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defon
2 years ago
7

Use the given Maclaurin series to evaluate the limit

Mathematics
1 answer:
nekit [7.7K]2 years ago
7 0

The "given series" should be for \cos(x), not x, so that

\cos(x) = 1 - \dfrac{x^2}2 + \dfrac{x^4}{24} - \dfrac{x^6}{720} + \cdots

In the limit (which should say x\to\infty, not n), we have

\displaystyle \lim_{x\to\infty} \frac{\frac{x^2}{1+\cos(x)}}{x^4} = \lim_{x\to\infty} \frac{1}{x^2\left(2 - \frac{x^2}2 + \frac{x^4}{24} - \cdots\right)} = \boxed{0}

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Solve the problem. Use the Central Limit Theorem.The annual precipitation amounts in a certain mountain range are normally distr
bazaltina [42]

Answer:

0.8944 = 89.44% probability that the mean annual precipitation during 25 randomly picked years will be less than 112 inches.

Step-by-step explanation:

To solve this question, we use the normal probability distribution and the central limit theorem.

Normal Probability Distribution:

Problems of normal distributions can be solved using the z-score formula.

In a set with mean \mu and standard deviation \sigma, the z-score of a measure X is given by:

Z = \frac{X - \mu}{\sigma}

The Z-score measures how many standard deviations the measure is from the mean. After finding the Z-score, we look at the z-score table and find the p-value associated with this z-score. This p-value is the probability that the value of the measure is smaller than X, that is, the percentile of X. Subtracting 1 by the p-value, we get the probability that the value of the measure is greater than X.

Central Limit Theorem

The Central Limit Theorem establishes that, for a normally distributed random variable X, with mean \mu and standard deviation \sigma, the sampling distribution of the sample means with size n can be approximated to a normal distribution with mean \mu and standard deviation s = \frac{\sigma}{\sqrt{n}}.

For a skewed variable, the Central Limit Theorem can also be applied, as long as n is at least 30.

Mean of 109.0 inches, and a standard deviation of 12 inches.

This means that \mu = 109, \sigma = 12

Sample of 25.

This means that n = 25, s = \frac{12}{\sqrt{25}} = 2.4

What is the probability that the mean annual precipitation during 25 randomly picked years will be less than 112 inches?

This is the p-value of Z when X = 112. So

Z = \frac{X - \mu}{\sigma}

By the Central Limit Theorem

Z = \frac{X - \mu}{s}

Z = \frac{112 - 109}{2.4}

Z = 1.25

Z = 1.25 has a p-value of 0.8944.

0.8944 = 89.44% probability that the mean annual precipitation during 25 randomly picked years will be less than 112 inches.

7 0
3 years ago
A barrel of oil was filled at a constant rate of 7.9 gal/min. The barrel had 12 gallons before filling began.
jenyasd209 [6]

The equation that models this situation is z = 7.9y + 12.

A linear equation is a function that has a single variable raised to the power of 1.  An example is x = 4y + 2.

Where:

  • 2 is the constant
  • x = dependent variable
  • y = independent variable  

From the equation given, 12 would be the constant, the independent variable would be 7.9 and z would be the dependent variable.

z = 7.9y + 12

Here is the complete question: A barrel of oil was filled at a constant rate of 7.9 gal/min. The barrel had 12 gallons before filling began. write an equation in standard form to model the linear situation.

A similar question was answered here: brainly.com/question/2238405

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2 years ago
You have 6 reindeer, Rudy, Jebediab, Ezekiel, Lancer, Gloopin, and Balthazar, and you want to have 5 fly your sleigh. You always
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You can arrange your reindeer 30 different ways.
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3 years ago
Two men are decorating a room. One has painted 20 m² and the
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Answer:

Your answer is 120m4

Step-by-step explanation:

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Which physical property do the two rocks share?
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Answer:

Step-by-step explanation:

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