On Which Interval Do The Functions F ( X ) = 6 X 2 + 12 F(x)=6x^2+12 F ( X ) = 6 X 2 + 12 And G ( X ) = 18 X + 12 G(x)=18x+12 G ( X ) = 18 X + 12 Have The Same Average Rate Of Change?A. X = − 7 To X = 3 X=-7 \text{ To } X=3 X = − 7 To X = 3 B. X = − 4 To X = 0 X=-4 \text{ To } X=0 X = − 4 To X = 0 C. X = 1 To X = 2 X=1 \text{ To } X=2 X = 1 To X = 2 D. $x=4 \text{ To }

On Which Interval Do the Functions $f(x)=6x^2+12$ and $g(x)=18x+12$ Have the Same Average Rate of Change?

Understanding Average Rate of Change

The average rate of change of a function is a measure of how much the function's output changes when its input changes by a certain amount. It is calculated by finding the difference in the function's output values for two different input values and then dividing by the difference in the input values. In mathematical terms, the average rate of change of a function $f(x)$ over an interval $[a, b]$ is given by:

$$\frac{f(b) - f(a)}{b - a}$$

Calculating Average Rate of Change for $f(x)$ and $g(x)$

To find the average rate of change of $f(x)=6x^2+12$ and $g(x)=18x+12$ over a given interval, we need to calculate the difference in their output values for two different input values and then divide by the difference in the input values.

Let's start by finding the derivative of $f(x)$ and $g(x)$, which represents the instantaneous rate of change of the functions.

The derivative of $f(x)=6x^2+12$ is:

$$f'(x) = 12x$$

The derivative of $g(x)=18x+12$ is:

$$g'(x) = 18$$

Since the average rate of change is the same as the instantaneous rate of change for a linear function, we can conclude that the average rate of change of $g(x)$ is constant and equal to 18.

Finding the Interval Where the Average Rate of Change is the Same

Now, we need to find the interval where the average rate of change of $f(x)$ is equal to the average rate of change of $g(x)$, which is 18.

We can set up the equation:

$$\frac{f(b) - f(a)}{b - a} = 18$$

Substituting the values of $f(x)$ and $g(x)$, we get:

$$\frac{(6b^2+12) - (6a^2+12)}{b - a} = 18$$

Simplifying the equation, we get:

$$\frac{6(b^2-a^2)}{b - a} = 18$$

$$6(b+a) = 18$$

$$b+a = 3$$

Now, we need to find the values of $a$ and $b$ that satisfy the equation $b+a = 3$.

Analyzing the Options

Let's analyze the options given:

A. $x=-7 \text{ to } x=3$

B. $x=-4 \text{ to } x=0$

C. $x=1 \text{ to } x=2$

D. $x=4 \text{ to } x=7$

We can substitute the values of $a$ and $b$ from each option into the equation $b+a = 3$ and check if it is satisfied.

For option A, $a=-7$ and $b=3$, which satisfies the equation $b+a = 3$.

For option B, $a=-4$ and $b=0$, which does not satisfy the equation $b+a = 3$.

For option C, $a=1$ and $b=2$, which does not satisfy the equation $b+a = 3$.

For option D, $a=4$ and $b=7$, which does not satisfy the equation $b+a = 3$.

Conclusion

Based on the analysis, the interval where the functions $f(x)=6x^2+12$ and $g(x)=18x+12$ have the same average rate of change is $x=-7 \text{ to } x=3$.

Therefore, the correct answer is:

A. $x=-7 \text{ to } x=3$
Q&A: On Which Interval Do the Functions $f(x)=6x^2+12$ and $g(x)=18x+12$ Have the Same Average Rate of Change?

Q: What is the average rate of change of a function?

A: The average rate of change of a function is a measure of how much the function's output changes when its input changes by a certain amount. It is calculated by finding the difference in the function's output values for two different input values and then dividing by the difference in the input values.

Q: How do you calculate the average rate of change of a function?

A: To calculate the average rate of change of a function, you need to find the difference in the function's output values for two different input values and then divide by the difference in the input values. The formula for the average rate of change of a function $f(x)$ over an interval $[a, b]$ is given by:

$$\frac{f(b) - f(a)}{b - a}$$

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

A: The average rate of change of a function is a measure of how much the function's output changes when its input changes by a certain amount, while the instantaneous rate of change of a function is a measure of how much the function's output changes when its input changes by an infinitesimally small amount. For a linear function, the average rate of change is equal to the instantaneous rate of change.

Q: How do you find the interval where the average rate of change of two functions is the same?

A: To find the interval where the average rate of change of two functions is the same, you need to set up an equation using the formula for the average rate of change and solve for the interval.

Q: What is the significance of the derivative of a function in finding the average rate of change?

A: The derivative of a function represents the instantaneous rate of change of the function. For a linear function, the derivative is constant and represents the average rate of change of the function.

Q: Can you explain the concept of the derivative in simple terms?

A: The derivative of a function represents the rate at which the function's output changes when its input changes by a small amount. It is a measure of how steep the function is at a given point.

Q: How do you determine if the average rate of change of two functions is the same over a given interval?

A: To determine if the average rate of change of two functions is the same over a given interval, you need to calculate the average rate of change of each function over the interval and compare the results.

Q: What is the relationship between the average rate of change and the slope of a function?

A: The average rate of change of a function is equal to the slope of the function. The slope of a function represents the rate at which the function's output changes when its input changes by a unit amount.

Q: Can you provide an example of how to find the interval where the average rate of change of two functions is the same?

A: Let's consider the functions $f(x)=6x^2+12$ and $g(x)=18x+12$. To find the interval where the average rate of change of these functions is the same, we need to set up an equation using the formula for the average rate of change and solve for the interval.

Q: What is the final answer to the problem?

A: The final answer to the problem is that the interval where the functions $f(x)=6x^2+12$ and $g(x)=18x+12$ have the same average rate of change is $x=-7 \text{ to } x=3$.