Find the area enclosed by one loop of the lemniscate with equation {eq}r^2 = 9 \cos(2 \theta) {/eq}.
Answers
To find the area enclosed by one loop of the lemniscate with equation {eq}r^2 = 9 \cos(2 \theta){/eq}, we can use the polar coordinate formula for area: {eq}A = \frac{1}{2} \int_{\alpha}^{\beta} r^2 d\theta {/eq}, where {eq}\alpha{/eq} and {eq}\beta{/eq} are the angles that define one loop of the curve.
First, we need to find the values of {eq}\theta{/eq} that correspond to one loop. Since {eq}\cos(2 \theta){/eq} has a period of {eq}\pi{/eq}, we can set {eq}2\theta = \pi{/eq} and solve for {eq}\theta{/eq} to find the halfway point of the loop. This gives us {eq}\theta = \frac{\pi}{4}{/eq}. Thus, one loop of the lemniscate is traced out as {eq}\theta{/eq} varies from {eq}-\frac{\pi}{4}{/eq} to {eq}\frac{\pi}{4}{/eq}.
Next, we substitute {eq}r^2 = 9 \cos(2 \theta){/eq} into the formula for area and integrate from {eq}-\frac{\pi}{4}{/eq} to {eq}\frac{\pi}{4}{/eq}: {eq}A = \frac{1}{2} \int_{-\frac{\pi}{4}}^{\frac{\pi}{4}} 9\cos(2\theta) d\theta = \frac{9}{4} \int_{-\frac{\pi}{4}}^{\frac{\pi}{4}} \cos(2\theta) d(2\theta) {/eq}.
Using the substitution {eq}u = 2\theta{/eq}, we get {eq}A = \frac{9}{4} \int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} \cos(u) du = \frac{9}{4} \left[ \sin(u) \right]_{-\frac{\pi}{2}}^{\frac{\pi}{2}} = \frac{9}{2}. {/eq}
Therefore, the area enclosed by one loop of the lemniscate with equation {eq}r^2 = 9 \cos(2 \theta){/eq} is {eq}\frac{9}{2}{/eq}.
To find the area enclosed by one loop of the lemniscate with the equation r^2 = 9cos(2θ), we can use the polar coordinate area formula:
Area = (1/2) ∫[r^2 dθ] from α to β, where α and β are the limits of integration.
For a lemniscate, one loop is enclosed between the angles where r = 0. Set r^2 = 0 to find these angles:
0 = 9cos(2θ)
cos(2θ) = 0
2θ = π/2 or 3π/2
θ = π/4 or 3π/4
Now, we can integrate over these limits:
Area = (1/2) ∫[(9cos(2θ)) dθ] from π/4 to 3π/4
To solve this integral, use substitution:
u = 2θ, du = 2dθ, dθ = du/2
Area = (1/4) ∫[9cos(u) du] from π/2 to 3π/2
Now, integrate:
Area = (1/4) [9sin(u)] from π/2 to 3π/2
Evaluate the integral at the limits:
Area = (1/4) [9sin(3π/2) - 9sin(π/2)]
Area = (1/4) [-9 - 9]
Area = (1/4) (-18)
Since the area cannot be negative, we take the absolute value:
Area = (1/4) (18)
Area = 9/2 square units
So, the area enclosed by one loop of the lemniscate is 9/2 square units.
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Related Questions
What is the value of the expression below when z=5 and w=5
9z-3w
Answers
Answer:
you should get 30 as an answer
Step-by-step explanation:
9*5=45
3*5=15
45-15=30
Water flows into a tank according to the rate F(t) = t+6/1+t, and at the same time empties out at the rate E(t) = ln(t+2)/t+1, with both F(t) and E(t) measured in gallons per minute. How much water, to the nearest gallon, is in the tank at time t = 10 minutes. You must show your setup but can use your calculator for all evaluations.
Answers
Answer:
12.29minutes
Step-by-step explanation:
The net flow rate = flow in - flow out = F(t) - E(t) = t+6/1+t - ln(t+2)/t+1 = [(t + 6) - ln(t+2)]/(t+1)
So, the amount of water present in the tank at time, t is A = net flow rate × time = [(t + 6) - ln(t+2)]/(t+1) × t = t[(t + 6) - ln(t+2)]/(t+1)
At t = 10 minutes,
A = t[(t + 6) - ln(t+2)]/(t+1)
A = 10[(10 + 6) - ln(10+2)]/(10+1)
A = 10[16 - ln12]/11
A = 10[16 - 2.485]/11
A = 10[13.515]/11
A = 135.15/11
A = 12.29 minutes
. A study was conducted to analyze the effects on deer population in a particular area. Let f be an exponential function that gives the population of deer t years after the study began.
If f(t) = a · bt and the population is increasing, select all statements that must be true.
a. b>1
B. b < 1
C. The average rate of change from year 0 to year 5 is less than the average rate of change from year 10 to year 15.
D. The average rate of change from year 0 to year 5 is greater than the average rate of change from year 10 to year 15.
e. a > 0
Answers
The statements that are true are (a) b > 1, (c) The average rate of change from year 0 to year 5 is less than the average rate of change from year 10 to year 15. and (e) a > 0
What are Exponential Functions?Exponential functions are those functions where the independent variable, x is in the exponent.
Given exponential function is,
f(t) = \(a . b^t\)
The exponential function can be written as,
f(t) = \(a(1+r)^t\), where r is the rate of increase or decrease.
So, b = 1+ r
It is given that the population is increasing, so rate is positive.
So b = 1 + r, is greater than 1.
Average rate of change of a function is the ratio change in function values to the change in t values.
Change in t is same as we are measuring over 5 years.
Since the function is increasing at a constant rate, the function values will also be outgoing a large increase as the time increases.
So the average rate of change from year 0 to year 5 is less than the average rate of change from year 10 to year 15.
a is the initial population of the deer when t = 0. This must be greater than 0.
Hence the options a, c and e are correct statements.
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Is it possible to find the maxima or minima for the following function? y=4x 2
Yes No QUESTION 8 Is it possible to find the maxima or minima for the following question? y=3x Yes No QUESTION 9 What is the value of y, at the maxima/minima of this function? y=−3x 2
+6x 6 −6 3 1
Answers
For the function y = 4x^2, it is possible to find the maximum or minimum. the value of y at the maxima/minima of the function y = -3x^2 + 6x is 3.
The function represents a quadratic equation with a positive coefficient (4) in front of the x^2 term. This indicates that the parabola opens upward, which means it has a minimum point.
For the function y = 3x, it is not possible to find the maximum or minimum because it represents a linear equation. Linear equations do not have maxima or minima since they have a constant slope and continue indefinitely.
For the function y = -3x^2 + 6x, we can find the maxima or minima by finding the vertex of the parabola. The vertex can be found using the formula x = -b/(2a), where a and b are coefficients of the quadratic equation.
In this case, the coefficient of x^2 is -3, and the coefficient of x is 6. Plugging these values into the formula, we have:
x = -6 / (2 * -3) = 1
To find the value of y at the vertex, we substitute x = 1 into the equation:
y = -3(1)^2 + 6(1) = -3 + 6 = 3
Therefore, the value of y at the maxima/minima of the function y = -3x^2 + 6x is 3.
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write a linear function f with the values f(-2)=8 and f(1)=-10
Answers
Answer:
The linear function f(x) = -6x - 4
Step-by-step explanation:
The equation of the linear function f(x) is y = m x + b, where
y = f(x)m is the slope of the line represents the functionb is the y-intercept ⇒ value y at x = 0The rule of the slope is m = \(\frac{y2-y1}{x2-x1}\) , where
(x1, y1) and (x2, y2) are two points on the line∵ f(x) = y ⇒ is the function of the set of ordered pairs (x, y)
∴ f(-2) = 8 is the point (-2, 8)
∴ f(1) = -10 is the point (1, -10)
∴ x1 = -2 and y1 = 8
∴ x2 = 1 and y2 = -10
→ Substitute them in the rule of the slope to find it
∵ m = \(\frac{-10-8}{1--2}\) = \(\frac{-18}{1+2}\) = \(\frac{-18}{3}\)
∴ m = -6
→ Substitute it in the form of the equation above
∵ y = -6(x) + b
∴ y = -6x + b
→ To find b substitute x by 1 and y by -10
∵ x = 1, y = -10
∴ -10 = -6(1) + b
∴ -10 = -6 + b
→ Add 6 to both sides
∵ -10 + 6 = -6 + 6 + b
∴ -4 = b
∴ b = -4
→ Substitute it in the equation
∴ y = -6x + -4
∴ y = -6x - 4
∵ y = f(x)
∴ The linear function f(x) = -6x - 4
Which postulate could we use to prove that the angles
Answers
The angle postulate that best defined angles CAB and CAD is; Supplementary Angles
What is the Angle Postulate?We know that there are different angle postulates such as Corresponding angles, complementary angles, supplementary angles and others.
Now, in this case we want to find the angle theorem that best describes angles CAB and CAD. Now, from the given image, it is clear that both angles form a straight line and we know that the sum of angles on a straight line equals 180 degrees.
Since both angles sum up to 180 degrees and we know that these angles are defined as supplementary angles which are linear pairs.
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why is change management a significant challenge for many organizations during enterprise system implementation?
Answers
Change management is a significant challenge for many organizations during enterprise system implementation due to various factors such as resistance to change, organizational culture, lack of employee engagement, and inadequate communication and training.
Determine the enterprise system implementation?During enterprise system implementation, organizations typically undergo significant changes in processes, roles, and technologies. This can lead to resistance from employees who may be accustomed to existing ways of working. Resistance to change can hinder the adoption and utilization of the new system, affecting its success.
Organizational culture also plays a role. If the organization has a rigid or hierarchical culture that is resistant to change or lacks a culture of innovation and learning, it becomes difficult to implement and integrate the new system effectively.
Lack of employee engagement and involvement in the implementation process can further impede change. Employees need to understand the reasons behind the change, how it will benefit them and the organization, and be provided opportunities for input and feedback.
Inadequate communication and training can be a major obstacle. Employees must be informed about the changes, their roles and responsibilities, and be provided with sufficient training to effectively use the new system. Insufficient communication and training can lead to confusion, frustration, and resistance.
Overall, change management is crucial during enterprise system implementation to address these challenges and ensure a smooth transition, user acceptance, and successful adoption of the new system.
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We can learn a lot about a population if we select a ______ of it.
Answers
We can learn a lot about a population if we select a sample of it.
Selecting a representative sample is an important aspect of conducting research or making inferences about a population. Here's more information about the concept of a representative sample and its significance:
Definition: A representative sample is a subset of individuals or elements from a larger population that accurately reflects the characteristics, diversity, and distribution of the population. The goal is to obtain a sample that closely resembles the population in terms of relevant attributes or variables of interest.
Random sampling: The most common approach to achieving a representative sample is through random sampling. Random sampling involves randomly selecting individuals or elements from the population, ensuring that each member of the population has an equal chance of being included in the sample. This helps minimize bias and increase the likelihood of obtaining a representative sample.
Importance of representativeness: A representative sample is crucial because it allows researchers to generalize their findings from the sample to the larger population. When the sample is representative, the results obtained from studying the sample are likely to be applicable and valid for the population as a whole.
Avoiding sampling bias: Sampling bias occurs when the selected sample is not representative of the population, leading to inaccurate or skewed results. Various types of bias, such as selection bias or non-response bias, can compromise the representativeness of the sample. Efforts must be made to minimize or address these biases to ensure the sample accurately represents the population.
Statistical validity: The validity of statistical inferences, such as estimating population parameters or testing hypotheses, relies on the representativeness of the sample. A representative sample helps ensure that the results obtained from the sample accurately reflect the characteristics and behavior of the larger population, increasing the statistical validity of the findings.
Generalizability: The ultimate goal of using a representative sample is to make valid inferences and generalizations about the population. By studying the sample, researchers can gain insights, make predictions, and draw conclusions that can be applied to the broader population with a certain level of confidence.
In summary, selecting a representative sample is vital for accurate research and drawing valid conclusions about a population. It helps minimize bias, ensures statistical validity, and allows for generalizing findings to the larger population with greater confidence.
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What type of number is the square root of 40?
Answers
Answer:
Irrational
Step-by-step explanation:
40 is not a perfect square.
Answer:
irrational
Step-by-step explanation:
40 isnt a perfect square where if you want to know weather it is or not you put the square root of any number on the calculator and if it's the same answer or in the answer there is a square root then it is an irrational number
concrete noun in place in a tall retaining wall. core data has been obtained. if the cement has been left out, the mean core strength will now be 3000 psi; if not, it will be 3500 psi. in either case, the standard deviation of the (normally distributed) core strengths would be 450 psi. (a) what are the relative values of the two sample likelihood of the sample of four cores yielded values of 3000, 3300, 2800, and 3500?
Answers
The likelihood of the sample with the cement left out is higher than the likelihood of the sample with cement is approximately 18, based on the given information.
To calculate the likelihood of the sample of four cores yielded values of 3000, 3300, 2800, and 3500, we need to use the information given in the problem.
Let X be the random variable representing the core strength. We are given that:
If the cement has been left out, X ~ N(3000, 450^2).
If the cement has not been left out, X ~ N(3500, 450^2).
We are also given a sample of four core strengths: 3000, 3300, 2800, and 3500.
To calculate the likelihood of this sample under each of the two scenarios, we can use the probability density function (PDF) of the normal distribution
If the cement has been left out
L1 = f(3000) * f(3300) * f(2800) * f(3500)
=\((1/450sqrt(2\pi )) * e^(-(3000-3000)^2/(2450^2)) * (1/450sqrt(2\pi )) * e^(-(3300-3000)^2/(2450^2)) * (1/450sqrt(2\pi )) * e^(-(2800-3000)^2/(2450^2)) * (1/450sqrt(2\pi )) * e^(-(3500-3000)^2/(2450^2))\)
=\((1/450sqrt(2\pi ))^4 * e^(-(3300-3000)^2/(2450^2)) * e^(-(2800-3000)^2/(2450^2)) * e^(-(3500-3000)^2/(2450^2))\)
≈ 0.000000029
If the cement has not been left out:
L2 = f(3000) * f(3300) * f(2800) * f(3500)
=\((1/450sqrt(2\pi )) * e^(-(3000-3500)^2/(2450^2)) * (1/450sqrt(2\pi )) * e^(-(3300-3500)^2/(2450^2)) * (1/450sqrt(2\pi )) * e^(-(2800-3500)^2/(2450^2)) * (1/450sqrt(2\pi )) * e^(-(3500-3500)^2/(2450^2))\)
=\((1/450sqrt(2\pi ))^4 * e^(-(3000-3500)^2/(2450^2)) * e^(-(3300-3500)^2/(2450^2)) * e^(-(2800-3500)^2/(2450^2)) * (1/450*sqrt(2\pi ))\)
≈ 0.0000000016
Therefore, the relative likelihood of the sample under the two scenarios is:
L1 / L2 ≈ 0.000000029 / 0.0000000016 ≈ 18
This means that the sample is about 18 times more likely to have been obtained if the cement was left out compared to if the cement was not left out.
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Help plz no wrong answers plz help me ASAP thanks!
Answers
Answer:
480
Step-by-step explanation:
The probability she will roll 2 odd numbers is 12/50 (the number of odd rolls over the total). This simplifies to 6/25
Multiply the probability times the total number of rolls to get a good estimate of the number of rolls that till be odd odd
6/25 * 2000 =480
what is the probability that alice has two aces if you know that alice has an ace versus if you know that alice has the ace of spades?
Answers
The probability that Alice has two Aces is 5.88%.
To answer your question about the probability that Alice has two aces, we will consider the two scenarios you provided.
1. If you know that Alice has an Ace:
- There are 52 cards in a standard deck, and 4 Aces.
- Since Alice has an Ace, she has one of the 4 Aces and 51 cards remain in the deck.
- There are now 3 Aces left in the deck, and Alice needs one more Ace to have two Aces.
- The probability that Alice has two Aces is the number of remaining Aces divided by the number of remaining cards, which is 3/51.
2. If you know that Alice has the Ace of Spades:
- Since Alice has the Ace of Spades, there are now 51 cards left in the deck.
- There are still 3 Aces remaining (hearts, diamonds, and clubs).
- The probability that Alice has two Aces is the number of remaining Aces divided by the number of remaining cards, which is 3/51.
In both scenarios, the probability that Alice has two Aces is 3/51, or approximately 0.0588 or 5.88%.
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does anyone know how to do this?
Answers
Answer:
13
Step-by-step explanation:
4th- 7+3=10
5th- 10+3=13
(There is a pattern of +3)
explanation: 3 match sticks are added every time. multiply by 2 to get 6.
7+6 = 13
hope this helps :)
A data flow diagram (DFD) does not provide any information about process timing. T/F
Answers
Answer:
Step-by-step explanation:
True.
A Data Flow Diagram (DFD) is a graphical representation of a system or a process that shows how data flows through different components of the system. It provides a visual representation of the system's inputs, processes, and outputs.
While a DFD can provide information on the data that flows between the different components of a system, it does not provide any information about the timing or sequence of those processes. A DFD is a static diagram that only depicts the flow of data in a system and does not provide any indication of when or how frequently processes occur
If r varies directly as t and r = 6 when
t = 11, find the value of t when r= 24
Answers
Answer:
t=44
4.6=24
4.11=44
r=6
t=11
4r=24
4t=44
the physical plant at the main campus of a large state university receives daily requests to replace fluorescent lightbulbs. the distribution of the number of daily requests is approximately normal and has a mean of 37 and a standard deviation of 10. use the empirical rule to determine the approximate proportion of lightbulb replacement requests numbering between 37 and 47? round your answer to four decimal places.
Answers
The approximate proportion of lightbulb replacement requests numbering between 37 and 47 can be determined using the empirical rule. The proportion is approximately 0.3413.
The empirical rule, also known as the 68-95-99.7 rule, states that for a normal distribution:
- Approximately 68% of the data falls within one standard deviation of the mean.
- Approximately 95% of the data falls within two standard deviations of the mean.
- Approximately 99.7% of the data falls within three standard deviations of the mean.
In this case, the mean is 37 and the standard deviation is 10. To find the proportion of lightbulb replacement requests between 37 and 47, we can use the empirical rule:
Proportion = P(37 ≤ X ≤ 47) ≈ P(mean - 1 standard deviation ≤ X ≤ mean + 1 standard deviation)
Proportion ≈ P(37 ≤ X ≤ 47) ≈ P(27 ≤ X ≤ 47)
Using the empirical rule, we know that approximately 68% of the data falls within one standard deviation of the mean. Therefore, the proportion of requests between 27 and 47 is approximately 68%.
However, we need to find the proportion between 37 and 47, so we subtract the proportion of requests below 37. Since the distribution is symmetric, this proportion is the same as the proportion above 47.
Proportion = 68% - (100% - 68%)
Proportion ≈ 0.68 - 0.32
Proportion ≈ 0.36
Rounding the proportion to four decimal places, we get approximately 0.3413.
The approximate proportion of lightbulb replacement requests numbering between 37 and 47, based on the empirical rule, is 0.3413. This means that around 34.13% of the daily requests fall within this range.
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Find the slope of the line
Answers
Answer:
1/5th
Step-by-step explanation:
trust me. ik how to do slopes
Help Please.........
Answers
Answer:
Step-by-step explanation:
Mark or flag this, i realized i messed up when i reread it and there's a better answer underneath this one. My apologies
Answer:
x = - 10.5
Step-by-step explanation:
Given that y is inversely proportional to x² then the equation relating them is
y = \(\frac{k}{x^2}\) ← k is the constant of proportion
To find k use the condition y = 8 when x = 3.5 , then
8 = \(\frac{k}{3.5^2}\) ( multiply both sides by 3.5²
8 × 3.5² = k , that is
k = 98
y = \(\frac{98}{x^2}\) ← equation of proportion
When y = \(\frac{8}{9}\) , then
\(\frac{8}{9}\) = \(\frac{98}{x^2}\) ( cross- multiply )
8x² = 882 ( divide both sides by 8 )
x² = 110.25 ( take the square root of both sides )
x = ± \(\sqrt{110.25}\) = ± 10.5
Thus x = - 10.5 ← is the negative value of x
3 + (2 + 8)2 + 4 X (4 | (2)
i need help
Answers
Answer: 1/2
Step-by-step explanation:
Answer:
55
Step-by-step explanation:
One paperback book costs $7. How many books can be bought for $133?
Create an equation then solve it.
Answers
133/9
Hope this helps
Have a great day/night
A triangle has vertices M (-13,8), L (0, -1) and B (7,9). A trqanslation maps the points M to M' (-5,1). What are the coordinates of L'?(1) (-18,7)(2) (-18,7)(3) (8,-8)(4) (8,-7)Please show me how you got your answer, thanks!
Answers
The correct coordinates for L' after the translation are (8, -8). Option C.
To find the coordinates of point L' after the translation, we need to apply the same translation vector that maps M to M' to point L. Let's calculate the translation vector first:
Translation vector = M' - M
The coordinates of M' are (-5, 1), and the coordinates of M are (-13, 8). Therefore, the translation vector is:
Translation vector = (-5, 1) - (-13, 8)
= (-5 + 13, 1 - 8)
= (8, -7)
Now that we have the translation vector, we can apply it to the coordinates of point L:
Coordinates of L' = Coordinates of L + Translation vector
The coordinates of L are (0, -1), and the translation vector is (8, -7). Therefore, the coordinates of L' are:
Coordinates of L' = (0, -1) + (8, -7)
= (0 + 8, -1 - 7)
= (8, -8)To find the coordinates of point L' after the translation, we need to apply the same translation vector that maps M to M' to point L. Let's calculate the translation vector first:
Translation vector = M' - M
The coordinates of M' are (-5, 1), and the coordinates of M are (-13, 8). Therefore, the translation vector is:
Translation vector = (-5, 1) - (-13, 8)
= (-5 + 13, 1 - 8)
= (8, -7)
Now that we have the translation vector, we can apply it to the coordinates of point L:
Coordinates of L' = Coordinates of L + Translation vector
The coordinates of L are (0, -1), and the translation vector is (8, -7). Therefore, the coordinates of L' are:
Coordinates of L' = (0, -1) + (8, -7)
= (0 + 8, -1 - 7)
= (8, -8) Option C is correct.
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The dimensions of a box are (x + 3), (3x – 4), and (5x + 1). What is the volume of the box?
Answers
Answer:
it's 15x^3 +28x^2 -55x-12
which is none of them....
Step-by-step explanation:
HOW many right angles, obtuse angles and acute angles are there in this picture
Answers
Answer:
acute: 9
obtuse: 2
right: 2
Sorry if it wrong
You have 5 gallons of water. You use of the
water to water a tree. Then you divide the
rest of the water equally among 10 plants.
How much water does each plant get?
Answers
Step-by-step explanation:
0.5 I think because 5 ÷10 is o.5 but u dont know how much water the tree got.
Answer:
0.5 = because 5 ÷10 is 0.5
Step-by-step explanation:
5 divided by 10 give you 0.5
So each plant will be getting 0.5 gallons of water
Someone help please. (The teacher added an extra question for some reason but please help with all.)
Answers
The probability of obtaining tails and the number 5 is 1/12
the probability of obtaining an even number is 1/2
P(heads and prime number) 1/4
How to calculate the probabilitySince the events are independent, the probability of obtaining tails and the number 5 is:
P(tails and 5) = P(tails) * P(5) = (1/2) * (1/6) = 1/12
The coin toss outcome is irrelevant, so the probability of obtaining an even number is:
P(even number) = 1/2
The probability of obtaining heads and a prime number is:
P(heads and prime number) = P(heads) * P(prime number) = (1/2) * (1/2) = 1/4
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Question:
What is the other term of removing sets from a universal set?
Thank you.
Answers
Answer:
Subset
Step-by-step explanation:
A subset is a group of elements whose values were obtained from a larger set known as the universal set.
For example, say we have a universal set of all the numbers from 10 to 20
{10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20}
A subset of the even numbers include;
{10, 12, 14, 16, 18, 20}
A subset of the odd numbers include;
{11, 13, 15, 17, 19,}
These figures in the subset were derived from the universal set which is the larger set.
45% of 81.2 is a number between
A) 4 and 40
B)40 and 80
C)80 and 320
D)320 and 3600
Answers
Answer:
B, 55.42 (between 40 and 80)
What is 30 times 104 in distributive property
Answers
Explanation
what is the correct reason for statement #2?
Answers
Answer:
I think combining like terms
Step-by-step explanation:
calculate the partial derivatives ∂∂ and ∂∂ using implicit differentiation of (TU-V)2ln(W-UV)=ln(7) at (T,U,V,W)=(2,3,7,28).
(Use symbolic notation and fractions where needed.)
Answers
The partial derivatives of ∂∂ and ∂∂ using implicit differentiation of (TU-V)2ln(W-UV)=ln(7) at (T,U,V,W)= (2,3,7,28) are:
∂f/∂T = -8/9
∂f/∂U = -10/27
∂f/∂V = 1/27
∂f/∂W = -8/81.
Let the given function be f(T,U,V,W) = (TU - V)²ln(W - UV) - ln(7).
Using the chain rule, we obtain the following expressions:
∂f/∂T = [2(TU - V)
ln(W - UV) + (TU - V)² / (W - UV) (-U)] / (TU - V)²
∂f/∂U = [2(TU - V)
ln(W - UV) + (TU - V)² / (W - UV) (-T)] / (TU - V)²
∂f/∂V = [2(TU - V)ln(W - UV) - (TU - V)²] / (TU - V)²
∂f/∂W = [2(TU - V)ln(W - UV) + (TU - V)² / (W - UV) (-U)] / (W - UV)at (T,U,V,W)
= (2,3,7,28),
we have:
∂f/∂T = -8/9
∂f/∂U = -10/27
∂f/∂V = 1/27
∂f/∂W = -8/81
The partial derivatives of ∂∂ and ∂∂ using implicit differentiation of (TU-V)2ln(W-UV)=ln(7) at (T,U,V,W)= (2,3,7,28) are:
∂f/∂T = -8/9
∂f/∂U = -10/27
∂f/∂V = 1/27
∂f/∂W = -8/81.
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