text: ### Home > APCALC > Chapter 6 > Lesson 6.2.2 > Problem6-77
  
  6-77.
  
  The graph of $y=\frac{1}{x}$ is shown at right. The tangent line at $(2,0.5)$, along with the coordinate axes, form a shaded triangle.
  
  1. Write the equation of the tangent line.
  
  Find $y^\prime$ and use point slope form to find the equation.
  
  Point slope form: $y=m(x−x_1)+y_1$
  
  2. Explain why the tangent line will always give an under approximation of the curve using the second derivative.
  
  If the graph was concave down, would the tangent line be an over or under-approximation?
  
  3. Calculate the area of the shaded triangle.
  
  The base and height are given on the graph.
  
  4. Calculate the area of the triangle formed when the tangent is instead placed at $(1,1)$.
  
  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.
  
  $\text{For a point }\left (a, \frac{1}{a} \right ),\text{use the result of } y'(a) \text{ and point slope form to find the equation of the tangent line.}$
  
  The vertices of the triangle made by the tangent line are $(0,0)$, $(0,t(0))$, and ($x$-intercept of $\left(t(x),0\right)$, where $t(x)$ is the equation of the tangent line.
  
  Calculate the area of the triangle using the vertices to determine the dimensions.
explanation: The document contains a series of math questions related to the graph of \( y = \frac{1}{x} \). The questions involve finding the equation of the tangent line, explaining the concavity and its effect on the approximation, calculating the area of the shaded triangle, and proving that the area of the triangle formed by the tangent line is constant regardless of the point of tangency.
distilabel_metadata.content: Evaluate the user given text extract and determine if it contains a math question and optionally and answer to this question.
  
  Output rules:
  
  1. Briefly describe the document and mention if it contains a question or a question and answer.
  2. Finalize the output with: Label: no-question,question,question-and-answer,multiple-question,multiple-question-and-answer
distilabel_metadata.role: system
distilabel_metadata.content: ### Home > APCALC > Chapter 6 > Lesson 6.2.2 > Problem6-77
  
  6-77.
  
  The graph of $y=\frac{1}{x}$ is shown at right. The tangent line at $(2,0.5)$, along with the coordinate axes, form a shaded triangle.
  
  1. Write the equation of the tangent line.
  
  Find $y^\prime$ and use point slope form to find the equation.
  
  Point slope form: $y=m(x−x_1)+y_1$
  
  2. Explain why the tangent line will always give an under approximation of the curve using the second derivative.
  
  If the graph was concave down, would the tangent line be an over or under-approximation?
  
  3. Calculate the area of the shaded triangle.
  
  The base and height are given on the graph.
  
  4. Calculate the area of the triangle formed when the tangent is instead placed at $(1,1)$.
  
  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.
  
  $\text{For a point }\left (a, \frac{1}{a} \right ),\text{use the result of } y'(a) \text{ and point slope form to find the equation of the tangent line.}$
  
  The vertices of the triangle made by the tangent line are $(0,0)$, $(0,t(0))$, and ($x$-intercept of $\left(t(x),0\right)$, where $t(x)$ is the equation of the tangent line.
  
  Calculate the area of the triangle using the vertices to determine the dimensions.
distilabel_metadata.role: user
distilabel_metadata.raw_output_q_a_detector_0: The document contains a series of math questions related to the graph of \( y = \frac{1}{x} \). The questions involve finding the equation of the tangent line, explaining the concavity and its effect on the approximation, calculating the area of the shaded triangle, and proving that the area of the triangle formed by the tangent line is constant regardless of the point of tangency.
  
  Label: multiple-question