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2 years ago

Find the average value of f over the region D. f(x, y) = 3xy, D is the triangle with vertices (0, 0), (1, 0), and (1, 9)

Mathematics

1 answer:

MakcuM [25]2 years ago

8 0

The average value of f over the region D is 243/4

To answer the question, we need to know what the average value of a function is

<h3>What is the average value of a function?</h3>

The average value of a function f(x) over an interval [a,b] is given by

$\frac{1}{b - a} \int\limits^b_a {f(x)} \, dx$

Now, given that we require the average value of f(x,y) = 3xy over the region D where D is the triangle with vertices (0, 0), (1, 0), and (1, 9).

x is intergrated from x = 0 to 1 and the interval is [0,1] and y is integrated from y = 0 to y = 9

So, $\frac{1}{b - a} \int\limits^b_a {f(x,y)} \, dA = \frac{1}{1 - 0} \int\limits^1_0 \int\limits^9_0 {3xy} \, dxdy \\= \frac{3}{1} \int\limits^1_0 {x} \,dx\int\limits^9_0 {y} \,dy\\ = \frac{3}{1} [\frac{x^{2} }{2} ]^{1}_{0}[\frac{y^{2} }{2} ]^{9}_{0} \\= 3[\frac{1^{2} }{2} - \frac{0^{2}}{2} ] [\frac{9^{2} }{2} - \frac{0^{2}}{2} ] \\= 3[\frac{1}{2} - 0 ][\frac{81}{2} - 0 ]\\= \frac{81}{2} X3 X \frac{1}{2} \\= \frac{243}{4}$

So, the average value of f over the region D is 243/4

Learn more about average value of a function here:

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For example, a common economic problem is the consumer choice decision. Households are selecting consumption of various goods. However, consumers are not allowed to spend more than their income (otherwise they would buy infinite amounts of everything!!). Let’s set up the consumer’s problem:

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4 years ago

Suppose the ice Q scores in one region are normally distributed within a standard deviation of 13. Suppose also that exactly 51%

elena-14-01-66 [18.8K]

Answer:

$z=-0.025$

And if we solve for $\mu$ we got

$\mu=100 +0.025*13=100.325$

Step-by-step explanation:

Previous concepts

Normal distribution, is a "probability distribution that is symmetric about the mean, showing that data near the mean are more frequent in occurrence than data far from the mean".

The Z-score is "a numerical measurement used in statistics of a value's relationship to the mean (average) of a group of values, measured in terms of standard deviations from the mean".

Solution to the problem

Let X the random variable that represent the scores of a population, and for this case we know the distribution for X is given by:

$X \sim N(\mu,13)$

Where $\mu=?$ and $\sigma=13$

And the best way to solve this problem is using the normal standard distribution and the z score given by:

$z=\frac{x-\mu}{\sigma}$

For this case we know these conditions:

$P(X>100)=0.51$ (a)

$P(X$ (b)

Both conditions are equivalent on this case. We can use the z score again in order to find the value a.

As we can see on the figure attached the z value that satisfy the condition with 0.49 of the area on the left and 0.51 of the area on the right it's z=-0.025. On this case P(Z<-0.025)=0.49 and P(z>-0.025)=0.51

If we use condition (b) from previous we have this:

$P(X$

$P(z$

But we know which value of z satisfy the previous equation so then we can do this:

$z=-0.025$

And if we solve for $\mu$ we got

$\mu=100 +0.025*13=100.325$

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For example: if I want to find how long side AB is, I would use the point A (-5, 2) and B (2 1/2, 2) and plug them into the distance formula, where (-5, 2) is (x1, x2) and (2 1/2, 2) is (x2, y2) and solve that.

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Step-by-step explanation:

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