What is the average value of the function # f(x) = 4x ln (2x) # on the interval #[1,4]#?

Answer 1

#94/3ln(2)-5#

The average value #s# of a function #f# on the interval #[a,b]# can be found through the formula
#s=1/(b-a)int_a^bf(x)dx#

So here, the average value is equal to

#s=1/(4-1)int_1^4 4xln(2x)dx#

Let's first find this integral without the bounds.

#I=int4xln(2x)dx#
To do this integral, we'll need to use integration by parts. This technique takes the form #intudv=uv-intvdu#. Here, we want to choose a value of #u# that becomes simpler when differentiated and a value of #dv# that can be integrated.
For the given integral, let #u=ln(2x)# and #dv=4xcolor(white).dx#. Differentiating #u# and integrating #dv# we see that:
#{(u=ln(2x),=>,(du)/dx=1/x,=>,du=1/xdx),(dv=4xcolor(white).dx,=>,intdv=int4xcolor(white).dx,=>,v=2x^2):}#

So:

#I=uv-intvdu=2x^2ln(2x)-int2x^2 1/xdx#

Simplifying and integrating:

#I=2x^2ln(2x)-int2xcolor(white).dx=2x^2ln(2x)-x^2=x^2(2ln(2x)-1)#

This is without the constant go integration because we will use this to evaluate the integral. Returning to the average value, we see that

#s=1/(4-1)int_1^4 4xln(2x)dx=1/3[x^2(2ln(2x)-1)]_1^4#

Now evaluating in full:

#s=1/3 16(2ln(8)-1)-1/3 1(2ln(2)-1)#
#s=32/3ln(8)-16/3-2/3ln(2)+1/3#
Note that #ln(8)=ln(2^3)=3ln(2)#:
#s=32ln(2)-2/3ln(2)-16/3+1/3#
#s=94/3ln(2)-5#
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Answer 2

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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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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