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Q15.

Expert-verifiedFound in: Page 922

Book edition
7th Edition

Author(s)
James Stewart, Lothar Redlin, Saleem Watson

Pages
948 pages

ISBN
9781337067508

**Find the equation of the tangent line to the curve at the given point. Graph the curve and tangent line.**

.$y=\frac{x}{x-1},\text{}at\text{}(2,2)$

The equation of the tangent line to the curve at a given point is$y=-x+4$ .

The graph of the curve and tangent line:

The function here given is,

$\begin{array}{l}y=\frac{x}{x-1}\\ po\mathrm{int}=(2,2)\end{array}$

The tangent line to the curve$y=f\left(x\right)$ at the point $P(a,f(a\left)\right)$ is the line through p with slope,

$m=\underset{x\to a}{\mathrm{lim}}\frac{f\left(x\right)-f\left(a\right)}{x-a}$.

provided that this limit exists.

Equation of tangent line:

$y=m(x-a)+f\left(a\right)$

Let $y=\frac{x}{x-1}$, then the slope of the tangent line at $(2,2)$ is,

$\begin{array}{c}m=\underset{x\to 2}{\mathrm{lim}}\frac{f\left(x\right)-f\left(2\right)}{x-2}\\ =\underset{x\to 2}{\mathrm{lim}}\frac{\left(\frac{x}{x-1}\right)-\left(\frac{2}{2-1}\right)}{x-2}\\ =\underset{x\to 2}{\mathrm{lim}}\frac{\frac{x}{x-1}-\frac{2}{1}\cdot \frac{x-1}{x-1}}{x-2}\\ =\underset{x\to 2}{\mathrm{lim}}\frac{\frac{x-2x+2}{x-1}}{x-2}\\ =\underset{x\to 2}{\mathrm{lim}}\frac{-\overline{)(x-2)}}{\overline{)(x-2)}(x-1)}\\ =-\frac{1}{2-1}\\ =-1\end{array}$

The slope of the tangent line is -1.

Here, the slope of the tangent line is$m=-1$ . Equation of the tangent line is written as,

$\begin{array}{l}y=-1(x-2)+2\\ y=-x+2+2\\ y=-x+4\end{array}$ ,

Therefore, the equation of the tangent line is$y=-x+4$ .

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