题目

Problem

A curve has equation $y = 2x + \frac{12}{x^2}$ .

Find the equation of the tangent to the curve at the point $(-2, -1)$ . Give your answer in the form $y = mx + c$ .

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题目中文翻译

一条曲线的方程为 $y = 2x + \frac{12}{x^2}$ 。

求曲线在点 $(-2, -1)$ 处的切线方程。答案写成 $y = mx + c$ 的形式。

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解答

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先对 $12x^{-2}$ 求导，再把 $x=-2$ 代入。用切点和斜率写成斜截式。

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The derivative is $y'=2-24x^{-3}$ . At $x=-2$ , it is $2-24/(-8)=5$ . The tangent through $(-2,-1)$ is $y+1=5(x+2)$ , or $y=5x+9$ .