The average value of g(t) over the interval [0,3] is given by
A. (g(3) +g(0))/2.
B. (g(3)−g(0))/3.
C.1/3∫[3,0]g(t)dt.
D. 3∫[3,0]g(t)dt.
E. None of the above
You know the area under the curve is ∫[3,0]g(t) dt
You want the height of a rectangle with the same area.
So divide the area by the width, and you get choice B
Thanks for your help!
That is actually C I think.
To find the average value of a function g(t) over an interval [a,b], you need to compute the definite integral of g(t) over that interval and then divide it by the length of the interval (b - a). In this case, the interval is [0,3].
So, to find the average value of g(t) over [0,3], we need to compute the definite integral of g(t) from 0 to 3 and divide it by the length of the interval, which is 3 - 0 = 3.
Let's go through the options one by one:
A. (g(3) + g(0))/2: This option suggests taking the average of g(3) and g(0). However, this does not involve integrating g(t) over the given interval or dividing it by the length of the interval. Therefore, this option is not correct.
B. (g(3) - g(0))/3: Similar to option A, this option suggests taking the difference of g(3) and g(0) and dividing by 3. Again, this does not involve integrating g(t) over the given interval or dividing it by the length of the interval. Therefore, this option is not correct.
C. 1/3∫[3,0]g(t)dt: This option correctly represents the average value formula. It suggests finding the definite integral of g(t) from 0 to 3 (which is denoted by ∫[3,0]g(t)dt) and multiplying it by 1/3. Therefore, this option is correct.
D. 3∫[3,0]g(t)dt: This option incorrectly suggests multiplying the integral by 3, which is not the correct approach according to the average value formula. Therefore, this option is not correct.
E. None of the above: Since option C, 1/3∫[3,0]g(t)dt, is the correct representation of the average value formula, the correct answer is E. None of the above.
Therefore, the correct answer is E. None of the above.