what is average value of cos(x) on the interval [-3,5]?
Well you know that cos(x) is sketched between 1 and -1 on the graph. And you are looking at the average value between two points.
To find the average value of cos(x) on the interval [-3,5], we need to calculate the definite integral of cos(x) over that interval and then divide it by the length of the interval.
The definite integral of cos(x) over the interval [-3,5] can be calculated as follows:
∫[-3,5] cos(x) dx = [sin(x)] [-3,5] = sin(5) - sin(-3)
Now, let's evaluate sin(5) and sin(-3):
sin(5) ≈ 0.9589
sin(-3) ≈ -0.1411
Therefore, the definite integral of cos(x) over the interval [-3,5] is approximately:
∫[-3,5] cos(x) dx ≈ 0.9589 - (-0.1411) = 1.1
The length of the interval is 5 - (-3) = 8.
Now, we can calculate the average value of cos(x) on the interval [-3,5] by dividing the definite integral by the length of the interval:
Average value = (1.1) / 8 ≈ 0.1375
Hence, the average value of cos(x) on the interval [-3,5] is approximately 0.1375.
To find the average value of a function on an interval, you need to integrate the function over that interval and then divide by the width of the interval. In this case, you want to find the average value of the function f(x) = cos(x) on the interval [-3,5].
To calculate the average value, you can follow these steps:
1. Find the definite integral of cos(x) over the interval [-3,5]. The integral of cos(x) is sin(x), so you need to evaluate sin(x) at the upper (5) and lower (-3) limits and subtract the result:
Integral of cos(x)dx = sin(x) | from -3 to 5
= sin(5) - sin(-3)
2. Calculate the difference between sin(5) and sin(-3). To do this, you can use a calculator or refer to a trigonometric table.
3. Divide the result by the width of the interval, which is 5 - (-3) = 8.
Average value = (sin(5) - sin(-3)) / 8
Now you have the formula to calculate the average value of cos(x) on the interval [-3,5]. Just substitute the values of sin(5) and sin(-3), and perform the calculation to get the final answer.
the average value of f(x) on the interval [a,b] is
∫[a,b] f(x) dx
-----------------------
(b-a)
so just plug in your values.