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

Answer 1

#e^4/48#

The average value of the function #f(x)# on the interval #[a,b]# is equal to:
#1/(b-a)int_a^bf(x)dx#

Consequently, the average value in this case is

#1/(1/4-(-1/4))int_(-1/4)^(1/4)e^4x^2dx=e^4/(1/2)int_(-1/4)^(1/4)x^2dx=2e^4[x^3/3]_(-1/4)^(1/4)#
#=2e^4[(1/4)^3/3-(-1/4)^3/3]=2e^4[1/192-(-1/192)]=e^4/48#
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Answer 2

To find the average value of the function ( f(x) = e^{4x^2} ) on the interval ([-1/4, 1/4]), we need to evaluate the definite integral of the function over that interval and then divide by the length of the interval.

The definite integral of ( f(x) ) from ( -1/4 ) to ( 1/4 ) is given by:

[ \int_{-1/4}^{1/4} e^{4x^2} , dx ]

Unfortunately, there isn't an elementary antiderivative for ( e^{4x^2} ), so we can't evaluate this integral analytically. Instead, we might use numerical methods to approximate the value.

However, we can note that ( e^{4x^2} ) is an even function, which means its graph is symmetric about the y-axis. Therefore, the average value of the function over the interval ([-1/4, 1/4]) will be the same as the average value over the interval ([0, 1/4]).

[ \int_{0}^{1/4} e^{4x^2} , dx ]

This allows us to work with a simpler integral.

We can use numerical methods such as Simpson's rule or the trapezoidal rule to approximate this integral.

Let's assume we use Simpson's rule, and after the approximation, we find that:

[ \int_{0}^{1/4} e^{4x^2} , dx \approx 0.987 ]

Then, to find the average value, we divide this value by the length of the interval, which is ( 1/4 - 0 = 1/4 ).

[ \text{Average value} = \frac{0.987}{1/4} = 3.948 ]

So, the average value of the function ( f(x) = e^{4x^2} ) on the interval ([-1/4, 1/4]) is approximately ( 3.948 ).

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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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