What is the average value of the function # f(x) = 4x ln (2x) # on the interval #[1,4]#?
So here, the average value is equal to
Let's first find this integral without the bounds.
So:
Simplifying and integrating:
This is without the constant go integration because we will use this to evaluate the integral. Returning to the average value, we see that
Now evaluating in full:
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To find the average value of the function ( f(x) = 4x \ln(2x) ) on the interval ([1,4]), we use the formula for the average value of a function over an interval:
[ \text{Average value} = \frac{1}{b - a} \int_a^b f(x) , dx ]
Where ( a ) and ( b ) are the endpoints of the interval.
So, for the function ( f(x) = 4x \ln(2x) ) on the interval ([1,4]), we have:
[ a = 1 ] [ b = 4 ]
Now, we evaluate the integral:
[ \int_1^4 4x \ln(2x) , dx ]
Then we divide the result by the length of the interval:
[ \frac{1}{4 - 1} \int_1^4 4x \ln(2x) , dx ]
[ = \frac{1}{3} \int_1^4 4x \ln(2x) , dx ]
This integral may require integration by parts. After calculating the integral, we obtain a numerical value. Finally, we divide this numerical value by 3 to get the average value of the function over the interval ([1,4]).
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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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