Show that if f(z) is a nonconstant analytic function on a domain D, then the image under f(z) of any open set is open. Remark. This is the open mapping theorem for analytic functions. The proof is easy when f ′
(z)
=0, since the Jacobian of f(z) coincides with ∣f ′
(z)∣ 2
. Use Exercise 9 to deal with the points where f ′
(z) is zero.
Answers
The open mapping theorem for analytic functionsThe open mapping theorem for analytic functions can be expressed as follows:
If f(z) is a non-constant analytic function in a domain D, then the image under f(z) of any open set is open.Proof of the open mapping theorem for analytic functions:Let f(z) be an analytic function in a domain D. Suppose that f ′(z) ≠ 0 for all z ∈ D and let S be any open set in D. Let w be a point in the image of S, i.e., there exists z ∈ S such that w = f(z).
Consider any point ζ in the image of S. Since f(z) is analytic and f′(z) ≠ 0 in D, we can use the inverse function theorem to conclude that there exists a neighborhood N of z in D such that f(z) is a one-to-one analytic function of z in N.Let u = f′(z). Since u ≠ 0, it follows that u has a continuous inverse v in a neighborhood of f(z). We can choose the branch of v such that v(f(z)) = z. Then, we have f(v(w)) = w for all w in a neighborhood of f(z). Hence, w is an interior point of the image of S. Therefore, the image of S is open.If there exists a point z0 in D such that f′(z0) = 0, then we use Exercise 9 to deal with this point.
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Related Questions
This is 2 parts of one of my practice problems. The current age used for the first question is 30 and the retirement age is 58. The amount wanted to save is $1,060,123.
a) You and your family would like to have a $X saving at the end of the year you retire. You are planning to retire at the age of Y. Given your age today (please specify an age, which doesn’t have to reflect your true age), and planning to make $400 monthly deposits, what rate should you earn annually to reach your retirement goal? (Hint: Use Rate function)
b) You would like to buy a car with a loan that charges APR of 3.69% per year compounded monthly, (3.69%/12 per month). You borrow $40,000 and promised to pay monthly in 5 years (5*12=60 months). What would be your monthly payments?
Thank you!
Answers
A retirement savings goal of $1,060,123 by the age of 58, while starting at the age of 30 and making monthly deposits of $400, an annual interest rate of 3.69% compounded monthly and agrees to make monthly payments over a period of 5 years.
a) To determine the required annual interest rate to reach the retirement savings goal, the Rate function can be used in financial calculations. The known values in this scenario are the starting age (30), the retirement age (58), the desired savings amount ($1,060,123), and the monthly deposits ($400). By using the Rate function, the interest rate required to achieve the goal can be calculated. The formula for the Rate function is Rate(Nper, PMT, PV, FV). In this case, Nper represents the number of periods (in years), PMT represents the monthly deposit amount, PV represents the present value (initial savings), and FV represents the future value (retirement savings goal). By plugging in the given values, the function can determine the required interest rate.
b) To calculate the monthly payments for a car loan, the known values are the borrowed amount ($40,000), the annual percentage rate (APR) of 3.69%, and the loan term of 5 years (or 60 months). The monthly interest rate is calculated by dividing the APR by 12 (to reflect monthly compounding). Using the loan formula for monthly payments, which is PMT = (P * r * (1 + r)^n) / ((1 + r)^n - 1), where PMT represents the monthly payment, P represents the principal amount (borrowed amount), r represents the monthly interest rate, and n represents the number of periods (in this case, the total number of months).
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f(x) = x2 + 1 and g(x) = x2 - 1.Step 1 of 3: Find five ordered pairs that satisfy the function, f(x) = x2 + 1.AnswerHow to enter your answer (opens in new window){(1,0)(C).O.O.O).O.O.O.O.O.O)})C
Answers
To get the ordered pairs that satisfy the function x, we will assign any value to x in order to get corresponding values of f(x).
Since there is no restriction on the values to assign to x from the question,
let x = -4, -2, 0, 2, 4
\(\begin{gathered} when\text{ x = -4} \\ f(x)\text{ = \lparen-4\rparen}^2\text{ + 1} \\ f(x)\text{ = 16 + 1 = 17} \\ \\ when\text{ x = -2} \\ f(x)\text{ = \lparen-2\rparen}^2\text{ + 1} \\ f(x)\text{ = 4+ 1 = 5} \end{gathered}\)\(\begin{gathered} when\text{ x = 0} \\ f(x)\text{ = 0}^2\text{ + 1} \\ f(x)\text{ = 0 +1 = 1} \\ \\ when\text{ x = 2} \\ f(x)\text{ = 2}^2\text{ + 1} \\ f(x)\text{ = 4+ 1 = 5} \\ \\ when\text{ x = 4} \\ f(x)\text{ = 4}^2\text{ + 1 } \\ f(x)\text{ = 16+ 1 = 17} \end{gathered}\)The ordered pair will be written in form: (x, f(x))
\(\lbrace(-4,\text{ 17\rparen, \lparen-2, 5\rparen, \lparen0, 1\rparen, \lparen2, 5\rparen, 4, 17\rparen\textbraceright}\)suppose that the population of one endangered species decreases at a rate of 4% per year. in one habitat, the current population of the species is 143. after how long will the population drop below 30? round the answer to the nearest tenth.
Answers
The population of the species in the habitat will drop below 30 in approximately 38.5 years.
To calculate this, we can use the exponential decay formula:
\(N(t) = N_0 * e^{-rt}\)
Where N(t) is the population at time t, N₀ is the initial population, r is the annual rate of decrease (as a decimal), and e is Euler's number (approximately 2.71828).
In this case, N₀ = 143, r = 0.04, and we want to find t when N(t) = 30. So we can solve for t as follows:
\(30 = 143 * e^{-0.04t}\)
\(ln(30/143) = -0.04t\)
\(t = -ln(30/143) / 0.04\)
t ≈ 38.5
Therefore, the population will drop below 30 in approximately 38.5 years. It's important to note that this is just an estimate based on the given rate of decrease, and external factors could affect the actual rate of decline. Additionally, conservation efforts could potentially slow or reverse the decline of the species.
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how do I set this up and how do I know what am I looking for
Answers
We have to calculate the number of seconds for the flare to return to the sea.
We have an initial vertical speed of 448 ft/s.
The only force acting on the flare is gravity.
Then, we can model the heigth in function of time using the equation:
\(h(t)=h_0+v_yt-\frac{g}{2}t^2\)where h0 = 0 ft, vy = 448 ft/s and g = 32 ft/s².
Then, we can replace and calculate t for h(t) = 0, that is the time it took the flare to come back to the initial height (that is 0 ft):
\(\begin{gathered} h(t)=0 \\ h_0+v_yt-\frac{g}{2}t^2=0 \\ 0+448t-\frac{32}{2}t^2=0 \\ 448t=16t^2 \\ 448=16t \\ t=\frac{448}{16} \\ t=28 \end{gathered}\)Answer: it will take 28 seconds for the flare to return to the sea [Third option].
If 2x² – 2y = 125 and x - y = 2.5, find the value of
x + y
Answers
We can solve this using substitution.
because x-y=2.5
x=y+2.5
Now substitute x into other equation.
2(y+2.5)^2-2y=125
2y^2+10y+12.5-2y=125
2y^2+8y-112.5=0
Now comes the ugly part
y=-4+-sqrt(16+900)
y=-4+-sqrt916
y=-4+-2sqrt229
x=-1.5+-2sqrt229
NEED HELP ASAP THIS IS OVERDUE !!!
Answers
Answer:
A
Step-by-step explanation:
The slope goes up by 11 and over by 1.
Answer: D
Step-by-step explanation:
y= \(\frac{1}{55}\)×
. the fraction of the variation in the values of a response y that is explained by the least-squares regression of y on x isquestion 15 options:a) the correlation coefficient.b) the slope of the least-squares regression line.c) the coefficient of determination.d) the intercept of the least-squares regression line.
Answers
The coefficient of determination measures the fraction of variability in the values of y that is explained by the least-square regression of y on x.
The fraction of variance of the regressand y cannot be explained, i.e., which is not correctly predicted by the explanatory variables x.
Fraction of variance forms the ratio of mean squared prediction error over the total variance of the response (except subtracting off an estimate of trial-to-trial variability from both) and subtracts this ratio from one.
Regression involves the determination of the degree of relationship in the patterns of variation of two or more variables through the calculation of the coefficient of correlation, r.
Therefore the fraction of the variation in the values of a response y that is explained by the least-squares regression of y on x is the coefficient of determination.
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Ethan serves 6.7 litres of soda at the drive-thru each hour. How many litres of soda will Ethan serve in 5 hours?
Answers
Answer:
33.5 litres
Step-by-step explanation:
6.7 × 5 = 33.5
Answer:
33.5 litres of soda
Step-by-step explanation:
Amount of litres of soda Ethan serves at the drive-thru per hour= 6.7 L
Amount of litres of soda Ethan serves at the drive-thru in 5 hours= 6.7 x 5
= 33.5 L
∴ Ethan serves 33.5 lires of soda in 5 hours.
suppose babies born in a large hospital have a mean weight of 3215 grams, and a variance of 84,681 . if 67 babies are sampled at random from the hospital, what is the probability that the mean weight of the sample babies would be less than 3174 grams? round your answer to four decimal places.
Answers
The probability that the mean weight of the sample babies would be less than 3174 grams is 0.1237 (rounded to four decimal places).
Given that the mean weight of babies born in a large hospital is 3215 grams and the variance is 84681. A sample of 67 babies is chosen at random from the hospital. We need to find the probability that the mean weight of the sample babies is less than 3174 grams.
To solve this, we can use the central limit theorem, which states that the sample means of a large sample (n > 30) taken from a population with a mean μ and a standard deviation σ will be approximately normally distributed with a mean μ and a standard deviation σ / √n.
Here,
n = 67,
μ = 3215 and
σ² = 84681.
σ = √σ² = √84681 = 290.8191
σ / √n = 290.8191 / √67 = 35.4465
To find the probability that the sample mean weight of the babies is less than 3174 grams, we need to find the z-score.
z = (x - μ) / (σ / √n) = (3174 - 3215) / 35.4465 = -1.1572
From the standard normal distribution table, we find that the probability of z being less than -1.1572 is 0.1237.
Therefore, the probability that the mean weight of the sample babies would be less than 3174 grams is 0.1237 (rounded to four decimal places).
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Jaxon must sell at least 49 rolls of wrapping paper to support the robotics club fundraiser. He has already sold 24 rolls of wrapping paper. Which inequality best represents the number of rolls of wrapping paper Jaxon still needs to sell
Answers
Answer:
x≥25
Step-by-step explanation:
Step one:
given data
Jaxon must sell at least 49 rolls of wrapping paper to support the robotics club fundraiser
We are told that he has already sold 24 rolls of wrapping paper.
Step two:
let x be the number of rolls remaining to be sold,
hence,
x≥49-24
x≥25
Therefore the inequality for the situation is x≥25
which reads that x must be greater than or equal to 25
Rearrange to make x the subject
Answers
t3/b=x
Step by step explanation
t = bx/3
Take b/3 to the left hand side
= x = 3t/b
Hope it helped !!
Comment below
Answer:
\(x = \frac{3t}{b} \)
Using the table below, determine the unit rate in beats per minute.
minutes
beats
3
171
4
228
5
285
Answers
Answer: Where is the table?
Step-by-step explanation: i wish i could help but im afraid there isnt a table
Find the equation of the line that
is perpendicular to the line y = 3x - 1 and passes through the point (7, 4).
Answers
Answer:
y=-1/3 x + 19/3
Step-by-step explanation:
Solve for x, using the tangent lines
46 x
Check image below:
Answers
The value of x using the tangent lines = 96°
What is arc length?Arc length is more accurately described as the length along the perimeter of any circle or curve (arc). The arc length is the measurement of any distance along the curved path that forms the arc. Arc refers to a segment of a curve or the circumference of a circle.
Each of them is always shaped like a curve. Every straight line between the endpoints of an arc's length is longer than the length of the arc itself (a chord).
x = 1/2 (Major Arc - Minor Arc)
Minor Arc = 84°
Major Arc = 360° - 84° = 276°
Now,
x = 1/2 (276° - 84°)
= 1/2 × 192 °
= 96°
Therefore, the value of x using the tangent lines = 96°.
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Simplify the expression -6w + (-8) + 1 + (-7w)
Answers
Answer:
The expression in simplified form is -13w - 7
Step-by-step explanation:
Given expression is;
-6w + (-8) + 1 + (-7w)
Rearranging the expression and combining alike terms;
-6w + (-7w) + (-8) + 1
-6w - 7w -8 + 1
-13w - 7
The expression in simplified form is -13w - 7
Hence,
The expression in simplified form is -13w - 7
Probabilities. Please help
Answers
In throwing a dice, there are 6 possible outcomes, namely 1, 2, 3, 4, 5 and 6.
P=(no. of favourable outcomes)/(no. of possible outcomes)
When we multiply the probability of an event with no. of trials, the we get the expected frequency of that event.
1) P(getting 1)=1/6
expected frequency of getting 1
=P(getting)1)×no. of trials
=1/6×600=100
But it is given that one is scored 200 times, so it makes a large difference from expected frequency, which is 100.
Therefore, the dice is not fair.
2) P(getting a tail)=0.3
estimate for the number of times the coin will land on a tail
=expected frequency of getting a tail
=P(getting a tail)×no. of trials
=0.3×150=45
3)P(getting a six)=2/3
expected frequency of getting a six
=P(getting a six)×no. of trials
=2/3×300=200
4)P(getting a three)=0.5
expected frequency of getting a three
=P(getting a three)×350
=0.5×350=175
A hiker begins at sea level and descends 25 feet every minute. How long will it take to get to an elevation of -750 feet?
Answers
Answer:
it would take about 7 1/2 minutes.
Step-by-step explanation:
please give brainliest
Answer:
30 minutes
Step-by-step explanation:
The hiker starts at sea level or 0 feet. He descends 25 feet every minute. This can be written in an equation as so, 0 acting as the starting point, f being feet, and m being minutes:
f = 0 - 25m
This simplifies to:
f = - 25m
We can substitute f in for -750
-750 = - 25m
Now, we divide by - 25 by both sides of the equation
-750 ÷ - 25 = 750 ÷ 25 = 30
30 = m
It would take 30 minutes for the hiker to get to - 750 feet in elevation.
I Hope That This Helps! :)
What is 1+2_#_+#+#+#+3?
Answers
Answer:
Noice
Step-by-step explanation:
PLEASE EXPLAIN I DONT KNOW HOW TO DO THIS!Using a standard normal distribution.
Find the probability of getting a value between 1.10 and 1.50.
1) 0.0689
2) 0.9332
3) 0.1357
4) 0.8643
Answers
The probability of getting a value between 1.10 and 1.50 using a standard normal distribution is 0.0689. The correct answer is option 1.
To find the probability of getting a value between 1.10 and 1.50 using a standard normal distribution, we will use the Z-table to look up the corresponding probabilities.
Step 1: Look up the probability for Z = 1.10 in the Z-table. The value you find is the probability of getting a value less than 1.10. Let's call this P(Z < 1.10).
Step 2: Look up the probability for Z = 1.50 in the Z-table. The value you find is the probability of getting a value less than 1.50. Let's call this P(Z < 1.50).
Step 3: To find the probability of getting a value between 1.10 and 1.50, subtract the probability of getting a value less than 1.10 from the probability of getting a value less than 1.50: P(Z < 1.50) - P(Z < 1.10).
Using the Z-table:
P(Z < 1.10) = 0.8643
P(Z < 1.50) = 0.9332
Now, subtract the probabilities:
0.9332 - 0.8643 = 0.0689
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a. what fraction of the total goal distance did sarah run? b. tori ran 2 −3 of the total goal distance. who ran farther – sarah or tori?
Answers
The total goal distance Sarah run is 7/8.
The distance covered by Sarah is found to be more than Tori.
What is the exponents?A number's exponent indicates what many times a number has been multiplied by itself. For instance, 2×2×2×2 can be composed as 2⁴.
If a fraction does have a negative exponent, we take its reciprocal to make this same exponent positive.
If a fraction does have a negative exponent, we take its reciprocal to create the exponent positive.
Some properties of exponents are-
Law of Product: \(a^m \times a^n=a^{m+n}\)Law of Negative Exponent: \(a^{-m}=1 / a^m\)Law of Power of a Power: \(\left(a^m\right)^n=a^{m n}\)Law of Power of a Product: \((a b)^m=a^m b^m\)Law of Quotient: \(a^m / a^n=a^{m-n}\)Now, as pet the given question;
Tori has covered 2⁻³ of the total distance;
Thus, it can be written as;
= 1/2³
= 1/8
Lets consider the total distance covered by both of them is 1.
Then, the distance covered by Sarah will be;
= 1 - 1/8
= 7/8
Thus, the distance covered by Sarah is 7/8.
As, 7/8 > 1/8
Therefore, the distance covered by Sarah is more than Tori.
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Zoe has joined a club that meets and plays fortnite. Membership is $20 for a startup fee and $15 per month
Answers
Answer:
y=15x+20 slope interceot form
PLease help meeeeeeeeeee
Answers
Answer:
1)
1/2 does not belong to the set
All other numbers belong to the Integers (Z).
2)
√7 does not belong to the set.
All other numbers are rational numbers (Q).
dellana company tested 50 products for 75 hours each. during this time, it experienced four breakdowns. compute the number of failures per hour. what is the mean time between failures?
Answers
dellana company tested 50 products for 75 hours each. during this time, it experienced four breakdowns. compute the number of failures per hour, 937.50 hours is the mean time between failures.
What is mean time?The span of time before something occurs or before a given time frame expires. We can utilise the current computers while waiting for the new ones to arrive next week.
Given that,
dellana company tested 50 products
time =75 hours
Thus, total products in hours
= 50 products × 75 hours each
= 3750 product hours
Also said that, 4 breakdowns in 3750 product hours,
so, (4/3750) = 1.067 × 10⁻³ = 0.001067 breakdowns per hour.
Also, an average of 0.001067 breakdowns per hour = λ
therefore, mean time between failures:
= 1/ λ
= 1/0.001067
= 937.50 hours
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There is a line through the origin that divides the region bounded by the parabola y=5x−3x^2 and the x-axis into two regions with equal area. What is the slope of that line?
Answers
The slope of the line that divides the region bounded by the parabola \(y=5x-3x^2\)and the x-axis into two regions with equal area is 5.
To find the slope of the line that divides the region into two equal areas, we need to determine the point of intersection between the parabola and the x-axis. Since the line passes through the origin, its equation will be y = mx, where m represents the slope.
Setting the equation of the parabola equal to zero, we find the x-values where the parabola intersects the x-axis. By solving the equation\(5x - 3x^2 = 0\), we get x = 0 and x = 5/3.
To divide the region into two equal areas, the line must pass through the midpoint between these x-values, which is x = 5/6. Plugging this value into the equation of the line, we have y = (5/6)m.
Since the areas on both sides of the line need to be equal, we can set up an equation using definite integrals. By integrating the equation of the parabola from 0 to 5/6 and setting it equal to the integral of the line from 0 to 5/6, we can solve for m. After performing the integration, we find that m = 5.
Therefore, the slope of the line that divides the region into two equal areas is 5.
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Tom wants to buy his wife a bouquet of flowers for their anniversary. He decides to get them delivered to her work. Tom spends 20 dollars on the roses. He is charged $1.25 per rose and an additional $5 for a card. How many roses did Tom send his wife?.
Answers
Tom sent 16 roses to his wife
Step-by-step explanation:
20/1.25=16
first you have to get the cost of each rose then divide that by 20 to get the total amount of roses received
if two samples a and b had the same mean and sample size, but sample b had a larger standard deviation, which sample would have the wider 95% confidence interval? g
Answers
If two samples have the same mean but different standard deviations, the sample with the larger standard deviation will have a wider 95% confidence interval.
This is because the standard deviation is a measure of the variability of the data, and a larger standard deviation indicates that the data points are more spread out from the mean.
The confidence interval is a range of values that we can be reasonably confident contains the true population mean. The width of the confidence interval depends on the standard error of the mean, which is calculated as the standard deviation of the sample divided by the square root of the sample size.
Since the larger standard deviation in sample b implies a larger standard error of the mean, the confidence interval for sample b will be wider than that of sample a, even though both samples have the same mean and sample size. We will be less precise in estimating the true population mean with sample b than with sample a.
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Simplify the rational
Answers
Answer:
x+8 / x-4
Step-by-step explanation:
Factor the top and bottom (numerator and denominator).
Simplify (cancel what's the same on top and bottom)
See image.
If X =85, standard deviation = 8 and n = 64 construct a 95% confidence interval estimate for the population mean.
Answers
95% confident that the true population mean μ lies between 83.04 and 86.96
Confidence Interval
A confidence interval in statistics refers to the probability that a population parameter will fall between a set of values for a certain proportion of times. Analysts often use confidence intervals than contain either 95% or 99% of expected observations.
First need to determine whether dealing with means or proportions in this problem. The sample and population mean, that are dealing with means.
One sample mean, this mean creating a confidence interval for one sample (1 Sample t Interval)
Normally we would check for conditions, but since this is not formulated as a real-world scenario type problem, it is hard to check for randomness and independence. Therefore excluding conditions from this answer.
The formula for constructing a confidence interval for means is as follows:
x ± t ( σ / \(\sqrt{n}\) )
The variables are:
x = 85
σ = 8
n = 64
Plug these values into the formula for the confidence interval
85 ± t ( 8 / \(\sqrt{64}\))
Finding the Critical Value (t)
In order to find t, use this formula:
\(\frac{1 - C}{2} = A\)
The z-score associated with A will give us t
So plug in the confidence interval 95% (.95) into the formula
\(\frac{1 - 0.95}{2} = 0.25\)
Use the calculator or a t-table to find the z-score associated with this area under the curve
t = 1.96
Constructing Confidence Interval
Finish the confidence interval created
= 85 ± 1.96 ( 8 / \(\sqrt{64}\) )
The confidence interval using this formula, to be
(83.04, 86.96)
Interpreting the Confidence Interval
95% confident that the true population mean μ lies between 83.04 and 86.96
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Joan Smith earns $42,500.00; is married with one child. The tax rate is 2½%. Calculate the tax. Federal exemption is $3,000.00
Answers
Answer:
$987.50
Step-by-step explanation:
To calculate Joan Smith's tax, we need to follow these steps:
Step 1: Determine the taxable income
Taxable Income = Joan Smith's earnings - Federal exemption
Taxable Income = $42,500.00 - $3,000.00 = $39,500.00
Step 2: Calculate the tax amount
Tax amount = Taxable Income * Tax rate
Tax amount = $39,500.00 * 2.5% = $987.50
Therefore, Joan Smith's tax amount is $987.50.
f(x)= -3x^2 - 6x + 1
Answers
The value of the function f(8) would be equal to -239
What is a function?Function is a type of relation, or rule, that maps one input to specific single output.
In mathematics, a function is an expression, rule, or law that describes the relationship between one variable (the independent variable) and another variable (the dependent variable) (the dependent variable). In mathematics and the physical sciences, functions are indispensable for formulating physical relationships.
We are given that the function is
f(x)= -3x² - 6x + 1
Therefore, we have;
-3(8) ²-6(8)+1
-19²-48+1
-240+1
=-239
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The complete question is;
F(x)=−3x ² −6x+1 Find f(8)