Calculating Average Value of y=x^2√x3+1 from 0 to 2

In summary, the conversation is about finding the average value of a function using the mean value theorem and the definition of average value. The final answer is 6.
  • #1
tandoorichicken
245
0
Hmmm... got this one wrong

What is the average value of [tex] y = x^{2}\sqrt{x^3+1} [/tex] on the interval [0,2] ?

Okay, so another average value problem right?

[tex] f(b) - f(a) = f'(c)(b-a) [/tex]
[tex] f(2) - f(0) = 2f'(c) [/tex]
[tex] 12 = 2f'(c) [/tex]
So then the average value is 6 right?
 
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  • #2
Your problem is that you're trying to find the average value of a function using the mean value theorem. Try using the definition of the average value of a function:

[tex] \bar y = \frac{\int_a^b{f(x)dx}}{b-a} = \frac 1 2 \int_0^2{x^2\sqrt{x^3+1}dx} [/tex]

(Simple change in variable to solve from here.)
 
  • #3
Simple use the following to find the Average Value of a function:

1\b-a[tex] \int_a^b{f(x)dx} [/tex]
 
Last edited:

1. What is the formula for calculating the average value of y=x^2√x3+1 from 0 to 2?

The formula for calculating the average value of a function over a given interval is: Average value = (1/b-a)∫abf(x)dx, where a and b are the limits of the interval and f(x) is the given function. For the function y=x^2√x3+1, the average value over the interval [0,2] would be calculated as: Average value = (1/2-0)∫02(x^2√x3+1)dx.

2. What is the unit of measurement for the average value of y=x^2√x3+1 from 0 to 2?

The unit of measurement for the average value of y=x^2√x3+1 would depend on the unit of measurement for the function itself. For example, if the function represents distance in meters, then the average value would be in meters. If the function represents weight in kilograms, then the average value would be in kilograms.

3. Can the average value of y=x^2√x3+1 from 0 to 2 be negative?

Yes, the average value of a function over an interval can be negative if the function has both positive and negative values over that interval. The average value is simply a measure of the overall trend of the function over the given interval and does not necessarily have to be positive.

4. What is the significance of calculating the average value of a function?

Calculating the average value of a function can provide useful information about the overall behavior of the function over a given interval. It can help in understanding the general trend and can also be used in various applications such as calculating the average rate of change or the average value of a physical quantity.

5. Can the average value of y=x^2√x3+1 from 0 to 2 be greater than the maximum value of the function over that interval?

Yes, the average value of a function can be greater than the maximum value of the function over a given interval. This can happen if the function has a very steep increase or decrease at certain points within the interval, resulting in a higher average value. The maximum value of a function only represents the highest point of the function over that interval, while the average value takes into account the entire interval.

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