# Prove these examples are correct: a) What is the area of the largest rectangle that fits inside of the ellipse x^{2} + 2y^{2} = 1? b) Prove the following: Let c in (a, b). If f is continuous on [a, b], differentiable on (a, b)?

Question
Transformation properties
Prove these examples are correct:
a) What is the area of the largest rectangle that fits inside of the ellipse
$$x^{2}\ +\ 2y^{2} = 1?$$
b) Prove the following: Let c in (a, b). If f is continuous on $$[a,\ b],$$ differentiable on (a, b)?

2021-02-26
a) Сonsider this figure:

From the figure it can be seen that:
Area $$(A) = 4xy$$
And also,
$$x^{2}\ +\ 2y^{2} = 1$$
$$x = \sqrt{1\ -\ 2y^{2}$$
We've taken the positive value since we chose this point to be in the first quadrant
So now deciding:
$$A = 4xy$$
$$A = 4y \sqrt{1\ -\ 2y^{2}$$
Differentiating the above function with respect to "y":
$$\frac{dA}{dy}=\frac{d}{dy}\left(4y\sqrt{1\ -\ 2y^{2}}\right)$$
$$\frac{dA}{dy}=4y\frac{d}{dy}\sqrt{1\ -\ 4y^{2}}\ +\ \sqrt{1\ -\ 4y^{2}}\frac{d}{dy}(4y)$$
$$\frac{dA}{dy}=4y\frac{1}{2\sqrt{1\ -\ 4y^{2}}}(-8y)\ +\ 4\sqrt{1\ -\ 4y^{2}}$$
$$\frac{dA}{dy}=4\left[\frac{-8y^{2}\ +\ 2\ -\ 8y^{2}}{2\sqrt{1\ -\ 4y^{2}}}\right]$$
$$\frac{dA}{dy}=4\left[\frac{2\ -\ 16y^{2}}{2\sqrt{1\ -\ 4y^{2}}}\right]$$
For maximize the area:
Put,
$$\frac{dA}{dy} = 0$$
$$\frac{dA}{dy}=4\left[\frac{2\ -\ 16y^{2}}{2\sqrt{1\ -\ 4y^{2}}}\right]$$
$$\left[\frac{2\ -\ 16y^{2}}{2\sqrt{1\ -\ 4y^{2}}}\right]=0$$
$$2\ -\ 16y^{2} = 0$$
$$y^{2} = \frac{1}{8}$$
$$y = \frac{1}{\sqrt{8}}$$
Corresponding to this,
$$x=\sqrt{1\ -\ 2\ \times\ \frac{1}{8}}$$
$$x = \sqrt {1\ -\ \frac{1}{4}}$$
$$x = \sqrt{\frac{3}{4}}$$
$$x = \frac{\sqrt{3}}{2}}$$
Hence the maximum area:
Area $$(A) = 4xy$$
$$Area_{max} = 4\ \times\ \frac{\sqrt{3}}{2}\ \times\ \frac{1}{\sqrt{8}}$$
$$Area_{max} = 2\ \times\ \frac{\sqrt{3}}{2 \sqrt{2}}$$
$$Area_{max} = \sqrt{\frac{3}{2}}$$
b)Prove the following: Let c in (a, b). If f is continuous on [a, b], differentiable on (a, b), and:
$$\lim_{x\ \rightarrow\ x}\ f'(x)=L\ then\ f'(c)=L$$
Properties used
$$\lim_{h\ \rightarrow\ 0}\ \frac{f'(x\ +\ h)\ -\ f(x)}{h}=f'(x)$$
Proof is given below:
Since:
$$\lim_{x\ \rightarrow\ c} f'(x) = L$$
By using the property:
$$\lim_{x\ \rightarrow\ c}\ \left[\lim_{h\ \rightarrow\ 0}\frac{f'(x\ +\ h)\ -\ f(x)}{h}\right]=L$$
$$\lim_{h\ \rightarrow\ 0}\ \left[\lim_{x\ \rightarrow\ c}\frac{f(x\ +\ h)\ -\ f(x)}{h}\right]=L$$
$$[:' f(x) is continuous]$$
$$\lim_{h\ \rightarrow\ 0}\ \left[\frac{f(c\ +\ h)\ -\ f(c)}{h}\right]=L$$
$$\left[\lim_{h\ \rightarrow\ 0}\ \frac{f(c\ +\ h)\ -\ f(c)}{h}\right]=L$$
$$f' (c) = L$$
a) $$Area_{max} = \sqrt{\frac{3}{2}}$$
b) The proof is given above.

### Relevant Questions

Prove that the metric area is defined as $$\displaystyle{P}\ {<}\ {x}_{{{1}}},\ {y}_{{{1}}}\ {>}\ {\quad\text{and}\quad}\ {Q}\ {<}\ {x}_{{{2}}},\ {y}_{{{2}}}\ {>}$$. If the proof of examples says that the first properties (positive definiteness and symmetry) are trivial. Prove the versatility of properties for a given space.
Let $$T\ :\ U\ \rightarrow\ U$$ be a linear transformatiom and let
$$\mathscr{B}$$ be a basis of U. Define the determanant det (T) of T as
$$det(T) = det([T]\ \mathscr{B}).$$
Show that det (T) is well=defined, i.e. that it does not depend on the choice of the basis $$\mathscr{B}.$$
Prove that T is invertible if and only if det $$(T) \neq\ 0.$$ If T is invertible, show that
$$det(T^{-1}) =\ \frac{1}{det(T)}.$$
Guided Proof Let $${v_{1}, v_{2}, .... V_{n}}$$ be a basis for a vector space V.
Prove that if a linear transformation $$T : V \rightarrow V$$ satisfies
$$T (v_{i}) = 0\ for\ i = 1, 2,..., n,$$ then T is the zero transformation.
To prove that T is the zero transformation, you need to show that $$T(v) = 0$$ for every vector v in V.
(i) Let v be the arbitrary vector in V such that $$v = c_{1} v_{1} + c_{2} v_{2} +\cdots + c_{n} V_{n}$$
(ii) Use the definition and properties of linear transformations to rewrite T(v) as a linear combination of $$T(v_{j})$$ .
(iii) Use the fact that $$T (v_{j}) = 0$$
to conclude that $$T (v) = 0,$$ making T the zero transformation.
Let $$T : U \rightarrow U$$ be a linear transformation and let beta be a basis of U Define the determinant det(T) of T as
det$$(T) = det([T]_{\beta}).$$
Show ta det (T) is well-defined, i. e. that it does not depend on the choice of the basis beta
Prove that T is invertible if and only if det $$(T) \neq 0.$$ If T is invertible, show that det $$(T^{-1}) = \frac{1}{det(T)}$$
1. Find each of the requested values for a population with a mean of $$? = 40$$, and a standard deviation of $$? = 8$$ A. What is the z-score corresponding to $$X = 52?$$ B. What is the X value corresponding to $$z = - 0.50?$$ C. If all of the scores in the population are transformed into z-scores, what will be the values for the mean and standard deviation for the complete set of z-scores? D. What is the z-score corresponding to a sample mean of $$M=42$$ for a sample of $$n = 4$$ scores? E. What is the z-scores corresponding to a sample mean of $$M= 42$$ for a sample of $$n = 6$$ scores? 2. True or false: a. All normal distributions are symmetrical b. All normal distributions have a mean of 1.0 c. All normal distributions have a standard deviation of 1.0 d. The total area under the curve of all normal distributions is equal to 1 3. Interpret the location, direction, and distance (near or far) of the following zscores: $$a. -2.00 b. 1.25 c. 3.50 d. -0.34$$ 4. You are part of a trivia team and have tracked your team’s performance since you started playing, so you know that your scores are normally distributed with $$\mu = 78$$ and $$\sigma = 12$$. Recently, a new person joined the team, and you think the scores have gotten better. Use hypothesis testing to see if the average score has improved based on the following 8 weeks’ worth of score data: $$82, 74, 62, 68, 79, 94, 90, 81, 80$$. 5. You get hired as a server at a local restaurant, and the manager tells you that servers’ tips are $42 on average but vary about $$12 (\mu = 42, \sigma = 12)$$. You decide to track your tips to see if you make a different amount, but because this is your first job as a server, you don’t know if you will make more or less in tips. After working 16 shifts, you find that your average nightly amount is$44.50 from tips. Test for a difference between this value and the population mean at the $$\alpha = 0.05$$ level of significance.
The table below shows the number of people for three different race groups who were shot by police that were either armed or unarmed. These values are very close to the exact numbers. They have been changed slightly for each student to get a unique problem.
Suspect was Armed:
Black - 543
White - 1176
Hispanic - 378
Total - 2097
Suspect was unarmed:
Black - 60
White - 67
Hispanic - 38
Total - 165
Total:
Black - 603
White - 1243
Hispanic - 416
Total - 2262
Give your answer as a decimal to at least three decimal places.
a) What percent are Black?
b) What percent are Unarmed?
c) In order for two variables to be Independent of each other, the P $$(A and B) = P(A) \cdot P(B) P(A and B) = P(A) \cdot P(B).$$
This just means that the percentage of times that both things happen equals the individual percentages multiplied together (Only if they are Independent of each other).
Therefore, if a person's race is independent of whether they were killed being unarmed then the percentage of black people that are killed while being unarmed should equal the percentage of blacks times the percentage of Unarmed. Let's check this. Multiply your answer to part a (percentage of blacks) by your answer to part b (percentage of unarmed).
Remember, the previous answer is only correct if the variables are Independent.
d) Now let's get the real percent that are Black and Unarmed by using the table?
If answer c is "significantly different" than answer d, then that means that there could be a different percentage of unarmed people being shot based on race. We will check this out later in the course.
Let's compare the percentage of unarmed shot for each race.
e) What percent are White and Unarmed?
f) What percent are Hispanic and Unarmed?
If you compare answers d, e and f it shows the highest percentage of unarmed people being shot is most likely white.
Why is that?
This is because there are more white people in the United States than any other race and therefore there are likely to be more white people in the table. Since there are more white people in the table, there most likely would be more white and unarmed people shot by police than any other race. This pulls the percentage of white and unarmed up. In addition, there most likely would be more white and armed shot by police. All the percentages for white people would be higher, because there are more white people. For example, the table contains very few Hispanic people, and the percentage of people in the table that were Hispanic and unarmed is the lowest percentage.
Think of it this way. If you went to a college that was 90% female and 10% male, then females would most likely have the highest percentage of A grades. They would also most likely have the highest percentage of B, C, D and F grades
The correct way to compare is "conditional probability". Conditional probability is getting the probability of something happening, given we are dealing with just the people in a particular group.
g) What percent of blacks shot and killed by police were unarmed?
h) What percent of whites shot and killed by police were unarmed?
i) What percent of Hispanics shot and killed by police were unarmed?
You can see by the answers to part g and h, that the percentage of blacks that were unarmed and killed by police is approximately twice that of whites that were unarmed and killed by police.
j) Why do you believe this is happening?
Do a search on the internet for reasons why blacks are more likely to be killed by police. Read a few articles on the topic. Write your response using the articles as references. Give the websites used in your response. Your answer should be several sentences long with at least one website listed. This part of this problem will be graded after the due date.
Which of the following are linear transformations from $$RR^{2} \rightarrow RR^{2} ?$$
(d) Rotation: if $$x = r \cos \theta, y = r \sin \theta,$$ then
$$\overrightarrow{T}(x,y)=(r \cos(\theta+ \varphi), r \sin (\theta+ \varphi))$$
for some constants $$\angle \varphi$$
(f) Reflection: given a fixed vector $$\overrightarrow{r} = (a, b), \overrightarrow{T}$$ maps each point to its reflection with
respect to $$\overrightarrow{r} \overrightarrow{T}(\overrightarrow{x})=\overrightarrow{x}-2\overrightarrow{x}_{r \perp}$$
$$=2 \overrightarrow{x}_{r}-\overrightarrow{x}$$
Let C be the ellipse contained in the xy plane whose equation is $$\displaystyle{4}{x}^{{2}}+{y}^{{2}}={4}$$, oriented clockwise. The force field F described by $$\displaystyle{F}{\left({x},{y},{z}\right)}={x}^{{2}}{i}+{2}{x}{j}+{z}^{{2}}{k}$$, moves a particle along C in the same direction as the curve orientation, performing a W job. C as the surface boundary S: $$\displaystyle{z}={4}-{4}{x}^{{2}}-{y}^{{2}},{z}\ge{0}$$ (with ascending orientation, that is, the component in the z direction equal to 1) and assuming $$\displaystyle\pi={3.14}$$, we can state what:
a) It is not necessary to apply Stokes' Theorem, as C is a closed curve and therefore W = 0.
b) Inverting the orientation of the surface S, we can apply Stokes' Theorem and conclude that W = 12.56.
c) We can apply Stokes' Theorem and conclude that W = 6.28
d) We can apply Stokes' Theorem and conclude that W = 12.56.
Assum T: R^m to R^n is a matrix transformation with matrix A. Prove that if the columns of A are linearly independent, then T is one to one (i.e injective). (Hint: Remember that matrix transformations satisfy the linearity properties.
Linearity Properties:
If A is a matrix, v and w are vectors and c is a scalar then
$$A 0 = 0$$
$$A(cv) = cAv$$
$$A(v\ +\ w) = Av\ +\ Aw$$
A random sample of $$\displaystyle{n}_{{1}}={16}$$ communities in western Kansas gave the following information for people under 25 years of age.
$$\displaystyle{X}_{{1}}:$$ Rate of hay fever per 1000 population for people under 25
$$\begin{array}{|c|c|} \hline 97 & 91 & 121 & 129 & 94 & 123 & 112 &93\\ \hline 125 & 95 & 125 & 117 & 97 & 122 & 127 & 88 \\ \hline \end{array}$$
A random sample of $$\displaystyle{n}_{{2}}={14}$$ regions in western Kansas gave the following information for people over 50 years old.
$$\displaystyle{X}_{{2}}:$$ Rate of hay fever per 1000 population for people over 50
$$\begin{array}{|c|c|} \hline 94 & 109 & 99 & 95 & 113 & 88 & 110\\ \hline 79 & 115 & 100 & 89 & 114 & 85 & 96\\ \hline \end{array}$$
(i) Use a calculator to calculate $$\displaystyle\overline{{x}}_{{1}},{s}_{{1}},\overline{{x}}_{{2}},{\quad\text{and}\quad}{s}_{{2}}.$$ (Round your answers to two decimal places.)
(ii) Assume that the hay fever rate in each age group has an approximately normal distribution. Do the data indicate that the age group over 50 has a lower rate of hay fever? Use $$\displaystyle\alpha={0.05}.$$
(a) What is the level of significance?
State the null and alternate hypotheses.
$$\displaystyle{H}_{{0}}:\mu_{{1}}=\mu_{{2}},{H}_{{1}}:\mu_{{1}}<\mu_{{2}}$$
$$\displaystyle{H}_{{0}}:\mu_{{1}}=\mu_{{2}},{H}_{{1}}:\mu_{{1}}>\mu_{{2}}$$
$$\displaystyle{H}_{{0}}:\mu_{{1}}=\mu_{{2}},{H}_{{1}}:\mu_{{1}}\ne\mu_{{2}}$$
$$\displaystyle{H}_{{0}}:\mu_{{1}}>\mu_{{2}},{H}_{{1}}:\mu_{{1}}=\mu_{{12}}$$
(b) What sampling distribution will you use? What assumptions are you making?
The standard normal. We assume that both population distributions are approximately normal with known standard deviations.
The Student's t. We assume that both population distributions are approximately normal with unknown standard deviations,
The standard normal. We assume that both population distributions are approximately normal with unknown standard deviations,
The Student's t. We assume that both population distributions are approximately normal with known standard deviations,
What is the value of the sample test statistic? (Test the difference $$\displaystyle\mu_{{1}}-\mu_{{2}}$$. Round your answer to three decimalplaces.)
What is the value of the sample test statistic? (Test the difference $$\displaystyle\mu_{{1}}-\mu_{{2}}$$. Round your answer to three decimal places.)
(c) Find (or estimate) the P-value.
P-value $$\displaystyle>{0.250}$$
$$\displaystyle{0.125}<{P}-\text{value}<{0},{250}$$
$$\displaystyle{0},{050}<{P}-\text{value}<{0},{125}$$
$$\displaystyle{0},{025}<{P}-\text{value}<{0},{050}$$
$$\displaystyle{0},{005}<{P}-\text{value}<{0},{025}$$
P-value $$\displaystyle<{0.005}$$
Sketch the sampling distribution and show the area corresponding to the P-value.
P.vaiue Pevgiue
P-value f P-value
...