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To find the linear approximation of a function, understanding how to calculate the derivative is crucial.

The derivative provides information on how a function changes at a specific point and is key to determining the slope of the tangent line.

For the function given, \[ f(x) = \frac{2}{x} \],we apply the quotient rule to compute its derivative. The quotient rule states that for a function expressed as \( \frac{u}{v} \), its derivative is given by \( \frac{u'v - uv'}{v^2} \).

For our function, this means differentiating \( 2 \) with respect to \( x \) and dividing by \( x^2 \), resulting in \( f'(x) = -\frac{2}{x^2} \).

• This derivative \( f'(x) = -\frac{2}{x^2} \) tells us how the function \( f(x) \) changes at any given point \( x \).
• At \( x = 1 \), plugging this value into the derivative gives us \( f'(1) = -2 \).

This specific value \(-2\) represents the slope of the tangent line to \( f(x) \) at \( x = 1 \).

Graphing functions and their linear approximations helps visualize how closely the linear model matches the actual function near the point of interest.

The original function, \( f(x) = \frac{2}{x} \), displays a hyperbolic shape. When graphing, remember:

• The function's curve approaches zero as \( x \) becomes large or very small.
• The function is undefined at \( x = 0 \).Because of this, the graph will show two branches: one in the positive \( x \) region and one in the negative \( x \) region.

On the other hand, the linear approximation \( L(x) = 2 - 2(x-1) \) represents a simple line.
To accurately graph:

• Start by plotting the point of tangency: \( (1, 2) \) where the function and its linear approximation intersect.
• Draw the curve of \( f(x) \) in both positive and negative \( x \) directions.

For visual clarity, use graphing tools that allow you to see both functions on the same axes. This will make it easy to observe how the linear approximation provides a good estimate near \( x = 1 \).

In calculus, the tangent line at any point on a function is the straight line that "touches" the curve at that point.

For \( f(x) = \frac{2}{x} \), the tangent line at \( x = 1 \) provides the best linear estimate of the function near this value.
To find the equation of the tangent line:

• Use the \( x_0 \) value where you want the tangent, \( x_0 = 1 \).
• Using the calculated derivative, \( f'(1) = -2 \), the slope of the tangent is \(-2\).
• The equation of the tangent line can be written using point-slope form: \( y = f(x_0) + f'(x_0)(x-x_0) \).
• With \( f(x_0) = 2, f'(x_0) = -2, \) and \( x_0 = 1 \),the equation becomes \( y = 2 - 2(x-1) \).

This line matches the linear approximation \( L(x) \).The tangent line shows how the linear approximation can be a powerful tool to simplify and understand the behavior of a function near a given point. It simplifies complex functions into manageable linear expressions.