At what point (or points) does the function g have an inflection point. Write "NONE" if appropriate. x = If x changes from a = 1 to 1.1, calculate the value of the differential dy, for y = g(x).

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Chapter2: Second-order Linear Odes
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2c and 2d.

**Graph Analysis and Transcription**

2) Suppose \( g \) is a continuous function on the interval \((-5, 4)\). The graph of \( g' \), the derivative of \( g \), is given in the figure below.

**Graph Description:**

- **Axes**: The horizontal axis is labeled from -5 to 4, and the vertical axis is labeled from -4 to 2.
- **Graph of \( g' \)**: The graph shows several distinct segments:
  - Starting at \((-5, 0)\), the curve rises steeply to \((-4, 2)\), indicating a positive slope.
  - It then descends to \((-2, -2)\), showing a negative slope.
  - The curve turns upwards slightly, reaching \((-1, 0)\).
  - There is then a sharp drop to \((0, -3)\).
  - From \((0, -3)\), the graph shows a straight increasing line up to \((4, 1)\).

- **Endpoints**: 
  - \((-5, 0)\), \((-1, 0)\), \((4, 1)\) are marked with open circles, indicating these points are not included in the graph.

This graph illustrates the derivative \( g' \) behavior over the given interval, showing where the function \( g \) is increasing or decreasing based on the positive or negative values of \( g' \).
Transcribed Image Text:**Graph Analysis and Transcription** 2) Suppose \( g \) is a continuous function on the interval \((-5, 4)\). The graph of \( g' \), the derivative of \( g \), is given in the figure below. **Graph Description:** - **Axes**: The horizontal axis is labeled from -5 to 4, and the vertical axis is labeled from -4 to 2. - **Graph of \( g' \)**: The graph shows several distinct segments: - Starting at \((-5, 0)\), the curve rises steeply to \((-4, 2)\), indicating a positive slope. - It then descends to \((-2, -2)\), showing a negative slope. - The curve turns upwards slightly, reaching \((-1, 0)\). - There is then a sharp drop to \((0, -3)\). - From \((0, -3)\), the graph shows a straight increasing line up to \((4, 1)\). - **Endpoints**: - \((-5, 0)\), \((-1, 0)\), \((4, 1)\) are marked with open circles, indicating these points are not included in the graph. This graph illustrates the derivative \( g' \) behavior over the given interval, showing where the function \( g \) is increasing or decreasing based on the positive or negative values of \( g' \).
### Inflection Points and Differentials

#### c) Inflection Points of Function \( g \)

The task is to determine the point or points where the function \( g \) exhibits an **inflection point**. An inflection point is where the function changes concavity, from concave up to concave down, or vice versa. If no inflection points exist, indicate as "NONE."

\[ x = \underline{\hspace{2cm}} \]

#### d) Calculating the Differential \( dy \)

If \( x \) changes from \( a = 1 \) to \( 1.1 \), calculate the value of the differential \( dy \), where \( y = g(x) \). The differential \( dy \) gives an approximation of how much \( y \) changes when \( x \) changes by a small amount.

\[ dy = \underline{\hspace{5cm}} \]

---
No graphs or diagrams are provided in this section.
Transcribed Image Text:### Inflection Points and Differentials #### c) Inflection Points of Function \( g \) The task is to determine the point or points where the function \( g \) exhibits an **inflection point**. An inflection point is where the function changes concavity, from concave up to concave down, or vice versa. If no inflection points exist, indicate as "NONE." \[ x = \underline{\hspace{2cm}} \] #### d) Calculating the Differential \( dy \) If \( x \) changes from \( a = 1 \) to \( 1.1 \), calculate the value of the differential \( dy \), where \( y = g(x) \). The differential \( dy \) gives an approximation of how much \( y \) changes when \( x \) changes by a small amount. \[ dy = \underline{\hspace{5cm}} \] --- No graphs or diagrams are provided in this section.
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