Answer:
The mean increases by 3.
Step-by-step explanation:
<u>What's an outlier?</u>
- An outlier is a data point that's very different from the other data points. In this example, the outlier is 50, as the other data points are around the number 80.
<u>What's the mean of a data set?</u>
- The mean is the average of a data set. It's found by adding up all the numbers in the set and then dividing by the number of data points there is.
<u>How do we solve this problem?</u>
First, we find the mean of the data set with the outlier, 50.
Next, we find the mean of the set without the outlier.
Lastly, we subtract 77 from 80 to find the difference.
Therefore, the answer is The mean increases by 3.
Formula for Riemann Sum is:
![\frac{b-a}{n} \sum_{i=1}^n f(a + i \frac{b-a}{n})](https://tex.z-dn.net/?f=%5Cfrac%7Bb-a%7D%7Bn%7D%20%5Csum_%7Bi%3D1%7D%5En%20f%28a%20%2B%20i%20%5Cfrac%7Bb-a%7D%7Bn%7D%29)
interval is [1,3] so a = 1, b = 3
f(x) = 3x , sub into Riemann sum
![\frac{2}{n} \sum_{i=1}^n 3(1 + \frac{2i}{n})](https://tex.z-dn.net/?f=%5Cfrac%7B2%7D%7Bn%7D%20%5Csum_%7Bi%3D1%7D%5En%203%281%20%2B%20%5Cfrac%7B2i%7D%7Bn%7D%29)
Continue by simplifying using properties of summations.
![= \frac{2}{n}\sum_{i=1}^n 3 + \frac{2}{n}\sum_{i=1}^n \frac{6i}{n} \\ \\ = \frac{6}{n}\sum_{i=1}^n 1 + \frac{12}{n^2}\sum_{i=1}^n i \\ \\ =\frac{6}{n} (n) + \frac{12}{n^2}(\frac{n(n+1)}{2}) \\ \\ =6+\frac{6}{n}(n+1) \\ \\ =12 + \frac{6}{n}](https://tex.z-dn.net/?f=%3D%20%5Cfrac%7B2%7D%7Bn%7D%5Csum_%7Bi%3D1%7D%5En%203%20%2B%20%20%5Cfrac%7B2%7D%7Bn%7D%5Csum_%7Bi%3D1%7D%5En%20%5Cfrac%7B6i%7D%7Bn%7D%20%5C%5C%20%20%5C%5C%20%3D%20%5Cfrac%7B6%7D%7Bn%7D%5Csum_%7Bi%3D1%7D%5En%201%20%2B%20%20%5Cfrac%7B12%7D%7Bn%5E2%7D%5Csum_%7Bi%3D1%7D%5En%20i%20%5C%5C%20%20%5C%5C%20%3D%5Cfrac%7B6%7D%7Bn%7D%20%28n%29%20%2B%20%5Cfrac%7B12%7D%7Bn%5E2%7D%28%5Cfrac%7Bn%28n%2B1%29%7D%7B2%7D%29%20%5C%5C%20%20%5C%5C%20%3D6%2B%5Cfrac%7B6%7D%7Bn%7D%28n%2B1%29%20%5C%5C%20%20%5C%5C%20%3D12%20%2B%20%5Cfrac%7B6%7D%7Bn%7D%20)
Now you have an expression for the summation in terms of 'n'.
Next, take the limit as n-> infinity.
The limit of
![\frac{6}{n}](https://tex.z-dn.net/?f=%5Cfrac%7B6%7D%7Bn%7D)
goes to 0, therefore the limit of the summation is 12.
The area under the curve from [1,3] is equal to limit of summation which is 12.
There are three rules of finding the horizontal asymptote depending on the orders of the numerator and denominator. If the degrees are equal for the numerator and the denominator, then the horizontal asymptote is equal to y = the ratio of the coefficients of the highest order from the numerator and the denominator. If the degree in the numerator is less than the degree in the denominator, then there the x axis is the horizontal asymptote. If on the other hand, the order in the numerator is greater than that of the denominator, then there is no horizontal asymptote.
Answer:
whats hgfghjkfdvdfvfdfvdfsfdvdfv
Step-by-step explanation: