### 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.
The text passage contains a series of questions related to the graph of \( y = \frac{1}{x} \) and the properties of the tangent line at different points. Each question builds upon the previous one, leading to a final proof about the area of the triangle formed by the tangent line.
multiple_questions

1. **write the equation of the tangent line.**
   - this is a mathematical question asking for the equation of the tangent line to the curve \( y = \frac{1}{x} \) at the point \((2, 0.5)\).

2. **explain why the tangent line will always give an under approximation of the curve using the second derivative.**
   - this is a conceptual question asking for an explanation based on the concavity of the curve.

3. **calculate the area of the shaded triangle.**
   - this is a mathematical question asking for the area of a specific triangle formed by the tangent line and the coordinate axes.

4. **calculate the area of the triangle formed when the tangent is instead placed at \((1,1)\).**
   - this is another mathematical question asking for the area of a triangle formed by the tangent line at a different point.

5.
Given a text passage, determine:
    1. If it contains a mathematical question (including word problems or calculations)
    2. If an answer to that mathematical question is provided in the text

First, describe the content structure you found in 1-2 sentences.

Then provide your analysis in the following format:
TYPE: <one of the following values>
- NO_QUESTION
- SINGLE_QUESTION
- SINGLE_QUESTION_WITH_ANSWER
- MULTIPLE_QUESTIONS
- MULTIPLE_QUESTIONS_WITH_ANSWERS
system
### 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.
user
The text passage contains a series of questions related to the graph of \( y = \frac{1}{x} \) and the properties of the tangent line at different points. Each question builds upon the previous one, leading to a final proof about the area of the triangle formed by the tangent line.

TYPE: MULTIPLE_QUESTIONS

1. **Write the equation of the tangent line.**
   - This is a mathematical question asking for the equation of the tangent line to the curve \( y = \frac{1}{x} \) at the point \((2, 0.5)\).

2. **Explain why the tangent line will always give an under approximation of the curve using the second derivative.**
   - This is a conceptual question asking for an explanation based on the concavity of the curve.

3. **Calculate the area of the shaded triangle.**
   - This is a mathematical question asking for the area of a specific triangle formed by the tangent line and the coordinate axes.

4. **Calculate the area of the triangle formed when the tangent is instead placed at \((1,1)\).**
   - This is another mathematical question asking for the area of a triangle formed by the tangent line at a different point.

5.