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2 years ago

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A data set contains an independent and a dependent variable. Which must be true of the data set if a linear function can be used

to represent the data?

1 answer:

Andrews [41]2 years ago

4 0

Answer:

the set must have a constant additive rate of change. the set must have a multiplicative rate of change.

Step-by-step explanation:

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More Calculus! (I'm so sorry)

Olenka [21]

Recall that converting from Cartesian to polar coordinates involves the identities

$\begin{cases}y(r,\phi)=r\sin\phi\\x(r,\phi)=r\cos\phi\end{cases}$

As a function in polar coordinates, $r$ depends on $\phi$ , so you can write $r=r(\phi)$ .

Differentiating the identities with respect to $\phi$ gives

$\begin{cases}\dfrac{\mathrm dy}{\mathrm d\phi}=\dfrac{\mathrm dr}{\mathrm d\phi}\sin\phi+r\cos\phi\\\\\dfrac{\mathrm dx}{\mathrm d\phi}=\dfrac{\mathrm dr}{\mathrm d\phi}\cos\phi-r\sin\phi\end{cases}$

The slope of the tangent line to $r(\phi)$ is given by

$\dfrac{\mathrm dy}{\mathrm dx}=\dfrac{\frac{\mathrm dy}{\mathrm d\phi}}{\frac{\mathrm dx}{\mathrm d\phi}}=\dfrac{\frac{\mathrm dr}{\mathrm d\phi}\sin\phi+r\cos\phi}{\frac{\mathrm dr}{\mathrm d\phi}\cos\phi-r\sin\phi}$

Given $r(\phi)=3\cos\phi$ , you have $\dfrac{\mathrm dr}{\mathrm d\phi}=-3\sin\phi$ . So the tangent line to $r(\phi)$ has a slope of

$\dfrac{\mathrm dy}{\mathrm dx}=\dfrac{-3\sin^2\phi+3\cos^2\phi}{-3\sin\phi\cos\phi-3\cos\phi\sin\phi}=\dfrac{3\cos2\phi}{-3\sin2\phi}=-\cot2\phi$

When $\phi=120^\circ=\dfrac{2\pi}3\text{ rad}$ , the tangent line has slope

$\dfrac{\mathrm dy}{\mathrm dx}=-\cot\dfrac{4\pi}3=-\dfrac1{\sqrt3}$

This line is tangent to the point $(r,\phi)=\left(-\dfrac32,\dfrac{2\pi}3\right)$ which in Cartesian coordinates is equivalent to $(x,y)=\left(\dfrac34,-\dfrac{3\sqrt3}4\right)$ , so the equation of the tangent line is

$y+\dfrac{3\sqrt3}4=-\dfrac1{\sqrt3}\left(x-\dfrac34\right)$

In polar coordinates, this line has equation

$r\sin\phi+\dfrac{3\sqrt3}4=-\dfrac1{\sqrt3}\left(r\cos\phi-\dfrac34\right)$
$\implies r=-\dfrac{3\sqrt3}{2\sqrt3\cos\phi+6\sin\phi}$

The tangent line passes through the y-axis when $x=0$ , so the y-intercept is $\left(0,-\dfrac{\sqrt3}2\right)$ .

The vector from this point to the point of tangency on $r(\phi)$ is given by the difference of the vector from the origin to the y-intercept (which I'll denote $\mathbf a$ ) and the vector from the origin to the point of tangency (denoted by $\mathbf b$ ). In the attached graphic, this corresponds to the green arrow.

$\mathbf b-\mathbf a=\left(\dfrac34,-\dfrac{3\sqrt3}4\right)-\left(0,-\dfrac{\sqrt3}2\right)=\left(\dfrac34,-\dfrac{\sqrt3}4\right)$

The angle between this vector and the vector pointing to the point of tangency is what you're looking for. This is given by

$\mathbf b\cdot(\mathbf b-\mathbf a)=\|\mathbf b\|\|\mathbf b-\mathbf a\|\cos\theta$
$\dfrac98=\dfrac{3\sqrt3}4\cos\theta$
$\implies\theta=\dfrac\pi6\text{ rad}=30^\circ$

The second problem is just a matter of computing the second derivative of $\phi$ with respect to $t$ and plugging in $t=2$ .

$\phi(t)=2t^3-6t$
$\phi'(t)=6t^2-6$
$\phi''(t)=12t$
$\implies\phi''(2)=24$

6 0

3 years ago

Would this be sin, cos, or tan?

Nataly [62]

Answer:

tan

Step-by-step explanation:

its the opposite over adjacent

7 0

3 years ago

Read 2 more answers

The mean number of words per minute (WPM) read by sixth graders is 97 with a standard deviation of 19 WPM. If 75 sixth graders a

andrey2020 [161]

Answer:

0.0174 = 1.74% probability that the sample mean would be greater than 101.63 WPM

Step-by-step explanation:

To solve this question, we need to understand the normal probability distribution and the central limit theorem.

Normal probability distribution

Problems of normally distributed samples are solved using the z-score formula.

In a set with mean $\mu$ and standard deviation $\sigma$ , the zscore 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 pvalue, we get the probability that the value of the measure is greater than X.

Central Limit Theorem

The Central Limit Theorem estabilishes 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.

In this problem, we have that:

$\mu = 97, \sigma = 19, n = 75, s = \frac{19}{\sqrt{75}} = 2.1939$

What is the probability that the sample mean would be greater than 101.63 WPM?

This is 1 subtracted by the pvalue of Z when X = 101.63. So

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

By the Central Limit Theorem

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

$Z = \frac{101.63 - 97}{2.1939}$

$Z = 2.11$

$Z = 2.11$ has a pvalue of 0.9826.

1 - 0.9826 = 0.0174

0.0174 = 1.74% probability that the sample mean would be greater than 101.63 WPM

5 0

3 years ago

(e) The cost of building the ship was $153 000 000. Write 153 000 000 in standard form.

Ostrovityanka [42]

Answer:

1.53 x 10 8

Step-by-step explanation: the 8 is small i just didnt know how to make it small lol

4 0

2 years ago

Consider the following system of equations

Vera_Pavlovna [14]

Hsisvskevwovewhsjsbaoaa

3 0

2 years ago