Francisca Rodden

2021-12-13

For the given function, (a) find the slope of the tangent line to the graph at the given point; (b) find the equation of the tangent line.
$h\left(x\right)=\frac{7}{x}atx=6$
(a) The slope of the tangent line at $x=$6 is ____.

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Step 1
The function is given b y
$h\left(x\right)=\frac{7}{x}atx=6$ To evaluate:
(a) The slope of the tangent line to the graph of the function at $x=6$
(b) The equation of the tangent line to the graph of the function at $x=6$
(a) Evaluation of the slope The slope of the tangent line to the graph of the function can be evaluat6ed as following
$m=\frac{dh}{dx}{\mid }_{x=6}$
$=\frac{d}{dx}\left(\frac{7}{x}\right){\mid }_{x=6}$
$=-\frac{7}{{x}^{2}}{\mid }_{x=6}$
$=-\frac{7}{{6}^{2}}$
$=-\frac{7}{36}$
Hence, the slope of the tangent line at $x=6is-736$

Ella Williams

(b) Evaluation of the equation of the tangent line
The equation of the tangent line can be evaluated as following
$y-h\left(6\right)=m\left(x-6\right)$
$⇒y-\frac{7}{6}=-\frac{7}{36}\left(x-6\right)$
$⇒y-\frac{7}{6}=-\frac{7}{3x+\frac{7}{6}}$
$⇒y=-\frac{7}{36}x+\frac{7}{6}+\frac{7}{6}$
$⇒y=-\frac{7}{36}x+\frac{7}{3}$
Hence, the equation of the tangent line at $x=6$ is
$y=-\frac{7}{36}x+\frac{7}{3}$

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