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Ahat [919]
3 years ago
9

To find the equation of a line, we need the slope of the line and a point on the line. Since we are requested to find the equati

on of the tangent line at the point (4, 2), we know that (4, 2) is a point on the line. So we just need to find its slope. The slope of a tangent line to f(x) at x = a can be found using the formula
Mathematics
1 answer:
murzikaleks [220]3 years ago
8 0

Answer:

x-4y+4=0

f(x)=\sqrt x and x=4

Step-by-step explanation:

We are given that a curve

y=\sqrt x

We have to find the equation of tangent at point (4,2) on the given curve.

Let y=f(x)

Differentiate w.r.t x

f'(x)=\frac{dy}{dx}=\frac{1}{2\sqrt x}

By using the formula \frac{d(\sqrt x)}{dx}=\frac{1}{2\sqrt x}

Substitute x=4

Slope of tangent

m=f'(x)=\frac{1}{2\sqrt 4}=\frac{1}{2\times 2}=\frac{1}{4}

In given question

m=\lim_{x\rightarrow a}\frac{f(x)-f(a)}{x-a}

\frac{1}{4}=\lim_{x\rightarrow 4}\frac{f(x)-f(4)}{x-4}

By comparing we get a=4

Point-slope form

y-y_1=m(x-x_1)

Using the formula

The equation of tangent at point (4,2)

y-2=\frac{1}{4}(x-4)

4y-8=x-4

x-4y-4+8=0

x-4y+4=0

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Answer:

z=\frac{0.7-0.07}{\sqrt{0.385(1-0.385)(\frac{1}{215}+\frac{1}{215})}}=13.42  

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So the p value is a very low value and using any significance level given \alpha=0.01 we se that p_v so we can conclude that we have enough evidence to reject the null hypothesis, and we can conclude that vinyl gloves have a greater virus leak rate than latex gloves at 1% of significance.

Step-by-step explanation:

Data given and notation  

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n_{L}=215 sample of latex gloves selected

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z would represent the statistic (variable of interest)  

p_v represent the value for the test (variable of interest)

Concepts and formulas to use  

We need to conduct a hypothesis in order to check if the vinyl gloves have a greater virus leak rate than latex gloves , the system of hypothesis would be:  

Null hypothesis:p_{V} \leq p_{L}  

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We need to apply a z test to compare proportions, and the statistic is given by:  

z=\frac{p_{V}-p_{L}}{\sqrt{\hat p (1-\hat p)(\frac{1}{n_{V}}+\frac{1}{n_{L}})}}   (1)

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Replacing in formula (1) the values obtained we got this:  

z=\frac{0.7-0.07}{\sqrt{0.385(1-0.385)(\frac{1}{215}+\frac{1}{215})}}=13.42  

Statistical decision

Since is a one right tailed test the p value would be:  

p_v =P(Z>13.42)\approx 0  

So the p value is a very low value and using any significance level given \alpha=0.01 we se that p_v so we can conclude that we have enough evidence to reject the null hypothesis, and we can conclude that vinyl gloves have a greater virus leak rate than latex gloves at 1% of significance.

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