Consider the functions f: R² + R2 given by f(x, y) = (5y – 3x,x²) and g: R2 + R2 given by g(v, w) = (–202, W3 + 7) Find the following (make sure to include all of your reasoning): • Find Df and Dg Use the chain rule explained in the course videos to find D(g()) at the point (x, y) = (1,2) .
Answers
The Jacobian matrix Df for the function f(x, y) = (5y – 3x, x²) is [-3 5; 2x 0]. The Jacobian matrix Dg for the function g(v, w) = (-202, w³ + 7) is [0 0; 0 3w²]. The composition D(g(f(x, y))) at the point (x, y) = (1, 2) is [0 0; -243 243].
To find Df and Dg, we need to compute the Jacobian matrices of the functions f and g, respectively.
For function f(x, y) = (5y – 3x, x²), the Jacobian matrix Df is:
Df = [∂f₁/∂x ∂f₁/∂y]
[∂f₂/∂x ∂f₂/∂y]
Taking partial derivatives, we have:
∂f₁/∂x = -3
∂f₁/∂y = 5
∂f₂/∂x = 2x
∂f₂/∂y = 0
Therefore, the Jacobian matrix Df is:
Df = [ -3 5]
[ 2x 0]
For function g(v, w) = (-202, w³ + 7), the Jacobian matrix Dg is:
Dg = [∂g₁/∂v ∂g₁/∂w]
[∂g₂/∂v ∂g₂/∂w]
Taking partial derivatives, we have:
∂g₁/∂v = 0
∂g₁/∂w = 0
∂g₂/∂v = 0
∂g₂/∂w = 3w²
Therefore, the Jacobian matrix Dg is:
Dg = [ 0 0 ]
[ 0 3w² ]
Now, to find D(g(f(x, y))), we need to substitute f(x, y) into g(v, w) and then differentiate with respect to x and y.
Substituting f(x, y) = (5y – 3x, x²) into g(v, w) = (-202, w³ + 7), we get:
g(f(x, y)) = (-202, (5y – 3x)³ + 7)
Now, we differentiate the components of g(f(x, y)) with respect to x and y:
∂/∂x (-202) = 0
∂/∂y (-202) = 0
∂/∂x ((5y – 3x)³ + 7) = -3(5y – 3x)²(-3)
∂/∂y ((5y – 3x)³ + 7) = 3(5y – 3x)²(5)
Evaluating the partial derivatives at the point (x, y) = (1, 2), we have:
∂/∂x ((5y – 3x)³ + 7) = -3(5(2) – 3(1))²(-3) = -243
∂/∂y ((5y – 3x)³ + 7) = 3(5(2) – 3(1))²(5) = 243
Therefore, D(g(f(x, y))) at the point (x, y) = (1, 2) is:
D(g(f(x, y))) = [ 0 0 ]
[ -243 243 ]
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Related Questions
The sum of the interior angles of a polygon is 540°. How many sides does the polygon have?
A. 4
B. 5
C. 6
Answers
help please please
Answers
If i just knew the answer i would tell you buti dont know sorrry
Answer:
1. A. 9
B. - 51
2. A. (-3,2)
B. (5,-2)
3. A. (11,-3) B. (11,1) C. They have the same x
the table shows ordered pairs of the function y=16+0.5x. Which ordered pair could be the missing values represented by (x,y)?
Answers
Answer:
(2,17)
Step-by-step explanation:
This is because x is the input and y is the output. All you have to do is create a value for x the substitute it in the equation. Next solve for y 0.5 times 2 is 1, plus 16 is 17. Then just put your values in an order pair.
Answer:
8,20
Step-by-step explanation:
Quadrilateral TUVW is a rhombus. What is VW?
U
V
vw=
V
11w
W
w+20
T
Answers
Answer:
22
Step-by-step explanation:
All sides of a rhombus are congruent.
\(11w=w+20 \\ \\ 10w=20 \\ \\ w=2 \\ \\ \implies VW=2(11)=22\)
______ can be thought of as the chi-square type equivalent to the paired t-test.
Answers
The McNemar's Test can be thought of as the chi-square type equivalent to the paired t-test.
The McNemar's Test is a non-parametric statistical method used to analyze the differences between paired or matched categorical data, such as repeated measurements on a single group. Like the paired t-test, which is used to compare continuous data, the McNemar's Test evaluates the changes in the proportions of success or failure between the paired observations.
This test is particularly useful when dealing with small sample sizes or when the assumptions of normality and homogeneity of variances required for the paired t-test are not met. By using the chi-square distribution, McNemar's Test provides a way to determine the significance of the differences between paired categorical data, while accounting for the dependency between the observations.
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help how do yall find this easy i cannot-
Answers
Answer:
153/50 or 3.06
Step-by-step explanation:
Just convert them into improper fractions and put parenthesis around the numbers with a division symbol in the middle of the two numbers.
Ex. (3/2)*(4/3)
On a 3-night camping
trip. 9 campers each
slept 4 hours a night.
(This is because of all
the creatures in their
tents!) What is the
reduced ratio of total
hours of sleep to nights?
a. 3 to 4
b. 36 to 1
c. 4 to 12
d. 27 to 36
Answers
Answer:
c is s the answer because I've already did this
Thurman put $90 into a CD that pays 4.4% interest. According to the rule of
72, approximately how long will it take for his money to double?
Answers
Answer:
B-16.4 years
Step-by-step explanation:
NEED HELP ASAP PLEASE!!!
An airplane is headed due south with an airspeed of 300 kph. A wind is blowing from the west at 120 kph. What is the ground speed and direction of the plane?
323 kph with a directional bearing of S 22° E
328 kph with a directional bearing of S 68° E
420 kph with a directional bearing of S 22° E
420 kph with a directional bearing of S 68° E
Answers
Answer:
a) 323 kph with a directional bearing of S 22° E
Step-by-step explanation:
to find the ground speed:
just do the pythagorean theorem.
c²=300²+120²
c²=104400
c=323
to find the directional bearing:
use the law of sines!
sin(90)/323 = sin(θ)/120
(*explanation: since the plane is heading due south and the wind is from due west, it makes a 90 degree angle; and the side opposite of the 90 degree angle is 323. set it equal to sin theta over 120; theta is the angle u r trying to find, and the side opposite of it is 120)
cross multiply:
120sin90=323sin(θ)
divide 323 on both sides:
sin(θ)=0.3715
use inverse sin to find out degree measure:
sin⁻¹(0.3715)=21.8
rounds up to 22.
The ground speed and direction of the plane is 323 kph with a directional bearing of S 22° E. Therefore, option A is the correct answer.
To find the ground speed of the plane, we need to know what a vector is
What is a vector?
A vector is a variable that has both magnitude and direction.
Since the airplane is headed due south with an airspeed of 300 kph, its vector is V = (-300 kph)j.
Also, the wind is blowing from the west at 120 kph. Its vector is v = (-120 kph)i
Now, the ground speed of the plane V' is the resultant vector of the airspeed and wind speed.
What is a resultant vector?A resultant vector is the sum of two or more vectors.
So, V' = v + V
V' = (-120 kph)i + (-300 kph)j
So, its magnitude V' = √(x² + y²) where
x = -120 kph and
y = -300 kph.
So, V' = √[(-120 kph)² + (-300 kph)²]
= √[14400 kph² + 90000 kph²]
= √(104400 kph²)
= 323.11 kph
≅ 323 kph
The direction of the plane
Its direction, Ф = tan⁻¹(y/x)
Ф = tan⁻¹(-300 kph/-120 kph)
Ф = tan⁻¹(2.5)
Ф = 68.2°
From the South-line we have α = 90° - Ф = 90° - 68.2° = 21.8° ≅ 22°
So, the directional bearing of the plane is S22°E.
So, the ground speed and direction of the plane is 323 kph with a directional bearing of S 22° E.
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Which equation best represents the line graphed above?
Answers
Answer: x=2 because for any y value the x value is always 2.
Can pls someone help me with my homework plsss
Answers
Step-by-step explanation:
Given that,
Hypotenuse, H = 10
The angle of elevation is 44°
We know that,
\(\sin\theta=\dfrac{P}{H}\)
P is perpendicular
\(P=H\times \sin\theta\\\\P=10\times \sin(44)\\\\P=6.94\)
Using Pythagoras theorem,
\(RT=\sqrt{10^2-6.94^2}\\\\RT=7.1\)
The sum of angle of a triangle is equal to 180.
44+90+S=180
S=180-134
S = 46°
Hence, this is the required solution.
Dustin wants to build a rectangular pen for his animals. One side of the pen will be against the barn; the other three sides will be enclosed with wire fencing. If Dustin has 900 feet of fencing, what dimensions would maximize the area of the pen
Answers
According to the question to maximize the area of the pen, the dimensions should be a length of 450 feet and a width of 225 feet.
To maximize the area of the rectangular pen, we need to find the dimensions that use up the entire 900 feet of fencing.
Let's assume the length of the pen is L and the width is W. Since there are three sides that need fencing, we have \(2W + L = 900.\)
To maximize the area, we need to express the area A in terms of a single variable. The area of a rectangle is given by \(A = L * W.\)
From the equation \(2W + L = 900\), we can solve for L to get \(L = 900 - 2W.\)
Substituting this value of L in the equation for the area, we have \(A = (900 - 2W) * W.\)
Expanding and simplifying, we get \(A = 900W - 2W^2.\)
To find the dimensions that maximize the area, we need to find the vertex of the quadratic function \(A = -2W^2 + 900W\). The vertex can be found using the formula \(W = -b / (2a)\), where \(a = -2\) and \(b = 900.\)
\(W = -900 / (2 * -2) = 225.\)
Substituting this value of W back into the equation for L, we have \(L = 900 - 2(225) = 450.\)
Therefore, to maximize the area of the pen, the dimensions should be a length of 450 feet and a width of 225 feet.
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how to find the missing length of a right triangle
Answers
Answer:
see below
Step-by-step explanation:
Use pythagorean theorem:
\(a^{2} +b^{2} =c^{2} \\\)
For example, if we are given the lengths:
Short side: 3
Hypotenuse: 5
and we have to find the other side length:
\(a^{2} +b^{2} =c^{2} \\3^{2} +b^{2} =5^{2} \\9+b^{2} =25\\b^{2} =16\\b=4\)
So, the missing side length would be 4.
Hope this helps! :)
the length and breadth of a rectangular plot is in the ratio of 4:3. if the perimeter is 140m, find the length and breadth of a rectangle
Answers
Answer:
l = 40 m and b = 30 m
Step-by-step explanation:
Let the length and breadth of a rectangle be 4x and 3x.
We know that,
The perimeter of a rectangle is given by :
P = 2(l+b)
140 = 2(4x+3x)
70 = 7x
x = 10
Length, l = 4x = 4(10) = 40 m
Breadth, b = 3x= 3(10) = 30 m
So, the length and breadth of a rectangle are 40 m and 30 m respectively.
. (3 pts.) Let's say you have written the shellcode program (see below) that overflows a character buffer to give you root shell access. What are the implications of having a null bytes "\x00" in your shellcode? Select all that apply.
char shellcode [] =
"\xac\xdf\x5e\x19\x85\x0a\xc4\x36\x00\x00\xc3\x26\x0c\x00\x00\x00"
"\x00\xb8\x0b\x00\x00\x00\x89\xf3\x8d\x4e\x08\x8d\x56\x0c\xcd\x80"
"\xb8\x01\x00\x00\x00\xbb\x00\x00\x00\x00\xcd\x80\xc8\xe1\xff\xff" "\xff\xdf\x64\x63\xce\x2f\x73\x68\x00\x89\xec\x5d\xc3";
void main() {
int *ret;
ret = (int *) &ret + 2;
(*ret) = (int) shellcode;
a. There are no implications; the shellcode cannot be modified.
b. The null byte indicates the end of a string which will terminate the function. c. There must be no null terminators in the shellcode in order for the exploit to work.
d. The exploit will succeed and the user will be given a root shell prompt.
ddress is approximately in the middle of the NOP sled,
Answers
The null byte indicates the end of a string that will terminate the function. Also, there must be no null terminators in the shellcode in order for the exploit to work. Therefore, the correct options are B and C.
When a shellcode program contains null bytes "\x00", the implications of having a null byte in your shellcode are that there must be no null terminators in the shellcode in order for the exploit to work and the null byte indicates the end of a string which will terminate the function.
The null byte indicates the end of a string which will terminate the function and There must be no null terminators in the shellcode in order for the exploit to work. In computer programming, the string is a sequence of characters that terminates with a null character. Null bytes inside the buffer cannot be included in the shellcode since it would be treated as a string-terminating character and the rest of the buffer would be ignored.
The shellcode program is written in C programming language. This shellcode program overflows a character buffer to give root shell access. By using this program, the user can gain access to the root shell prompt and run the desired commands.
So, the correct options are B and C.
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y’all help 3 giving brainlest if you help and give steps (send picture of your work if you can)
Answers
Answer 200 to 520 and 300 to 580.
Step-by-step explanation:
Each one increases by 60.
Answer:
They are adding 240 each time so; I'm a little unsure because the top is 160 and that's only 60 but...
Step-by-step explanation:
Could someone help????
Answers
Answer:
3 1/2 + 3 1/2 is 7
Step-by-step explanation:
Find the volume of the cone. Express your answer in terms of .
Radius of 6mm
Height if 28 mm
Answers
Answer:
The volume of the cone is approximately 1051.2 cubic millimeters (mm^3).
Step-by-step explanation:
The formula for the volume of a cone is given as V = (1/3) * π * r^2 * h, where r is the radius of the circular base of the cone, h is the height of the cone, and π is the mathematical constant pi.
Substituting the given values, we get:
V = (1/3) * π * (6mm)^2 * 28mm
V = (1/3) * π * 216mm^3 * 28mm
V = (1/3) * π * 6048mm^3
V = (π/3) * 6048mm^3
Hence, the volume of the cone is (π/3) * 6048mm^3, which is the exact answer. If you need an approximate decimal answer, you can use a calculator and substitute the value of π as 3.14159 to get V ≈ 20,107.06mm^3 (rounded to two decimal places).
Answer:V≈1.06×10-6m³
r= Radius 6mm
h= Height 28mm
Unit Conversion:
r=6×10-3m
h=0.028m
Solution
V=πr2h
3=π·6×10-32·0.028
3≈1.05558×10-6m³
- Dan had 268 dollars to spend on 8 books. After
buying them he had 12 dollars. How much did each book cost?
Answers
Answer:
$32
Step-by-step explanation:
Subtract 12 from 268, which makes 256, then divide 256 by 8 to get 32 :)
identify the meanings of the varibles in the point-slope form of a line
Answers
The variables in the point-slope form of a line (x₁,y₁) is a given point on the line, m the slope of the line and (x,y) is any point on the line.
Given that the variables in the point-slope form of a line.
The general form of the equation of a line is y - y₁ = m (x-x₁) where m is the slope of line, y₁ and x₁ is any point on the line
It is an equation of degree one x and y represent the coordinate of the point on the line represented in the coordinate plane
m = slope of the line
The general equation of a line is y = mx + c where
m = slope of line
(x₁ , y₁) = it is the given point on the line as data points are represented as (x₁,y₁) , (x₂,y₂) , (x₃,y₃) and so on
(x ,y) = it is any point line where x is any point on the x-axis and y is any point on y-axis
Hence, the variables in the point-slope form of a line where (x₁,y₁) is a given point on the line, m the slope of the line and (x,y) is any point on the line.
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Determine whether the geometric series is convergent or divergent. If it is convergent, find its sum.
[infinity]
∑
n = 0
1
(
√
1
7)n
.
Answers
To determine whether the geometric series ∑(n=0 to ∞) (√(1/7))^n is convergent or divergent, we can examine the common ratio (r) of the series.
The given series has a common ratio of (√(1/7)). For a geometric series to be convergent, the absolute value of the common ratio (|r|) must be less than 1.
In this case, |√(1/7)| = √(1/7) ≈ 0.377, which is less than 1. Therefore, the geometric series is convergent.
To find the sum of the geometric series, we can use the formula for the sum of an infinite geometric series:
Sum = a / (1 - r),
where 'a' is the first term and 'r' is the common ratio.
In this case, the first term 'a' is (√(1/7))^0 = 1, and the common ratio 'r' is √(1/7).
Substituting these values into the formula, we have:
Sum = 1 / (1 - √(1/7)).
To simplify this expression, we can rationalize the denominator:
Sum = 1 / (1 - √(1/7)) * (1 + √(1/7)) / (1 + √(1/7)).
Multiplying the numerators and the denominators, we get:
Sum = (1 + √(1/7)) / (1 - (1/7)).
Simplifying further:
Sum = (1 + √(1/7)) / (6/7).
Finally, multiplying the numerator by 7, we obtain the sum of the geometric series:
Sum = 7(1 + √(1/7)) / 6.
Therefore, the sum of the given geometric series is 7(1 + √(1/7)) / 6.
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Stefan sells Jin a bicycle for $149 and a helmet for $16. The total cost for Jin is 150% of what Stefan spent originally to buy the bike and helmet. How much did Stefan spend originally? How much money did he make by selling the bicycle and helmet to Jin?
Answers
Answer:
55
Step-by-step explanation:
149+16=165, 165/150=1.1, 1.1*100=110, 165-110=55
Answer:
55
Step-by-step explanation:
i used a calculator cause IK the equation but I couldn't get the right answer
What is the answer ?
Answers
Answer: 304
Step-by-step explanation:
8(8)=64
24(10)=240
240+64=304
I had $13 my mom gave $10 my dad gave $30 my aunt and uncle gave me $100 I had another $7 how much did I have
Answers
Answer:
you would have $160
Step-by-step explanation:
13+10+30+100+7= 160
The total amount $160 he have.
Given that,
He had amount already = $13
His mom gave him = $10
His dad give him = $30
His aunt and uncle give him = $100
He had another amount = $7
We have to determine,
How much amount he have..
According to the question,
To obtain the amount he had add all the amounts.
Therefore,
Total amount he had = He had amount already + Her mom gave him + His dad gave him + His aunt and uncle give him + He had another amount.
Total amount he had = $13+ $10+ $30+ $100 + $7
Total amount he had = $160
Hence, The required amount he had is $160.
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A computer store has marked up a $75 laptop to $110.06. what is the markup rate? round to the nearest hundredth.
Answers
The markup amount is $110.06 - $75 = $35.06.
The markup rate is the markup amount divided by the original cost, expressed as a percentage: $35.06/$75 = 0.4675 or 46.75% (rounded to the nearest hundredth).
Therefore, the markup rate is 46.75%.
How much profit did the store make on the laptop?To calculate the profit, we need to subtract the cost price from the selling price. The selling price of the laptop is $110.06 and the cost price is $75. Therefore, the profit made by the store is $110.06 - $75 = $35.06. This means that the store earned $35.06 on this laptop, which is the profit. It is important to note that this does not include any additional expenses that the store may have incurred, such as shipping or handling costs.
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4. Find the lowest number which is exactly divisible by 12 and 18.
Answers
Answer:
36
Step-by-step explanation:
36÷12=3
36÷18=2
What is the rule for multiplying by 6?.
Answers
Answer:
If you multiply 6 by an even number, it will end in the same digit. For example, 6 x 4 = 24, so 4 and the 4 in 24 are the same. The number in the tens place will be half the number in the ones place, so the 2 in 24 is half of 4
In a test out of 80 marks, pupils have to get 45% to pass the test. How many marks do pupils need to get to pass the test
Answers
Answer:
33 marks
Step-by-step explanation:
solution:
45%of 80 marks
45/100*50 marks
33 marks
A new phone costs
$850. Each year, its
value falls by 37.5%.
Answers
The value of the phone can be modeled with the exponential decay:
y = 850*(0.825)^x
How to find the value of the phone after x years?We know that the new phone costs $850 and its value decays at a rate of 37.5% per year
Then the value can be modeled by an exponential decay of the form:
y = A*(1 - r)^x
Where x is the number of years, A is the initial value, and r is the percentage in decimal form, then we will get:
A = 850
r = 0.375
Replacing that we will get the exponential decay:
y = 850*(1 - 0.375)^x
y = 850*(0.825)^x
That equation gives the value after x years.
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HELP ASAPPPPPPPPPPPPP
Answers
Answer:
Answer: A
Step-by-step explanation:
(go to attachment to see explanation)
hope this helped :D
