The graph of $y =\frac { 1 } { x }$ is shown below. The tangent line at $(2, 0.5)$, along with the coordinate axes, form a shaded triangle.

1. Find the equation of the tangent line.

   Hint 1(a):

   Find $y^\prime$ and use point slope form to find the equation.

   Hint 2(a):

   Point slope form: _{1 $y = m(x − x_1) + y_1$}

2. Explain why the tangent line will always give an under approximation of the curve using the second derivative.

   Hint (b):

   If the graph was concave down, would the tangent line be an over or under-approximation?

3. Find the area of the shaded triangle.

   Hint (c):

   The base and height are given on the graph.

4. Find the area of the triangle formed when the tangent is instead placed at $(1, 1)$.

   Hint (d):

   Compute $y^\prime(1)$ and use point slope form again to find the equation.

5. Prove that the area of the shaded triangle formed by a tangent to $y =\frac { 1 } { x }$ is always the same, regardless of the point of tangency.

   Step 1(e):

    For a point $\left (a, \frac{1}{a} \right )$, use the result of $y'(a)$ and point slope form to find the equation of the tangent line.

   Step 2(e):

   The vertices of the triangle made by the tangent line are $(0, 0)$, $(0, t(0)),$ and ($x$-intercept of $t(x)$, $0)$, where $t(x)$ is the equation of the tangent line.

   Step 3(e):

   Calculate the area of the triangle using the vertices to determine the dimensions.