Question

What is the average value of latex: y=\tan\left(\frac{x^2}{9}\right) y = tan ⁡ ( x 2 9 ) on the closed interval [1.25, 2]?

Answer 1

To find the average value of y = tan(x^2/9) on the closed interval [1.25, 2], evaluate the definite integral of the function over the interval and divide it by the length of the interval.

Step-by-step explanation:

To find the average value of y = tan(x^2/9) on the closed interval [1.25, 2], we need to evaluate the definite integral of the function over that interval and divide it by the length of the interval.

Step 1: Determine the definite integral of the function. The integral of tan(x^2/9) is not elementary, so we will need to use numerical methods or calculator software to approximate the value.

Step 2: Evaluate the integral using numerical methods or a calculator to find the area under the curve of the function on the interval [1.25, 2].

Step 3: Divide the area under the curve by the length of the interval (2 - 1.25 = 0.75) to find the average value.

Answer 2

For this case we have the following function:

`f(x) = tan((x^2)/(9))`
The average rate of change is given by:

`AVR = (f(x2) - f(x1))/(x2- x1)`
Evaluating the function for the given interval we have:
For x = 1.25:

`f(1.25) = tan((1.25^2)/(9))`

`f(1.25) = 0.18`
For x = 2:

`f(2) = tan((2^2)/(9))`

`f(2) = 0.48`
Then, replacing values we have:

`AVR = (0.48 - 0.18)/(2 - 1.25)`

`AVR = 0.4`
Answer:
the average value of on the closed interval [1.25, 2] is:

`AVR = 0.4`