The number of shirts Karla owns is in the table.

We are given that the function is defined as follows $f(x) = 9-x, x\leq 0$ and $f(x) = 9+12x, x>0$ and we want to check the continuity in the interval [-4,5]. Note that this a piecewise function whose only critical point (that might be a candidate of a discontinuity) x=0 since at this point is where the function "changes" of definition. Note that 9-x and 9+12x are polynomials that are continous over all $\mathbb{R}$ . So F is continous in the intervals [-4,0) and (0,5]. To check if f(x) is continuous at 0, we must check that

$\lim_{x\to 0^{-}} = f(0) = \lim_{x \to 0^{+} f(x)$ (this is the definition of continuity at x=0)

Note that if x=0, then f(x) = 9-x. So, f(0)=9. On the same time, note that

$\lim_{x\to 0^{-}} f(x) = \lim_{x\to 0^{-}} 9-x = 9$ . This result is because the function 9-x is continous at x=0, so the left-hand limit is equal to the value of the function at 0.

Note that when x>0, we have that f(x) = 9+12x. In this case, we have that

$\lim_{x\to 0^{+}} f(x) = \lim_{x\to 0^{+}} 9+12x = 9$ . As before, this result is because the function 9+12x is continous at x=0, so the right-hand limit is equal to the value of the function at 0.

Thus, $\lim_{x\to 0^{-}} = f(0) = \lim_{x \to 0^{+} f(x)=9$ , so by definition, f is continuous at x=0, hence continuous over the interval [-4,5].

5 0

3 years ago

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Identify any transformations (translations, reflections, rotations) that are used in each corporate logo below. Some logos may h

lord [1]

Answer:

Check below

Step-by-step explanation:

There are rigid transformations that produce congruent figures: translation, rotation, and one transformation that produces a similar figure, namely: Dilation.

1) Toyota: translation, and Dilation.

Two ellipses translated within a larger ellipse.

2) 2 Reflections, or 4 rotations of the pre-image

We have two ways of drawing this figure. Either by reflection of the right and the lower one or by rotation.

3) CBS: Dilation of the circumference, inside a rhombus.

4) Mitsubishi: Reflection about the y axis, Rotation for the figure up.

e) Reflection

f) Translation

6 0

3 years ago

A manufacturing company produces valves in various sizes and shapes. One particular valve plate is supposed to have a tensile st

ss7ja [257]

Answer:

Step-by-step explanation:

Hello!

The researcher wants to test if the valve plates manufactured have the expected tensile strength of 5 lbs/mm. So he took a sample of 42 valve plates and measured their tensile strength, obtaining a sample mean of X[bar]= 5.0611 lbs/mm and a sample standard deviation of S=0.2803 lbs/mm.

The study variable is:

X: tensile strength of a valve plate (lbs/mm)

The parameter of interest is the mean tensile strength of the valve plates, μ.

If the claim is that the valve plates of the sample have on average tensile strength of 5 lbs/mm, symbolically: μ = 5

a) The statistic hypotheses are:

H₀: μ = 5

H₁: μ ≠ 5

b) To determine the critical values and rejection region of a hypothesis test you need three to determine three factors of the hypothesis test:

1) The statistical hypothesis.

2) The significance level.

3) The statistic to use for the analysis.

The statistic hypothesis determines the number of critical values and the direction of the rejection region, in this case, the test is two-tailed you will have two critical values and the rejection region will be divided into two.

With the statistic, you will determine the distribution under which you will work and the significance level determines the probability of rejecting the null hypothesis.

To study the population mean you need that the variable of interest has at least a normal distribution, there is no information about the distribution of the study variable but the sample size is large enough n≥30, so you can apply the central limit theorem to approximate the distribution of the sample mean to normal: X[bar]≈N(μ;σ²/n)

Thanks to this approximation it is valid to use an approximation of the standard normal distribution for the test:

$Z= \frac{X[bar]-Mu}{\frac{Sigma}{\sqrt{n} } }$ ≈N(0;1)

The critical values are:

$Z_{\alpha /2}= Z_{0.05}= -1.648$

$Z_{1-\alpha /2}= Z_{0.95}= 1.648$

You will reject the null hypothesis if $Z_{H_0}$ ≤-1.648 or if $Z_{H_0}$ ≥1.648

You will not reject the null hypothesis if -1.648< $Z_{H_0}$ <1.648

c)

$Z_{H_0}= \frac{X[bar]-Mu}{\frac{S}{\sqrt{n} } }= \frac{5.0611-5}{\frac{0.2803}{\sqrt{42} } }= 1.41$

d) The value of the statistic is between the two critical values so the decision is to not reject the null hypothesis. Then using a significance level of 10% there is no significant evidence to reject the null hypothesis so the valve plates have on average tensile strength of 5 lbs/mm.

e) The p-value is defined as the probability corresponding to the calculated statistic if possible under the null hypothesis (i.e. the probability of obtaining a value as extreme as the value of the statistic under the null hypothesis). If the test is two-tailed, so is the p-value, you can calculate it as:

P(Z≤-1.41) + P(Z≥1.41)= P(Z≤-1.41) + (1 - P(Z≤1.41))= 0.079 + ( 1 - 0.921)= 0.158

p-value: 0.158

I hope it helps!

5 0

3 years ago