The area of [tex]$a$ is $\boxed{1200}$[/tex].
Consider the region $a$ in the complex plane that consists of all points $z$ such that both
$\frac{z}{40}$ and $\frac{40}{\overline{z}}$
have real and imaginary parts between $0$ and $1$, inclusive. We need to find the area of $a$.
First, let's simplify
$\frac{z}{40}$ and $\frac{40}{\overline{z}}$:
[tex]$$\frac{z}{40} = \frac{x + yi}{40} = \frac{x}{40} + \frac{y}{40}i$$$$\frac{40}{\overline{z}} = \frac{40}{x - yi} = \frac{40(x + yi)}{x^2 + y^2} = \frac{40x}{x^2 + y^2} - \frac{40y}{x^2 + y^2}i$$For $\frac{z}{40}$[/tex]
to have real and imaginary parts between $0$ and $1$, we need
$x \in [0, 40]$ and $y \in [0, 40]$.
For $\frac{40}{\overline{z}}$ to have real and imaginary parts between $0$ and $1$, we need $x^2 + y^2 \geq 40x$ (so that $\frac{40x}{x^2 + y^2} \in [0, 1]$) and $x^2 + y^2 \geq 40y$ (so that $\frac{-40y}{x^2 + y^2} \in [-1, 0]$). This simplifies to
[tex]$(x - 20)^2 + y^2 \geq 20^2$.[/tex]
We can plot these inequalities in the $xy$-plane to see that $a$ is a trapezoid with bases $40$ and $80$ and height $20$:The area of $a$ is the average of the lengths of the bases multiplied by the height:
[tex]$$\frac{(40 + 80)}{2} \cdot 20 = 1200$$[/tex]
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charlie and lucinda each have $50,000 invested in stock portfolios. charlie's has a beta of 1.2, an expected return of 10.8%, and a standard deviation of 25%. lucinda's has a beta of 0.8, an expected return of 9.2%, and a standard deviation that is also 25%. the correlation coefficient, r, between charlie's and lucinda's portfolios is zero. if charlie and lucinda marry and combine their portfolios, which of the following best describes their combined $100,000 portfolio?
The combined portfolio would have a weight of 56.2% in Charlie's portfolio and 43.8% in Lucinda's portfolio, with an expected return of 10.0% and a standard deviation of 22.6%. Whether this portfolio is preferable would depend on the investor's risk tolerance and investment objectives.
Based on the information you provided,
The combined portfolio of Charlie and Lucinda would have an expected return and standard deviation that depend on the weights of each portfolio in the combination.
One approach to solving this problem is to use the formula for the expected return and standard deviation of a portfolio that combines two assets:
E(Rp) = w1E(R1) + w2E(R2) σp
= √(w1²σ1² + w2²σ2² + 2w1w2σ1σ2ρ)
Where E(Rp) is the expected return of the combined portfolio,
E(R1) and E(R2) are the expected returns of the individual portfolios,
σ1 and σ2 are their standard deviations,
ρ is the correlation coefficient, and w1 and w2 are the weights of each portfolio in the combination (such that w1 + w2 = 1).
Using this formula, we can calculate the expected return and standard deviation of Charlie's portfolio separately:
E(Rc) = 10.8%
σc = 25%
And we can do the same for Lucinda's portfolio:
E(Rl) = 9.2%
σl = 25%
Since the correlation coefficient between the two portfolios is zero, we can assume that ρ = 0 in the formula.
This simplifies the formula to:
σp = √(w1²σ1² + w2²σ2²)
We can now use this formula to solve for the weights that would give the combined portfolio the highest Sharpe ratio:
Sharpe ratio = (E(Rp) - Rf) / σp
Where Rf is the risk-free rate, which we'll assume is 2%.
Plugging in the values we have, we get,
Sharpe ratio = (w1E(Rc) + w2E(Rl) - Rf) / √(w1²σc² + w2²σl²)
To maximize the Sharpe ratio, we need to take the partial derivatives of this formula with respect to w1 and w2, set them equal to zero, and solve for w1 and w2:
d(SR)/dw1 = (E(Rc) - Rf)σl² - (E(Rl) - Rf)σcσl / (σc² + σl² - 2w1σcσl)[tex]^{(3/2)}[/tex] = 0
d(SR)/dw2 = (E(Rl) - Rf)σc² - (E(Rc) - Rf)σcσl / (σc² + σl² - 2w1σcσl)[tex]^{(3/2)}[/tex]= 0
Solving these equations simultaneously, we get,
w1 = (E(Rc) - Rf)σl² / (E(Rc) - Rf)σl² + (E(Rl) - Rf)σc² w2
= (E(Rl) - Rf)σc² / (E(Rc) - Rf)σl² + (E(Rl) - Rf)σc²
Plugging in the values we have, we get:
w1 = 0.562
w2 = 0.438
Therefore, the combined portfolio would have a weight of 56.2% in Charlie's portfolio and 43.8% in Lucinda's portfolio.
The expected return and standard deviation of the combined portfolio would be:
E(Rp) = w1E(Rc) + w2E(Rl) = 10.0%
σp = √(w1²σc² + w2²σl²) = 22.6%
Therefore, the combined portfolio would have a lower expected return than Charlie's portfolio alone, but a lower standard deviation as well. Whether this portfolio is preferable would depend on the investor's risk tolerance and investment objectives
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Find a formula for the general term an of the sequence, assuming that the pattern of the first few terms continues. {10,6,10,6,10,6,…}
The general term an of the sequence is given by, an = 10 if n is even, an = 6 if n is odd.
The arithmetic mean, also known as the average, is a measure of central tendency used to determine the typical value of a set of numbers. It is calculated by adding up all the values in a set and dividing the sum by the total number of values.
To compute the arithmetic mean of a set of numbers, follow these steps:
Add up all the numbers in the set.
Count the total number of values in the set.
Divide the sum by the total number of values.
Mathematically, the arithmetic mean is represented as:
Mean = (Sum of all values) / (Total number of values)
The arithmetic mean provides a useful summary statistic that represents the "center" of a set of values. It is commonly used in various fields such as statistics, mathematics, economics, and everyday life to describe and analyze data.
The given sequence {10, 6, 10, 6, 10, 6, ...} alternates between the terms 10 and 6. We can observe that the terms 10 and 6 repeat after every two terms.
Therefore, we can express the general term an of the sequence using the concept of modular arithmetic. The remainder of dividing n by 2 determines whether the term should be 10 or 6.
If n is even (n = 2k, where k is an integer), then an = 10.
If n is odd (n = 2k + 1, where k is an integer), then an = 6.
In summary, the general term an of the sequence is given by:
an = 10 if n is even,
an = 6 if n is odd.
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Give regular expressions for the following sets of binary strings:
(a) Strings whose characters at even indexes are 1’s, where the first char- acter in a string is index 0. (Some valid strings include 1, 1111, and 101).
(b) Strings that do NOT end in 11
(a) Here are the regular expressions for the set of binary strings whose characters at even indexes are 1's: [tex]$$1(01)^{*}+0(01)^{*}1$$[/tex]
where [tex]$(01)^{*}$[/tex] represents zero or more occurrences of the pattern 01. The first part of the expression [tex]$1(01)^{*}$[/tex]specifies that the first character is 1 and every even indexed character after that is 1.
The second part of the expression[tex]$0(01)^{*}1$[/tex] specifies that the first character is 0 and every odd indexed character after that is 1.(b) Here is the regular expression for the set of binary strings that do NOT end in 11: [tex]$$0^{*}+1+10^{*}$$[/tex] where [tex]$0^{*}$[/tex] represents zero or more occurrences of 0 and [tex]$10^{*}$[/tex] represents zero or more occurrences of 10. The expression specifies that a string can either have zero or more 0's, or it can end in 1, or it can end in 10 followed by zero or more 0's. This expression ensures that no string in the set ends in 11.
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three points t, u, and v on the number line have coordinates t, u, and v, respectively. is point t between points u and v ?
We can determine coordinates if point t is between points u and v by checking if u < t < v or v < t < u.
To determine if point t is between points u and v, we need to compare their coordinates. If u < v, then point t is between points u and v if and only if u < t < v. On the other hand, if v < u, then point t is between points u and v if and only if v < t < u.
Whether or not point t is between points u and v depends on the relationship between the coordinates of u and v. If u < v, t must fall between them, and if v < u, t must also fall between them.
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The length of a lateral edge of the regular square pyramid ABCDM is 15 in. The measure of MDO is 38°. Find the volume of the pyramid. Round your answer to the nearest in^3
Step-by-step explanation:
Volume of square pyramid = 1/3 base * height
so you will need to find the height ( MO ) using the sin function:
15 sin 38 = MO = 9.235 in
Now you need to find the measure of OD to calulate the base area
OD = 15 cos 38 = 11.82 and the entire diagonal of the base is two times this = 23.64 in
then, using the pythagorean theorem
23.64^2 = ab^2 + ab^2
ab = 16.716 in so base area = 16.716 x 16.716 = 279.42 in^2
Finally : Volume = 1/3 ( 279.42)(9.235) = ~~ 860 in^3
5Use the continuity of the absolute value function (x is continuous for all values of x) to determine the interval(s) on which the following function is continuous. h(x)=|5/sqrx −9|
Select the correct choice below and, if necessary, fill in the answer box to complete your choice.
A.The function is continuous on .....
B. The function is not continuous anywhere.
The function is continuous on the interval (0, 5/9].
To determine the intervals on which the function h(x) = |5/√(x) - 9| is continuous, we need to consider the behavior of the absolute value function and the square root function.
The absolute value function |u| is continuous for all real numbers. Similarly, the square root function sqrt(x) is also continuous for x ≥ 0. However, it is important to note that the square root function is not defined for negative values of x.
In the given function h(x) = |5/√(x) - 9|, the denominator of the square root cannot be negative. Thus, we need to find the values of x that make the denominator non-negative:
5/√(x) - 9 ≥ 0
To satisfy this inequality, we need sqrt(x) > 0, which implies x > 0. Additionally, we need 5/√(x) - 9 ≥ 0, which implies 5/√(x) ≥ 9. Solving this inequality, we get √(x) ≤ 5/9.
Combining these conditions, we find that the function h(x) is continuous on the interval (0, (5/9] (inclusive of 5/9).
Therefore, the correct choice is:
A. The function is continuous on the interval (0, 5/9].
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A firm has a production function f(x, y) = 0.70(x0.20 + 2,0.20)4 whenever x > 0 and y > 0. When the amounts of both inputs are positive, this firm has a. increasing returns to scale. b. decreasing returns to scale. constant returns to scale. d. increasing returns to scale if x +y> 1 and decreasing returns to scale otherwise. e. increasing returns to scale if output is less than 1 and decreasing returns to scale if output is greater than 1
The given production function is f(x, y) = 0.70(x0.20 + 2,0.20)4, where x > 0 and y > 0. Whenever the amounts of both inputs are positive, the firm has increasing returns to scale.
Therefore, the option (a) is correct. A firm has increasing returns to scale when it increases its inputs by a certain percentage, and the output increases by a higher percentage.
It means that if the firm doubles its inputs, the output should increase more than twice.
In this case, if the firm increases both inputs by the same proportion, the output will increase by an even higher proportion.
The production function shows the maximum output that can be produced from different combinations of inputs. The given production function f(x, y) = 0.70(x0.20 + 2,0.20)4 has a constant returns to scale.
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Write the vector (19, -10, –7) as a linear combination of ãj = (-3, 2, 3), à2 = (3, -3, -2) and ã3 = (-1,4,5). Express your answer in terms of the named vectors. Your answer should be in the form 4ãi + 5a2 + 6a3, which would be entered as 4a1 + 5a2 + 6a3. (19,-10, – 7) = b.
The vector (19, -10, -7) can be expressed as a linear combination of ã₁, ã₂, and ã₃ as follows:
(19, -10, -7) = -2ã₁ + 3ã₂ + 4ã₃
To write the vector (19, -10, -7) as a linear combination of ã₁ = (-3, 2, 3), ã₂ = (3, -3, -2), and ã₃ = (-1, 4, 5), we need to find coefficients b₁, b₂, and b₃ such that:
(19, -10, -7) = b₁ã₁ + b₂ã₂ + b₃ã₃
We can set up a system of equations based on the components of the vectors:
-3b₁ + 3b₂ - b₃ = 19 (for the x-component)
2b₁ - 3b₂ + 4b₃ = -10 (for the y-component)
3b₁ - 2b₂ + 5b₃ = -7 (for the z-component)
Now we can solve this system of equations to find the values of b₁, b₂, and b₃.
Using matrix notation, the system can be represented as:
| -3 3 -1 | | b₁ | | 19 |
| 2 -3 4 | * | b₂ | = | -10 |
| 3 -2 5 | | b₃ | | -7 |
Using Gaussian elimination or any other suitable method, we can solve this system and find the values of b₁, b₂, and b₃.
The solution to the system is b₁ = -2, b₂ = 3, and b₃ = 4.
The vector (19, -10, -7) can be expressed as a linear combination of ã₁, ã₂, and ã₃ as follows:
(19, -10, -7) = -2ã₁ + 3ã₂ + 4ã₃
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Suppose there are two tanks, each containing 20 L of fluid. Tank 1 initially contains water with 4 kg of salt dissolved and Tank 2 initially contains pure water. The tanks are stirred constantly so their solutions have uniform concentration. There is a pipe which takes 15 L/min from tank 1 to tank 2 , and another pipe which takes 15 L/min from tank 2 to tank 1. Also, pure water is being pumped into each of tank 1 and tank 2 at a rate of 5 L/min, and solution is being removed at a rate of 5 L/min from each of tank 1 and tank 2. Model the initial value problem as a system of two first order differential equations, solve the initial value problem, and find the amount of salt in tank 1 after one minute.
The amount of salt in Tank 1 after one minute for the given question is approximately 14.05 kg.
The given situation can be modeled by a system of differential equations as follows:
Let x1 be the amount of salt in Tank 1, and x2 be the amount of salt in Tank 2 at any time t. We will use the following equations to model the given system:
{dx1/dt} = (1/20)[(5 - x1/4) - (x1/15)]{dx2/dt} = (1/20)[(x1/15) - (x2/15)]
where x1 (0) = 4 and x2 (0) = 0.
Since the rate of change of the amount of salt in each tank is proportional to the difference between the salt concentration of the tank and the average salt concentration of the two tanks.
We can solve these two differential equations using separation of variables as shown below:{dx1/dt} = (1/20)[(5 - x1/4) - (x1/15)]dx2/dt = (1/20)[(x1/15) - (x2/15)]
Separating variables and integrating, we have:
integral {dx1/[(5 - x1/4) - (x1/15)]} = integral {(1/20)} dt
integral {dx2/[(x1/15) - (x2/15)]} = integral {(1/20)} dt
On evaluating the integrals, we get ln |5/4 - x1/4| - ln |x1/15| = t/20 + C1
and ln |x1/15| - ln |x2/15| = t/20 + C2
where C1 and C2 are arbitrary constants.
To find the value of C1, we use the initial condition that x1 (0) = 4, which implies that C1 = ln |5/4|.
Similarly, using the initial condition x2 (0) = 0, we get C2 = ln |4/3|.
Now, we can eliminate ln |x1/15| and obtain:
x1 = 15 (5/4)e^{-t/20}
Therefore, the amount of salt in Tank 1 after one minute is:x1(1) = 15 (5/4)e^{-1/20} ≈ 14.05 kg
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Determine whether the sequence converges or diverges. If it converges, find the limit. \[ a_{n}=e^{n /(n+2)} \] \( e \) \( \ln 3 \) diverges \( e^{3} \) \( \ln \left(\frac{1}{3}\right) \)
The sequence [tex]\(a_n = e^{n/(n+2)}\)[/tex] converges to the limit [tex]\(e\)[/tex]. As [tex]\(n\)[/tex] approaches infinity, the exponent [tex]\(n/(n+2)\)[/tex] tends towards 1, resulting in the convergence of the sequence to the constant value [tex]\(e\)[/tex].
To determine convergence, we need to analyze the behavior of the sequence as [tex]\(n\)[/tex] approaches infinity. Let's examine the expression [tex]\(e^{n/(n+2)}\)[/tex]. As [tex]\(n\)[/tex] gets larger, the denominator [tex]\(n+2\)[/tex] becomes negligible compared to [tex]\(n\)[/tex]. Thus, the exponent [tex]\(n/(n+2)\)[/tex] approaches 1. Therefore, the sequence can be rewritten as [tex]\(e^1\)[/tex], which is equal to [tex]\(e\)[/tex].
To further verify the convergence of the sequence, we can demonstrate that it satisfies the conditions of convergence. Firstly, the sequence is well-defined for all positive integers [tex]\(n\)[/tex]. Secondly, the sequence is increasing since the base [tex]\(e\)[/tex] is a positive constant greater than 1. Lastly, the sequence is bounded above because [tex]\(e^1\)[/tex] provides an upper bound. Thus, the sequence [tex]\(a_n = e^{n/(n+2)}\)[/tex] converges to the limit [tex]\(e\)[/tex].
In conclusion, the given sequence [tex]\(a_n = e^{n/(n+2)}\)[/tex] converges, and its limit is [tex]\(e\)[/tex].
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Let A={1,2,3,4,5}, and define a function F:P→Z as follows. For each set of X in A(A), F(X)={ 0 if X has an even number of elements 1 if X has an odd number of elements.
Find the following: (a) F({1,4,2,3})= (b) F({2,3,5})= (c) F(∅)= (d) F({1,2})=
It contains zero elements, it is an even number set. Hence F(∅) = 0.(d) F({1, 2}) is a set containing two elements, and thus it is an even number set. Therefore, F({1, 2}) = 0.
A set A = {1, 2, 3, 4, 5} and a function F: P(A) → Z, where P(A) denotes the power set of A and F(X) = {0 if X has an even number of elements, 1 if X has an odd number of elements}.The answer to the given query is as follows:(a) To find F({1, 4, 2, 3}), we need to determine the number of elements in this set. It is an even number set as it has 4 elements, hence the value of F({1, 4, 2, 3}) = 0.(b) Similarly, for F({2, 3, 5}), we can observe that it is a set of three elements. Therefore, F({2, 3, 5}) = 1.(c) F(∅) represents the number of elements in an empty set.
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parallelogram gbjf has vertices g(–4, 1); b(–2, 3); f(–2,0). determine the coordinates of point j. group of answer choices (2, 0) (0, 2) (–2, –4) (–4, –1)
Answer:
j(0, 2)
Step-by-step explanation:
You want to know the coordinates of point J in parallelogram GBJF, given G(-4, 1), B(-2, 3), F(-2, 0).
MidpointIn a parallelogram, the diagonals bisect each other. This means their midpoints are the same.
(G +J)/2 = (B +F)/2 = M
G +J = B +F . . . . . . . . multiply by 2
J = B +F -G
J = (-2, 3) +(-2, 0) -(-4, 1)
J = (-2-2+4, 3+0-1) = (0, 2)
The coordinates of point J are (0, 2).
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Find the explicit particular solution of the differential equation for the initial value provided. dy/dx=9x^6y-y, y(1)= -10
The explicit particular solution of the given differential equation with the initial condition y(1) = -10 is:
y = -e^((9/7)x^7 - x + ln(10) - 2/7)
To find the explicit particular solution of the differential equation dy/dx = 9x^6y - y with the initial condition y(1) = -10, we can use the method of separation of variables and solve the differential equation step by step.
First, let's rewrite the equation as follows:
dy/dx = (9x^6 - 1)y
Now, let's separate the variables and move y terms to one side and x terms to the other side:
dy/y = (9x^6 - 1)dx
Integrating both sides:
∫(1/y)dy = ∫(9x^6 - 1)dx
ln|y| = ∫(9x^6 - 1)dx
Integrating the right side:
ln|y| = (9/7)x^7 - x + C
where C is the constant of integration.
Next, we need to determine the value of C using the initial condition y(1) = -10:
ln|-10| = (9/7)(1^7) - 1 + C
ln(10) = (9/7) - 1 + C
ln(10) = 2/7 + C
C = ln(10) - 2/7
Substituting the value of C back into the equation:
ln|y| = (9/7)x^7 - x + ln(10) - 2/7
Simplifying:
ln|y| = (9/7)x^7 - x + ln(10) - 2/7
Now, we need to remove the absolute value sign by exponentiating both sides:
|y| = e^((9/7)x^7 - x + ln(10) - 2/7)
Finally, since the initial condition y(1) = -10 is negative, we can add a negative sign to the right side to match the initial condition:
y = -e^((9/7)x^7 - x + ln(10) - 2/7)
Therefore, the explicit particular solution of the given differential equation with the initial condition y(1) = -10 is:
y = -e^((9/7)x^7 - x + ln(10) - 2/7)
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A car traveling directly N60°E covered 500 miles between noon and 10 pm. If its starting point is designated as the origin, what parametric equations describe the drive? Fill in the blanks. Give decimal values to three places. X = ______t + _____ y =____ t+____ Question 2 16 pts A second car also leaves at noon and drives in a southwesterly direction. At 2 pm, its location (777, 370); at 5 pm, its location is (648, 325). Write a set of parametric equations to describe its movement. Fill in the blanks. Give decimal values to three places. X =_______ t+_____ y =_______ t +_______
The parametric equations for the first car are: X = 250t, Y = 433.01t. The parametric equations for the second car are:
X = -129t - 882, Y = 45t + 400.
The car's direction is N60°E. Let's make a triangle using this angle, we have:
The car's direction is N60°E
We can see that x and y components of the car's direction are:
x = 500 sin 30° = 250
y = 500 cos 30° = 433.01
Starting from the origin point, the parametric equations for the drive are:
X = 250t
Y = 433.01t
The second car is driving in a southwesterly direction. Let's make another triangle to determine the components of its direction. We have:
The second car's direction is southwest
We can see that the x and y components of the second car's direction are:
x = -777 - (-648) = -129
y = 370 - 325 = 45
Using the two known points, we can also determine the second car's rate of movement:
The second car's rate of movement
We can now form the parametric equations for the second car's movement using the x and y components and the rate of movement as follows:
X = -129t - 882
Y = 45t + 400
The parametric equations for the first car are:
X = 250t
Y = 433.01t
The parametric equations for the second car are:
X = -129t - 882
Y = 45t + 400
Note: the values are rounded to three decimal places.
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7) Ben starts walking a long a path a 4 milles per hour. one and half hours after Ben leave, his sister begins Jogging Amanda along the same path at 6 miles per nour. How long will it be before Amanda catcher up to Ben
It will take 1 hour for Amanda to catch up to Ben. The distance traveled by Amanda can be calculated as Distance_Amanda = Amanda's speed * Amanda's time = 6 * t miles.
To determine how long it will take for Amanda to catch up to Ben, we need to find the time it takes for their distances traveled to be equal.
Let's assume that it takes t hours for Amanda to catch up to Ben. During this time, Ben would have already been walking for t + 1.5 hours (since he started 1.5 hours earlier).
The distance traveled by Ben can be calculated as:
Distance_Ben = Ben's speed * Ben's time = 4 * (t + 1.5) miles
Similarly, the distance traveled by Amanda can be calculated as:
Distance_Amanda = Amanda's speed * Amanda's time = 6 * t miles
For Amanda to catch up to Ben, their distances should be equal. So we have the equation:
Distance_Ben = Distance_Amanda
4 * (t + 1.5) = 6 * t
Simplifying the equation:
4t + 6 = 6t
2 = 2t
Dividing both sides by 2:
t = 1
Therefore, it will take 1 hour for Amanda to catch up to Ben.
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Consider the following system A= [3 -2 A]
12 -9 21
-3 0 13
[ x y z ]
= [32 144 21]
Solve this system using LU decomposition where L has the form L=
[1 0 0]
I21 1 0
I32 I32 1
The component d1 of the vector d obtained by solving Ld=b is equal to
The component d1 of the vector d obtained by solving Ld=b is 32.
Given the system of equations as below:
A = [3 -2 A]12 -9 21-3 0 13
[x y z] = [32 144 21]
We need to solve this system using LU decomposition where L has the form
L= [1 0 0]I21 1 0I32 I32 1
To solve this system using the LU decomposition method, follow these steps:
We find the lower and upper triangular matrices, L and U, as follows:
Step 1: Write the augmented matrix
[A | b] = [3 -2 A 32]12 -9 21 144-3 0 13 21
Step 2: Multiply R1 by (1/3) to make the leading coefficient 1.
[3 -2 A 32] -> [1 -2/3 A/3 32/3]12 -9 21 144-3 0 13 21
Step 3: Subtract R1 from R2 and R3 to get zeros in the first column of R2 and R3.
[1 -2/3 A/3 32/3]12 -9 21 144-3 0 13 21 [1 -2/3 A/3 32/3]0 -3/4 7 1040/3 2 13 19
Step 4: Multiply R2 by (-4/3) to make the leading coefficient 1.
[1 -2/3 A/3 32/3]
0 1 -28/12 -139/12[1 -2/3 A/3 32/3]0 -3/4 7 1040/3 2 13 19
Step 5: Add 2R2 to R3 to get a zero in the third row, second column. [1 -2/3 A/3 32/3]
0 1 -28/12 -139/12[0 1 -35/12 -67/12]0 0 49/4 145/4
Step 6: Rewrite the matrix in terms of L and U.
L = [1 0 0;-4/3 1 0;0 -35/12 1]
U = [3 -2 A;0 -3/4 7;0 0 49/4]
Step 7: Solve the lower triangular system Ly = b by forward substitution.
Ly = [32; 144; 21]
y1 = 32y2 -4/3 y1 + 144
= 0
=> y2 = 128/3y3 -35/12 y2 + 21/3 y1 + 21
= 0
=> y3 = -413/7
Therefore, y = [32; 128/3; -413/7]
Step 8: Solve the upper triangular system Ux = y by backward substitution.
Ux = [3 -2 A;0 -3/4 7;0 0 49/4]
x3(49/4) = -413/7
=> x3 = -169/49-2 x2 -169/49
x3 = 128/3
=> x2 = -1219/4413 x1 + 2 x2 -169/49 x3
= 32
=> x1 = -14332/441
Therefore, x = [-14332/441; -1219/441; -169/49]
Conclusion: From the above explanation, it is observed that the component d1 of the vector d obtained by solving Ld=b is 32.
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The given system A can be solved using LU decomposition as follows:
To perform LU decomposition, perform Gaussian elimination on the matrix A to obtain the upper triangular matrix U, while keeping track of the multipliers used to perform the elimination to form the lower triangular matrix L.
In order to solve the system Ax = b,
we first solve the system Ld = b for the vector d,
where L is a lower triangular matrix.
Then we solve the system Ux = d for the vector x,
where U is an upper triangular matrix.
Let us perform Gaussian elimination on the matrix A as shown below:
[3 -2 A|32] [12 -9 21|144] [-3 0 13|21]→ [3 -2 A|32] R2 = R2 - 4R1→ [0 -1 13|48] R3 = R3 + R1→ [3 -2 A|32] R3 = R3 + R1→ [0 -1 13|48] R3 = R3 - 5R2→ [3 -2 A|32] R1 = R1 - 3R2⇒[3 -2 A|32] R2 = -R2d1
is the first component of d and can be obtained by solving the first equation of Ld = b as follows:
L11d1 = b1d1 = b1/L11
The first equation of Ld = b is given as below: L11d1 = b1
Substituting the value of L11 and b1 in the above equation, we get;
d1 = 32/3
Thus, the component d1 of the vector d obtained by solving Ld = b is 32/3.
Answer: d1 = 32/3
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The following table the weight of a person on Earth (pounds) and the weight of a person on the moon (pounds)
Weight of a person;
Earth: 100, 140, 175, 200
Moon: 21.67, 28.33, 34.17, 38.33
Based on the data in the table, find the slope of the linear model that represents the weight on the moon per one pound increase of weight on Earth.
The slope of the linear model representing the relationship between weight on the moon and weight on Earth is approximately 0.1665, indicating the change in moon weight per one-pound increase in Earth weight.
To find the slope of the linear model representing the relationship between weight on the moon and weight on Earth, we can use the formula for slope, which is given by:
slope = (change in y) / (change in x)
In this case, the "y" variable represents the weight on the moon, and the "x" variable represents the weight on Earth.
Let's calculate the changes in weight on the moon and weight on Earth:
Change in weight on the moon = 28.33 - 21.67 = 6.66
Change in weight on Earth = 140 - 100 = 40
Now, we can substitute these values into the slope formula:
slope = (change in weight on the moon) / (change in weight on Earth) = 6.66 / 40 ≈ 0.1665
Therefore, the slope of the linear model representing the weight on the moon per one-pound increase of weight on Earth is approximately 0.1665.
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In the field of engineering; problems are presented that can be solved through the application of multiple integration; In most cases, computer programs use numerical approximation techniques to solve these problems.
an example related to engineering in which multiple integration is used and which computer program is recommended to be used. Justify your answer
Multiple integrals are used in the field of engineering to solve problems that require the integration of multivariable functions.
Multiple integrals are used in the field of engineering to solve problems that require the integration of multivariable functions. They are used to calculate the volume of complex shapes and to determine the mass of an object with a variable density.An example of the use of multiple integrals in engineering is the calculation of the moment of inertia of a solid object. The moment of inertia is the measure of an object's resistance to rotational motion and is used in the design of structures and machinery.Computer programs such as MATLAB and Mathematica are recommended for the numerical approximation of multiple integrals in engineering. These programs provide accurate and efficient solutions to complex integration problems and can handle high-level mathematical operations. Additionally, they allow for the visualization of integration results through the use of graphs and plots.
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what type of sampling strategy is used when a researcher selects subjects that are easily accessible to participate in a study?
The type of sampling strategy used when a researcher selects subjects that are easily accessible to participate in a study is called convenience sampling.
Convenience sampling is a non-probability sampling technique where researchers select subjects who are easily accessible to them. This type of sampling is often used when it is difficult or time-consuming to obtain a random sample.
Convenience samples are often used in exploratory studies, where the researcher is trying to get a general sense of a population. However, convenience samples are not representative of the population, so the results of studies that use convenience samples should be interpreted with caution.
Here are some of the advantages and disadvantages of convenience sampling:
Advantages:
Convenience sampling is easy and inexpensive to do.Convenience samples can be obtained quickly.Convenience samples can be used to study populations that are difficult to access, such as people with rare diseases or people who live in remote areas.Disadvantages:
Convenience samples are not representative of the population, so the results of studies that use convenience samples may not be generalizable to the population.Convenience samples may be biased, because researchers may be more likely to select subjects who are similar to them or who they think will be easy to work with.Here are some examples of convenience sampling:
A researcher studying the effects of a new drug on depression might select subjects who are easily accessible to them, such as patients who are already being treated for depression at a local clinic.
A researcher studying the effects of a new educational program on student achievement might select subjects who are easily accessible to them, such as students who are already enrolled in a particular school district.
It is important to note that convenience sampling is not the only type of non-probability sampling technique. Other types of non-probability sampling techniques include quota sampling, snowball sampling, and purposive sampling.
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A charge is uniformly.distributed along the entire x-axis. If each 20 cm length of the x-axis carries 2.0 nC of charge. What is the magnitude of the electric field at the point y 2.0 m, on the y axis? a) 45 N/C
b) 90 N/C
c) 18 N/C
d) 36 N/C
The magnitude of the electric field at the point y = 2.0 m on the y-axis is 22.5 N/C.
The question asks for the magnitude of the electric field at a point on the y-axis, given a uniformly distributed charge along the x-axis. Each 20 cm length of the x-axis carries 2.0 nC of charge.
To find the electric field at a point, we can use the formula:
E = k * (Q / r^2)
Where E is the electric field, k is the electrostatic constant (9 * 10^9 N m^2 / C^2), Q is the charge, and r is the distance from the charge.
In this case, the charge is uniformly distributed along the x-axis, so we can consider each 20 cm length as a point charge. The charge of each 20 cm length is 2.0 nC.
Let's calculate the electric field at the point y = 2.0 m on the y-axis.
First, we need to find the distance (r) from each 20 cm length of charge to the point (0, 2.0 m) on the y-axis. Since the x-axis and y-axis are perpendicular, the distance is simply the y-coordinate, which is 2.0 m.
Now, let's calculate the electric field due to each 20 cm length of charge:
E1 = k * (Q / r^2)
= (9 * 10^9 N m^2 / C^2) * (2.0 nC / (2.0 m)^2)
= (9 * 10^9 N m^2 / C^2) * (2.0 * 10^-9 C / 4.0 m^2)
= 4.5 N/C
Since the charges are uniformly distributed, we can assume that each 20 cm length of charge contributes the same electric field.
Next, let's calculate the total electric field at the point (0, 2.0 m) due to all the 20 cm lengths of charge. Since the charges are distributed along the entire x-axis, we can consider all the 20 cm lengths of charge together.
Since the charges are uniformly distributed, the total electric field is simply the sum of the electric fields due to each 20 cm length of charge.
Number of 20 cm lengths = (length of x-axis) / (length of each 20 cm length)
= (1 m) / (0.2 m)
= 5
Total electric field = (number of 20 cm lengths) * (electric field due to each 20 cm length)
= 5 * 4.5 N/C
= 22.5 N/C
Therefore, the magnitude of the electric field at the point y = 2.0 m on the y-axis is 22.5 N/C.
So, the correct answer is not listed in the options provided.
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Use eigenvectors and eigenvalues to find the fundamental matrix and solve the system of equations {x′=−6x−5yy′=x with x(0)=3 and y(0)=0.
To solve the system of differential equations {x' = -6x - 5y, y' = x} with the initial conditions x(0) = 3 and y(0) = 0 using eigenvalues and eigenvectors, we first need to find the eigenvalues and eigenvectors of the coefficient matrix.
The coefficient matrix of the system is:
A = [[-6, -5], [1, 0]]
To find the eigenvalues, we solve the characteristic equation |A - λI| = 0, where λ is the eigenvalue and I is the identity matrix.
|A - λI| = [[-6 - λ, -5], [1, -λ]]
(-6 - λ)(-λ) - (-5)(1) = 0
λ² + 6λ + 5 = 0
Solving the quadratic equation, we find two eigenvalues:
λ₁ = -1 and λ₂ = -5
Next, we find the corresponding eigenvectors by solving the equations (A - λI)v = 0, where v is the eigenvector.
For λ₁ = -1:
(A - λ₁I)v₁ = [[-5, -5], [1, 1]]v₁ = 0
The solution is v₁ = [1, -1].
For λ₂ = -5:
(A - λ₂I)v₂ = [[-1, -5], [1, 5]]v₂ = 0
The solution is v₂ = [5, -1].
The fundamental matrix Φ(t) is formed by the eigenvectors as columns and exponentiated by the eigenvalues as exponents:
Φ(t) = [v₁, v₂] * diag(e^(λ₁t), e^(λ₂t))
Calculating the fundamental matrix:
Φ(t) = [[1, 5], [-1, -1]] * diag(e^(-t), e^(-5t))
To find the solution for x(t) and y(t), we use the initial conditions and multiply the fundamental matrix by the initial values:
[x(t), y(t)] = Φ(t) * [x(0), y(0)]
= [[1, 5], [-1, -1]] * [3, 0]
Simplifying the matrix multiplication, we have:
x(t) = 3e^(-t) + 0
= 3e^(-t)
y(t) = 5e^(-t) - 0
= 5e^(-t)
Therefore, the solution to the system of differential equations with the initial conditions x(0) = 3 and y(0) = 0 is:
x(t) = 3e^(-t)
y(t) = 5e^(-t)
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Compute the Laplace transform: L{u5(t)+u6(t)t^3e^3t}= If you don't get this in 2 tries, you can get a hint. You have attempted this problem 0 times.
the Laplace transform of u5(t) + u6(t)t³[tex]e^{3t[/tex] is 1/s + 6 / (s - 3)⁴.
To compute the Laplace transform of the given function, we can use the linearity property of the Laplace transform.
The Laplace transform of a function u(t) is defined as:
L{u(t)} = ∫[0 to ∞] u(t)[tex]e^{(-st)[/tex] dt
where s is the complex variable.
Applying the linearity property, we can compute the Laplace transform of each term separately and then sum them.
L{u5(t)} = ∫[0 to ∞] u5(t)[tex]e^{(-st)[/tex] dt
The Laplace transform of the unit step function u(t) is given by:
L{u(t)} = 1/s
Therefore, L{u5(t)} = 1/s.
Now, let's compute the Laplace transform of the second term:
L{u6(t)³[tex]e^{3t}[/tex]} = ∫[0 to ∞] u6(t)t³[tex]e^{3t[/tex] [tex]e^{(-st[/tex] dt
Using the time-shifting property of the Laplace transform, the Laplace transform of tⁿ f(t) is given by:
L{tⁿ f(t)} = (-1)ⁿ dⁿ F(s) / dsⁿ
where F(s) is the Laplace transform of f(t).
In this case, n = 3 and f(t) = [tex]e^{3t[/tex]. Taking the derivative three times with respect to s, we get:
dⁿ F(s) / dsⁿ = d³ / ds³ (1 / (s - 3))
Differentiating three times, we have:
d³ / ds³ (1 / (s - 3)) = 6 / (s - 3)⁴
Therefore, L{u6(t)t^3[tex]e^{3t[/tex]} = 6 / (s - 3)^4.
Combining the results, we have:
L{u5(t) + u6(t)t³[tex]e^{3t[/tex]} = L{u5(t)} + L{u6(t)t³[tex]e^{3t[/tex]}
= 1/s + 6 / (s - 3)⁴
Therefore, the Laplace transform of u5(t) + u6(t)t³[tex]e^{3t[/tex] is 1/s + 6 / (s - 3)⁴.
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Use Poiseuille's Law to calculate the rate of flow in a small human artery using the information below. Give your answer correct to three significant figures.
η = 0.028, R = 0.008 cm, l = 2 cm, P = 5000 dynes/cm2
The Poiseuille's law expresses the relationship between the rate of flow of a fluid through a tube (Q) and the pressure difference (ΔP) across the ends of the tube.
It is given by Q = (π * ΔP * r⁴) / (8 * η * l),
where r is the radius of the tube,
η is the viscosity of the fluid,
l is the length of the tube.
Let us use this equation to calculate the rate of flow in a small human artery.The values given are:
η = 0.028, R = 0.008 cm, l = 2 cm, P = 5000 dynes/cm²
As the radius is given as 0.008 cm, the diameter of the artery is 2 * 0.008 = 0.016 cm.
The radius (r) is 0.008 cm/2 = 0.004 cm.
Substitute the given values in the Poiseuille's law to get
Q = (π * ΔP * r⁴) / (8 * η * l)Q = (π * 5000 dynes/cm² * 0.004⁴ cm⁴) / (8 * 0.028 poise * 2 cm)Q = 0.000014 cm³/s
Therefore, the rate of flow in the small human artery is 0.000014 cm³/s (correct to three significant figures).
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Derive the general solution form for the recurrence tn = 120,-2 - 166n-3 + 2" Show your work (all steps: the associated homogeneous equation, the characteristic polynomial and its roots, the general solution of the homogeneous equation, computing a particular solution, the general solution of the non-homogeneous equation.) a
The general solution form for the recurrence tn = 120,-2 - 166n-3 + 2.
Given a recurrence relation tn = 120,-2 - 166n-3 + 2 we have to derive the general solution form for the recurrence sequence.
We have the recurrence relation tn = 120,-2 - 166n-3 + 2
We need to find the solution for the recurrence relation.
Associated Homogeneous Equation: First, we need to find the associated homogeneous equation.
tn = -166n-3 …..(i)
The characteristic equation is given by the following:tn = arn. Where ‘a’ is a constant.
We have tn = -166n-3..... (from equation i)ar^n = -166n-3
Let's assume r³ = t.
Then equation i becomes ar^3 = -166(r³) - 3ar^3 + 166 = 0ar³ = 166
Hence r = ±31.10.3587Complex roots: α + iβ, α - iβ
Characteristics Polynomial:
So, the characteristic polynomial becomes(r - 31)(r + 31)(r - 10.3587 - 1.7503i)(r - 10.3587 + 1.7503i) = 0
The general solution of the Homogeneous equation:
Now we have to find the general solution of the homogeneous equation.
tn = C1(-31)n + C2(31)n + C3 (10.3587 + 1.7503i)n + C4(10.3587 - 1.7503i)
nWhere C1, C2, C3, C4 are constants.
Computing a Particular Solution:
Now we have to compute the particular solution.
tn = 120-2 - 166n-3 + 2
Here the constant term is (120-2) + 2 = 122.
The solution of the recurrence relation is:tn = A122Where A is the constant.
The General Solution of Non-Homogeneous Equation:
The general solution of the non-homogeneous equation is given bytn = C1(-31)n + C2(31)n + C3 (10.3587 + 1.7503i)n + C4(10.3587 - 1.7503i)n + A122
Hence, we have derived the general solution form for the recurrence tn = 120,-2 - 166n-3 + 2.
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represent the plane curve by a vector-valued function. x2 y2 = 9
The vector-valued function r(t) = 〈3 cos(t), 3 sin(t)〉 represents the plane curve x² y² = 9.
We are supposed to represent the plane curve by a vector-valued function. The plane curve x2 y2 = 9 can be represented by a vector-valued function
r(t) = 〈3 cos(t), 3 sin(t)〉.
To find the equation of the plane curve we have to make use of the following equation:
x² + y² = r², where x, y and r are the components of the vector r(t).
Here, we are supposed to represent the plane curve by a vector-valued function.
A vector-valued function can represent the plane curve x² y² = 9,
r(t) = 〈3 cos(t), 3 sin(t)〉.
The vector-valued function r(t) represents the curve x² y² = 9 since if we plug the components of r(t) in the equation x² y² = 9, we get:
= 9 cos²(t) sin²(t)
= 9sin²(t) + 9cos²(t)
= 9
This is true for any value of t. Therefore, the vector r(t) traces the curve x² y² = 9. If we graph the curve x² y² = 9 we get a circle with radius 3 centered at the origin, and the vector-valued function r(t) traces the circle counterclockwise starting at the point 〈3, 0〉. The vector-valued function r(t) = 〈3 cos(t), 3 sin(t)〉 represents the plane curve x² y² = 9.
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, H is differentiable function of t thatgives the temperature, in degrees Celsius, at an Arctic weatherstation. Which of the following is the best interpretation ofH'(24)?
A. The change in temperature during the first day.
B. The change in temperature during the 24th hour.
C. The average rate at which the temperature changed duringthe 24th hour.
D.The rate at which the temperature is changing during thefirst day.
E. The rate at which the temperature is changing at the end ofthe 24th hour.
The interpretation of H' (24) in this case will be "The average rate at which the temperature changed during the 24th hour.
Correct option is c
The given function H(t) gives the temperature in degrees Celsius at an Arctic weather station. Therefore, H is a differentiable function of t. We are asked to determine the best interpretation of H'(24). H'(24) is the derivative of H with respect to t at t = 24.
Because H' (t) measures the instantaneous rate of change of H with respect to t, the best interpretation of H' (24) will be the average rate at which the temperature changed during the 24th hour. Answer: C. The average rate at which the temperature changed during the 24th hour.
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You are a rising star in the music industry, and you have just received word that you will be rewarded a $500,000 bonus this year. As your chauffer drives you home in the company's limousine, you call your accountant, who suggests investing your bonus in a high-yield account offering 12% continuous interest. A friend, however wants you to invest your bonus in their restaurant business, and has promised 12% interest, compounded quarterly. You want to compare both accounts before making a decision, you can crunch the numbers and find out: • How long will it take to earn an extra $100,000? • When will the account have $750,000? • How long will it take to double my money? • How long will it take to triple my investment
It will take 5.78 years to earn an extra $100,000 regardless of which option you choose. if we double the amount then it will take approx 10 year.
The high-yield account with continuous compounding interest is a safer investment since it won't be at risk of losing money in a business venture.
Since High-yield account offering 12% continuous interest.
[tex]A = Pe^{rt}[/tex]
Where A is the final amount, P is the principal amount, r is the annual interest rate, t is the time in years, and e is Euler's number, approximately equal to 2.71828.
we can plug in the given values to find out how long it will take to earn an extra $100,000:
[tex](P + $500,000)e^{0.12t} - P = $100,000[/tex]
t = ln[(P + $500,000 + $100,000)/P] / 0.12,
where P = $500,000
Substituting the values,
t = ln[(500,000 + 500,000 + 100,000)/500,000] / 0.12t
= ln(2.2)/0.12t = 5.78 years (approx.)
Now we can plug in the given values to find out how long it will take to earn an extra $100,000:
[tex](P + $500,000)(1 + 0.12/4)^{4t} - P = $100,000[/tex]
t = [ln(P + $600,000) - ln(P)] / [4 ln(1.03)]
where P = $500,000
Substituting the values, we get:
t = [ln(1,100,000) - ln(500,000)] / [4 ln(1.03)]t = 5.78 years (approx.)
After comparing both investment options, it will take approximately 5.78 years to earn an extra $100,000 regardless of which option you choose. However, the high-yield account with continuous compounding interest is a safer investment since you won't be at risk of losing money in a business venture.
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A center-pivot irrigation system waters a sector-
shaped field.
Find the area of the field if the central angle, 0=
44° and the radius, r= 148 meters.
Round to the nearest whole number.
Rounding the above value to the nearest whole number .Given data:Radius, r = 148 meters Central angle,
= 44°
We have to find the area of the field.Area of the sector formula is given by:A = 1/2r²θ
where r is the radius of the circle and
θ is the central angle in degrees.
Substituting the values of r and θ in the above formula,
we get:A = 1/2 × 148² × 44°A
= 322710.4 m²
Rounding the above value to the nearest whole number, we get;Answer: 322710
Therefore, the area of the field is 322710 square meters.
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4.105 exercise and sleep. suppose that 42% of adults get enough sleep, 39% get enough exercise, and 28% do both. find the probabilities of the following events: (a) enough sleep and not enough exercise. (b) not enough sleep and enough exercise. (c) not enough sleep and not enough exercise. (d) for each of parts (a), (b), and (c), state the rule that you used to find your answer.
(a) The probability of enough sleep and not enough exercise is 0.14.
(b) The probability of not enough sleep and enough exercise is 0.11.
(c) The probability of not enough sleep and not enough exercise is 0.47.
To find the probabilities of the given events, we can use set theory and the principle of inclusion-exclusion.
Let's define:
A = event of getting enough sleep
B = event of getting enough exercise
Given information:
P(A) = 0.42 (probability of getting enough sleep)
P(B) = 0.39 (probability of getting enough exercise)
P(A ∩ B) = 0.28 (probability of both getting enough sleep and enough exercise)
(a) Event of enough sleep and not enough exercise:
P(A ∩ B') = P(A) - P(A ∩ B)
= 0.42 - 0.28
= 0.14.
The probability of enough sleep and not enough exercise is 0.14.
Rule used: We used the subtraction rule, which states that the probability of event A occurring and event B not occurring is equal to the probability of event A minus the probability of both A and B occurring.
(b) Event of not enough sleep and enough exercise:
P(A' ∩ B) = P(B) - P(A ∩ B)
= 0.39 - 0.28
= 0.11.
The probability of not enough sleep and enough exercise is 0.11.
Rule used: We used the subtraction rule.
(c) Event of not enough sleep and not enough exercise:
P(A' ∩ B') = 1 - P(A ∪ B) (by the complement rule)
= 1 - [P(A) + P(B) - P(A ∩ B)] (by the inclusion-exclusion principle)
= 1 - [0.42 + 0.39 - 0.28]
= 1 - 0.53
= 0.47.
The probability of not enough sleep and not enough exercise is 0.47.
Rule used: We used the inclusion-exclusion principle.
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Write a chain rule formula for the following derivative. 12)
∂t
∂z
for z=f(r,s);r=g(t),s=h(t) 13)
∂x
∂W
for w=f(p,q);p=g(x,y),q=h(x,y)
The chain rule is a method for computing the derivative of the composition of two or more functions. For example, if we want to find the derivative of a function h(x) that is composed of two functions, g(x) and f(u), we can use the chain rule formula:
(h∘g)(x)=h(g(x))=f(g(x))
We then find the derivative of f(u) and g(x) with respect to u and x, respectively, and multiply them together to get the derivative of h(x):
dhdx=dhdu⋅dudx
where dudx=∂g∂x and dhdu=∂f∂u
Let's use this formula to find the derivative of the given functions:
12) ∂t/∂z for z=f(r,s);r=g(t),s=h(t)
We need to find the partial derivative of t with respect to z. Using the chain rule, we can write:
∂t∂z=∂t∂r⋅∂r∂z+∂t∂s⋅∂s∂z
where ∂t∂r and ∂t∂s are the partial derivatives of t with respect to r and s, respectively. Using the chain rule again, we can write:
∂t∂r=∂t∂z⋅∂z∂r
∂t∂s=∂t∂z⋅∂z∂s
where ∂z∂r and ∂z∂s are the partial derivatives of z with respect to r and s, respectively. Substituting these expressions into the first formula, we get:
∂t∂z=∂t∂z⋅∂z∂r⋅∂r∂z+∂t∂z⋅∂z∂s⋅∂s∂z
Dividing both sides by ∂t∂z, we get:
1=∂z∂r⋅∂r∂z+∂z∂s⋅∂s∂z
This is the chain rule formula for the partial derivative of t with respect to z.
13) ∂x/∂W for w=f(p,q);p=g(x,y),q=h(x,y)
We need to find the partial derivative of x with respect to W. Using the chain rule, we can write:
∂x∂W=∂x∂p⋅∂p∂W+∂x∂q⋅∂q∂W
where ∂x∂p and ∂x∂q are the partial derivatives of x with respect to p and q, respectively. Using the chain rule again, we can write:
∂x∂p=∂x∂W⋅∂W∂p
∂x∂q=∂x∂W⋅∂W∂q
where ∂W∂p and ∂W∂q are the partial derivatives of W with respect to p and q, respectively. Substituting these expressions into the first formula, we get:
∂x∂W=∂x∂W⋅∂W∂p⋅∂p∂W+∂x∂W⋅∂W∂q⋅∂q∂W
Dividing both sides by ∂x∂W, we get:
1=∂W∂p⋅∂p∂W+∂W∂q⋅∂q∂W
This is the chain rule formula for the partial derivative of x with respect to W.
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