Tangent Line

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Cegep1

Mathematics

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162 words

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2 minutes

Line that touches a curve at exactly one point

The slope of the tangent line at $(a,f(a))$ to the curve given by $y=f(x)$ is

$$\underset{x\to a}{lim}\frac{f(x)-f(a)}{x-a}$$

[!example]- Find the equation of the tangent to the curve $y=x^2+2x$ at $x=1$.
The equation of the tangent is

$$y-y_o=m(x-x_o)$$

where

$$\begin{array}{rl}m& =\underset{x\to a}{lim}\frac{f(x)-f(a)}{x-a}\\ & =\underset{x\to 1}{lim}\frac{x^2+2x-3}{x-1}\\ & =\underset{x\to 1}{lim}\frac{(x-1)(x+3)}{x-1}\\ & =\underset{x\to 1}{lim}(x+3)\\ & =4\end{array}$$

Since the tangent shares the point $(1,f(1))=(1,3)$ with $f$,

$$\begin{array}{rl}y-3& =4(x-1)\\ y& =4x-4+3\\ y& =4x-1\end{array}$$

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Ajitani Hifumi

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6eaef0a-Vault update: 2024-10-01T18:02:12