For The Graphed Function $f(x)=(2)^{x+2}+1$, Calculate The Average Rate Of Change From $x=-1$ To $x=0$.A. -2 B. 2 C. 3 D. -3

Introduction

In mathematics, the average rate of change is a fundamental concept used to measure the rate at which a function changes over a given interval. It is a crucial tool in understanding the behavior of functions and is widely applied in various fields, including physics, engineering, and economics. In this article, we will explore how to calculate the average rate of change for a given function, using the graphed function $f(x)=(2)^{x+2}+1$ as a case study.

Understanding the Average Rate of Change

The average rate of change of a function $f(x)$ over an interval $[a,b]$ is defined as the ratio of the change in the function's output to the change in the input. Mathematically, it can be expressed as:

$$\text{Average Rate of Change} = \frac{f(b) - f(a)}{b - a}$$

This formula calculates the average rate at which the function changes over the interval $[a,b]$. It is a measure of the function's rate of change, and it can be used to determine the slope of the function over a given interval.

Calculating the Average Rate of Change for the Given Function

To calculate the average rate of change for the given function $f(x)=(2)^{x+2}+1$, we need to find the values of $f(-1)$ and $f(0)$.

Step 1: Find the value of $f(-1)$

To find the value of $f(-1)$, we substitute $x=-1$ into the function:

$$f(-1) = (2)^{(-1)+2} + 1$$

$$f(-1) = (2)^{1} + 1$$

$$f(-1) = 2 + 1$$

$$f(-1) = 3$$

Step 2: Find the value of $f(0)$

To find the value of $f(0)$, we substitute $x=0$ into the function:

$$f(0) = (2)^{(0)+2} + 1$$

$$f(0) = (2)^{2} + 1$$

$$f(0) = 4 + 1$$

$$f(0) = 5$$

Step 3: Calculate the Average Rate of Change

Now that we have the values of $f(-1)$ and $f(0)$, we can calculate the average rate of change using the formula:

$$\text{Average Rate of Change} = \frac{f(0) - f(-1)}{0 - (-1)}$$

$$\text{Average Rate of Change} = \frac{5 - 3}{0 + 1}$$

$$\text{Average Rate of Change} = \frac{2}{1}$$

$$\text{Average Rate of Change} = 2$$

Conclusion

In this article, we calculated the average rate of change for the graphed function $f(x)=(2)^{x+2}+1$ from $x=-1$ to $x=0$. We found the values of $f(-1)$ and $f(0)$, and then used the formula for the average rate of change to calculate the result. The average rate of change was found to be 2.

Key Takeaways

• The average rate of change is a measure of the rate at which a function changes over a given interval.
• It can be calculated using the formula: $\text{Average Rate of Change} = \frac{f(b) - f(a)}{b - a}$.
• The average rate of change can be used to determine the slope of a function over a given interval.

References

• [1] Calculus, 3rd edition, Michael Spivak.
• [2] Calculus, 2nd edition, James Stewart.

Discussion

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Introduction

In our previous article, we explored how to calculate the average rate of change for a given function, using the graphed function $f(x)=(2)^{x+2}+1$ as a case study. In this article, we will answer some frequently asked questions related to calculating the average rate of change.

Q: What is the average rate of change?

A: The average rate of change is a measure of the rate at which a function changes over a given interval. It is calculated using the formula: $\text{Average Rate of Change} = \frac{f(b) - f(a)}{b - a}$.

Q: How do I calculate the average rate of change?

A: To calculate the average rate of change, you need to follow these steps:

1. Find the values of $f(a)$ and $f(b)$ by substituting the values of $a$ and $b$ into the function.
2. Calculate the difference between $f(b)$ and $f(a)$.
3. Calculate the difference between $b$ and $a$.
4. Divide the difference between $f(b)$ and $f(a)$ by the difference between $b$ and $a$.

Q: What is the difference between the average rate of change and the instantaneous rate of change?

A: The average rate of change is a measure of the rate at which a function changes over a given interval, while the instantaneous rate of change is a measure of the rate at which a function changes at a specific point. The instantaneous rate of change is calculated using the derivative of the function.

Q: Can I use the average rate of change to determine the slope of a function?

A: Yes, you can use the average rate of change to determine the slope of a function over a given interval. The slope of a function is a measure of how steep the function is at a given point.

Q: What are some real-world applications of the average rate of change?

A: The average rate of change has many real-world applications, including:

• Physics: The average rate of change is used to calculate the velocity of an object over a given time interval.
• Engineering: The average rate of change is used to calculate the rate of change of a system's output over a given time interval.
• Economics: The average rate of change is used to calculate the rate of change of a country's GDP over a given time interval.

Q: Can I use a calculator to calculate the average rate of change?

A: Yes, you can use a calculator to calculate the average rate of change. Most calculators have a built-in function for calculating the average rate of change.

Q: What are some common mistakes to avoid when calculating the average rate of change?

A: Some common mistakes to avoid when calculating the average rate of change include:

• Not substituting the correct values of $a$ and $b$ into the function.
• Not calculating the difference between $f(b)$ and $f(a)$ correctly.
• Not calculating the difference between $b$ and $a$ correctly.
• Not dividing the difference between $f(b)$ and $f(a)$ by the difference between $b$ and $a$ correctly.

Conclusion

In this article, we answered some frequently asked questions related to calculating the average rate of change. We hope that this article has been helpful in clarifying any confusion you may have had about calculating the average rate of change.

Key Takeaways

• The average rate of change is a measure of the rate at which a function changes over a given interval.
• It can be calculated using the formula: $\text{Average Rate of Change} = \frac{f(b) - f(a)}{b - a}$.
• The average rate of change can be used to determine the slope of a function over a given interval.

References

• [1] Calculus, 3rd edition, Michael Spivak.
• [2] Calculus, 2nd edition, James Stewart.

Discussion

What are some other real-world applications of the average rate of change?