Find the average or mean slope of the function on a interval

• MHB
• jose1
You can then use the Mean Value Theorem to show that there exists a $c \in [-5, 6]$ such that $f'(c) = -8$. This does not mean that $c = -1/2$, as you seem to have assumed. In fact, the value of $c$ is irrelevant to the calculation of the average slope. In summary, you need to check your calculation and use the Mean Value Theorem correctly in order to find the average slope correctly.
jose1
Hello
I have the exercise below:
Consider the function f(x)=1−8x2 on the interval [−5,6]. Find the average or mean slope of the function on this interval, i.e.

[f(6)-f(-5)]/[6-(-5)]

according to the theorem of laGrange

the slope in a continues function which is derivable in an interval should equal to f'(c), So

[f(6)-f(-5)]/[6-(-5)] is equal to 8. According to the site this is wrong?

and

f'(x) = -16x. So, -16x=8 is x=-1/2. According to the site this is wrong?

Could you please give me some advice? where is my mistake?
I will appreciate any help

Thanks

jose said:
Hello
I have the exercise below:
Consider the function f(x)=1−8x2 on the interval [−5,6]. Find the average or mean slope of the function on this interval, i.e.

[f(6)-f(-5)]/[6-(-5)]

according to the theorem of laGrange

the slope in a continues function which is derivable in an interval should equal to f'(c), So

[f(6)-f(-5)]/[6-(-5)] is equal to 8. According to the site this is wrong?

and

f'(x) = -16x. So, -16x=8 is x=-1/2. According to the site this is wrong?

Could you please give me some advice? where is my mistake?
I will appreciate any help

Thanks
Hello jose, and welcome to MHB.

The answer to the question is $\dfrac {f(6) - f(-5)}{6 - (-5)}$. If you check your calculation for that, you should find that it gives the value $-8$, not $8$.

What is the definition of average or mean slope?

The average or mean slope of a function on a specific interval is the change in the function's output divided by the change in the function's input over that interval. It represents the average rate of change of the function on that interval.

How do you find the average or mean slope of a function?

To find the average or mean slope of a function on a specific interval, you need to choose two points on the function within that interval and calculate the slope between those two points using the slope formula: (change in y)/(change in x). This will give you the average or mean slope of the function on that interval.

What is the importance of finding the average or mean slope of a function?

Finding the average or mean slope of a function on a specific interval can help us understand the overall behavior of the function on that interval. It can also be used to determine the rate of change of the function, which is useful in many real-world applications such as economics, physics, and engineering.

How is the average or mean slope of a function different from the slope at a specific point?

The average or mean slope of a function on a specific interval represents the overall change of the function over that interval, whereas the slope at a specific point only represents the change of the function at that particular point. The average or mean slope is calculated using two points on the interval, while the slope at a specific point is calculated using the derivative of the function at that point.

What are some real-world applications of finding the average or mean slope of a function?

Finding the average or mean slope of a function can be useful in many real-world applications such as calculating average speed and acceleration in physics, determining average rates of change in economics, and finding the average rate of production in manufacturing. It can also be used to analyze trends and patterns in data.

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