Let f(x) be a function on the interval [a,b]. If we divided our interval into n equally sized intervals and took the sample at left endpoints f(x_{i}), then an approximation for the average value would be given by the formula:
Find the average value of the function over the given interval
Problem 1 :
Solution :
Finding average value of left piece :
f(x) = -x^{2} - 6x - 8 on [-4, -3]
a = -4, b = -3
Finding average value of right piece :
f(x) = (-x/2) - (1/2) on [-3, 3]
a = -3, b = 3
Adding these two values,
= 0.7 + (-2)
= -0.3
So, the average value of the given function in the given interval is -0.3.
Problem 2 :
Solution :
Finding average value of left piece :
f(x) = -1 on [0, 1]
a = 0, b = 1
Finding average value of right piece :
f(x) = -x^{2} + 4x -4 on [1, 3]
a = 1, b = 3
= -1 + (-0.35)
= -1.35
So, the average value is -1.35.
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