High School

In Exercise 76, find an equation of the tangent line to the graph of the function \( f \) through the point \((x_0, y_0)\) not on the graph. To find the point of tangency \((x, y)\) on the graph of \( f \), solve the equation.

76.
\[ f(x) = \frac{2}{x} \]
\( (x_0, y_0) = (5, 0) \)

Answer :

The most appropriate choice for tangent to a curve will be given by- 2x + 25y = 10 is the required equation of tangent.

What is tangent to a curve?
In geometry, a tangent line is a straight line that most closely resembles (or "clings to") a curve at a particular point. It could be thought of as the limiting role of straight lines that pass through the specified point and a nearby curve point as the second point gets closer to the first. If two curves share the same tangent at a given point, they are said to be tangent. Similar definitions apply to the tangent plane to an exterior at a point and two surfaces that are tangent at a point.



Equation of tangent to a curve at a point [tex](x_1, y_1)[/tex] is given by

[tex]y-y_1 = \frac{dy}{dx}|_{(x_1, y_1)}(x - x_1)[/tex]

Here,
f(x) = [tex]\frac{2}{x}[/tex]
f'(x) = [tex]\frac{d}{dx} (\frac{2}{x})[/tex]

= [tex]2(-1)x^{-1-1}\\-2x^{-2}[/tex]

[tex]f'(x)|_{(5, 0)} = -\frac{2}{5^2}[/tex]

[tex]= -\frac{2}{25}[/tex]

Equation of tangent

[tex]y - 0 = -\frac{2}{25}(x - 5)\\\\25y = -2x +10\\2x + 25y = 10[/tex]

To learn more about tangent to a curve
brainly.com/question/28199103
#SPJ13