**Exercise: Calculating the Average Rate of Change****Objective:** Use the graph below to find the average rate of change between the points $t = 3$ and $t = -1$.**Graph Description:**The graph displays a parabola, which is a symmetric curve that opens upwards. This parabola intersects the y-axis at the point (0, -9) and increases symmetrically on both sides of this point.**Key Points to Consider:**1. **Axes:** - The horizontal axis is labeled from -10 to 10, representing the t-axis. - The vertical axis is labeled from -10 to 10, representing the function's value axis.2. **Plot Points:** - At $t = -1$, the graph reaches a point on the parabola that appears to approximately intersect at (0, -8). - At $t = 3$, the graph intersects the parabola at a point approximately (0, 0).**Finding the Average Rate of Change:**The average rate of change of a function between two points $t_1$ and $t_2$ is given by the formula:\[ \text{Average Rate of Change} = \frac{f(t_2) - f(t_1)}{t_2 - t_1} \]- Identify function values at given t-values. - Estimate from graph: - $f(-1) = 0$ - $f(3) = 6$- Substitute into the formula: - $\frac{f(3) - f(-1)}{3 - (-1)} = \frac{6 - 0}{3 + 1} = \frac{6}{4} = 1.5 $Thus, the average rate of change between the points $t = 3$ and $t = -1$ is approximately $1.5$.

Answered: Use the graph below to find the average…

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Use the graph below to find the average rate of change between the points t=-3 and t=-1. 10 5- 10 -5 .5. 10 --5- --10-

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Question

**Exercise: Calculating the Average Rate of Change**

**Objective:** Use the graph below to find the average rate of change between the points $t = 3$ and $t = -1$.

**Graph Description:**

The graph displays a parabola, which is a symmetric curve that opens upwards. This parabola intersects the y-axis at the point (0, -9) and increases symmetrically on both sides of this point.

**Key Points to Consider:**

1. **Axes:**
- The horizontal axis is labeled from -10 to 10, representing the t-axis.
- The vertical axis is labeled from -10 to 10, representing the function's value axis.

2. **Plot Points:**
- At $t = -1$, the graph reaches a point on the parabola that appears to approximately intersect at (0, -8).
- At $t = 3$, the graph intersects the parabola at a point approximately (0, 0).

**Finding the Average Rate of Change:**

The average rate of change of a function between two points $t_1$ and $t_2$ is given by the formula:

\[ \text{Average Rate of Change} = \frac{f(t_2) - f(t_1)}{t_2 - t_1} \]

- Identify function values at given t-values.
- Estimate from graph:
- $f(-1) = 0$
- $f(3) = 6$
- Substitute into the formula:
- $\frac{f(3) - f(-1)}{3 - (-1)} = \frac{6 - 0}{3 + 1} = \frac{6}{4} = 1.5 $

Thus, the average rate of change between the points $t = 3$ and $t = -1$ is approximately $1.5$.

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