8\)

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