How do you find the average value of a function #f(x)=(x-1)^2# on the interval from x=1 to x=5?

Answer 1

The average value is #16/3#
(The average rate of change is #4#.)

The average value is requested in the question:

If we want the area of a rectangle to match the area under the function's graph, the height of the rectangle on that interval must equal the average value of the function over that interval.

To calculate that height, multiply the length of the interval by 1, or find the area under the graph and divide by its length.

Average value = # 1/(5-1) int_1^5 (x-1)^2 dx = 1/4 int_1^5 (x-1)^2 dx#.
We could evaluate the integral by expanding and evaluating the resulting quadratic expression, but I think this one is cleaner is we just substitute #u=x-1#
#1/4 (x-1)^3/3]_1^5 = 1/4[4^3/3 - 0/3] = 16/3#

Notably, the question was posted under "Average rate of change" even though it asks for the average value, which would be:

#(Deltay)/(Deltax) = (f(5)-f(1))/(5-1) = (16 - 0)/4 = 4#
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Answer 2

To find the average value of the function ( f(x) = (x - 1)^2 ) on the interval from ( x = 1 ) to ( x = 5 ), you can use the formula:

[ \text{Average value} = \frac{1}{b - a} \int_{a}^{b} f(x) , dx ]

Substitute the limits of integration and the function into the formula:

[ \text{Average value} = \frac{1}{5 - 1} \int_{1}^{5} (x - 1)^2 , dx ]

Then, integrate the function from ( x = 1 ) to ( x = 5 ):

[ \int_{1}^{5} (x - 1)^2 , dx = \left[ \frac{(x - 1)^3}{3} \right]_{1}^{5} ]

[ = \frac{(5 - 1)^3}{3} - \frac{(1 - 1)^3}{3} ]

[ = \frac{64}{3} ]

Finally, calculate the average value:

[ \text{Average value} = \frac{1}{4} \times \frac{64}{3} ]

[ = \frac{16}{3} ]

So, the average value of the function ( f(x) = (x - 1)^2 ) on the interval from ( x = 1 ) to ( x = 5 ) is ( \frac{16}{3} ).

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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