Average values Find the average value of the following functio...

Question

Average values Find the average value of the following functions on the given interval. Draw a graph of the function and indicate the average value.

ƒ(𝓍) = 𝓍ⁿ on [0,1] , for any positive integer n

Accepted Answer

Welcome back, everyone. Find the average value of the function F of X equals X to the power of N on the interval from 2 to 4 inclusive. Let's recall the average value formula. F average is going to be equal to 1 divided by B minus A. Now our A is 2 and our B is 4. We're considering the endpoints of the interval. So B minus A would be 4 minus 2. And then we have to multiply the result by the integral from A to B, meaning from 2 to 4 of F of XD X. So we are integrating X to the power of N. D X Now let's simplify. This is equal to 1/2 integral from 2 to 4 of X to the power of ND X, and to integrate, we can simply apply the power rule. We get 1/2 X to the power of N + 1 divided by N + 1, and we want to evaluate the result from 2 to 4. So let's go ahead and combine the constants. We have 1 divided by 2 in C N + 1. And we are evaluating 4 to the power of N + 1 minus 2 raise to the power of N + 1. And now what we can do is simply factor out 2. So we get 1 divided by 2 parencies and plus 1. And now considering our difference, we can essentially write it as 2. Raise to the power of 2 in N + 1, right? Based on the properties of exponents, we can write 4 S2 squad, and whenever we have 2 squared raises the power of N + 1, we basically multiply our exponents. This is how we arrive at our expression, and then we subtract 2 to the power of N + 1. And now to simplify, we are ready to factor out 2. So first of all, what we can do is just write 1 divided by 2 in CM plus 1, multiplied by. Now 2 to the power of 2 inres and plus 1 is equivalent to 2 to the power of 2 + 2, right? And essentially, if we want to factor out 2, we can write it as. 2 to the power of 2 + 1 multiplied by 2, right? Based on the properties of exponents, when we multiply 2 to the power of 2N plus 1 by 2 of 1, we get an exponent of 2N + 2. And then we subtract 2 to the power of N multiplied by 2 again because multiplying these yields an exponent of N + 1. Now we can clearly see that if we factor out 2. This cancels out the 2 in the denominator, and we can get our simplified form. So let's go ahead and write 2 of 2 N + 1 minus 2 power of N. Divided by N + 1. And if we want to completely factor that, we can also factor 2 to the power of N. Inparies, we're going to get 2 to the power of N. Plus 1, so let's check if this is correct. Basically, if we multiply these, we get N plus N + 1, which is 2 M + 1 in our exponent, and then we subtract 1. And then divide by M + 1. That would be our final answer for this problem, and thank you for watching.

Average values Find the average value of the following functions on the given interval. Draw a graph of the function and indicate the average value.
ƒ(𝓍) = 𝓍ⁿ on [0,1] , for any positive integer n

Key Concepts

Average Value of a Function

Definite Integral

Graphing Functions

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