Find the directions in which the function increases and decreases most rapidly at P0Then find the derivatives of the function in these directions.f(x,y) = x^2 + xy + y^2 , Po (-2,2)the direction in which the given function increases most rapidly at P_0(-2,2) is ?and decreases most rapidly in the direction -u=??(D_u f)_P0=
Answers
The function f(x, y) = x² + x y + y², decreases most rapidly in the direction opposite of the gradient vector at the point P₀(-2,2).
Consider a function f(x, y) = x² + x y + y²
∇f = (2x + y) i + (x + 2y ) j
The value of gradient of function f(x, y) at point (-2, 2) would be,
∇f (-2, 2) = - 2i + 2j
Then, the direction in which g increases most rapidly is:
\(\hat{u}\) = ∇f / |∇f|
\(\hat{u}=\begin{bmatrix}\frac{-1}{\sqrt{2} } \\\\\frac{1}{\sqrt{2} } \end{bmatrix}\)
The derivative in this direction is:
\(D=\begin{bmatrix}-2\\\\ 2 \end{bmatrix} . \begin{bmatrix}\frac{-1}{\sqrt{2} } \\\\\frac{1}{\sqrt{2} } \end{bmatrix}\\\\\\D=\begin{bmatrix}\sqrt{2} \\\\\sqrt{2} } \end{bmatrix}\)
The direction in which g decreases most rapidly is:
\(\hat{u}\) = - ∇f / |∇f|
\(\hat{u}=\begin{bmatrix}\frac{1}{\sqrt{2} } \\\\\frac{-1}{\sqrt{2} } \end{bmatrix}\)
The derivative in this direction is:
\(D=\begin{bmatrix}-2\\\\ 2 \end{bmatrix} . \begin{bmatrix}\frac{1}{\sqrt{2} } \\\\\frac{-1}{\sqrt{2} } \end{bmatrix}\\\\\\D=\begin{bmatrix}-\sqrt{2} \\\\ -\sqrt{2} } \end{bmatrix}\)
Hence the function decreases most rapidly in the direction of the vector v = <1/\(\sqrt{2}\), -1/\(\sqrt{2}\) > at the point P₀(-2,2).
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Related Questions
The triangles are similar, find the length of the unknown side. Round your answers to the nearest tenth (0.1), if necessary.
please help me
Answers
Answer:
\(22\)
Step-by-step explanation:
Similar shapes must have corresponding sides in a constant proportion. Therefore, we can set up the following equation and solve for \(?\):
\(\frac{28}{42}=\frac{?}{33}\) (divide corresponding sides)
\(\frac{28}{42}=\frac{?}{33},\\\\?=\frac{28\cdot 33}{42}=\boxed{22}\)
1) Drew is an artist. He paints portraits. The table below
shows the number of portraits painted in hours. Do the
numbers in the table represent a proportional relationship?
Answers
Answer:
Yes
Step-by-step explanation:
Their ratios are equivalent
Skylar invested $330 in an account paying an interest rate of 3 and 3/4% compounded continuously. Montraie invested $330 in an account paying an interest rate of 4 and 3/8% compounded annually. After 18 years, how much more money would Montraie have in his account than Skylar, to the nearest dollar?
Answers
The formula for annually compounded interest is:A=P*(1+r)^tA = 330*(1+0.04375)^18A ≈ $814.14Montraie has $814.14-$675.70 = $138.44 more in his account than Skylar, to the nearest dollar.
The problem is related to compound interest and it involves calculating the difference in the amount of money accumulated by Skylar and Mantri e after 18 years,
given that Skylar invested $330 in an account with an interest rate of 3 and 3/4% compounded continuously and Mantri e invested $330 in an account with an interest rate of 4 and 3/8% compounded annually.Compound interest formula is:
A=P(1+r/n)^(n*t)where A is the amount of money accumulated after t years P is the principal or initial investment r is the annual interest raten is the number of times interest is compounded in a year t is the time in Years For Skylar:Initial Investment,
P = $330Interest Rate, r = 3 and 3/4 %, which can be written as 3.75% in decimal form Time, t = 18 years Since the interest is compounded continuously, the number of times compounded, n = ∞.
The formula for continuously compounded interest is:
A=P*e^(r*t)
where e is the mathematical constant ≈ 2.71828A = 330*e^(0.0375*18)A ≈ $675.70
For Montraie:
Initial Investment, P = $330Interest Rate, r = 4 and 3/8 %,
which can be written as 4.375% in decimal From Time, t = 18 years Since the interest is compounded annually, the number of times compounded, n = 1.
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State whether the following statement is true or false. If false, tell why. If the elimination method leads to 0 = -2, the solution set of the system is {(0, -2)}.
A. True
B. False, the solution set is Ø.
C. False, the equations are equivalent and the solution set is one of the equations.
D. False, the solution set is {(-2,0)}.
Answers
Answer: B. False, the solution set is Ø.
Explanation:
If the elimination method leads to 0 = -2, then that's a false statement. This then has a chain reaction to cause the original system to be inconsistent, meaning there aren't any solutions.
We consider the solution set to be the empty set, denoted with the special symbol of Ø which unfortunately looks like the number 0, but they're different symbols.
Perhaps better notation is to write a pair of curly braces with nothing inside them. So you'd write { } to indicate the empty set.
Please help thanks!
Answers
Answer:
10
Step-by-step explanation:
=13+ 3^3 ÷9 we can change it to:
=13+ 3^3/-9
3^3 can be changed to 27
13+ 27/9
=13+−3
=10
Solve for x.
F
X =
B
H
(13x + 79)°
I (17x + 67)°
C
E
G
Answers
The value of the x is 7.133 degree.
What is parentheses?
A phrase, word, or sentence added to writing as more information or an afterthought is known as a parenthesis. It is emphasised with bracketed text, commas, or dashes. He had been a fan of Rockingham Rovers since he was five years old, as an illustration.
(13 x+79)+(17 x+67)=360
Remove parentheses: (a)=a
13 x+79+17 x+67=360
Group like terms
13 x+17 x+79+67=360
Add similar elements: 13 x+17 x=30 x
30 x+79+67=360
Add the numbers: 79+67=146
30 x+146=360
Move 146 to the right side
30 x=214
Divide both sides by 30
\(\frac{30 x}{30}=\frac{214}{30}\)
Simplify
\(x=\frac{107}{15}\)
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Help me with my math question im gonna give you 10 points just to answer that thankkkkkkkkkkkkkkkkkk you
Answers
Which set of algebra tiles represents the sentence below? Carlton earned 3 more stamps. 1 box contains x and 3 boxes contain plus signs. 1 box contains x and 3 boxes contain minus signs. 3 boxes contain x and 1 box contains a plus sign. 3 boxes contain negative x and 1 box contains a plus sign.
Answers
Answer:
The answer is A.
Step-by-step explanation:
Made a 100 on the quiz on E2020
Answer:
The correct answer is A
Step-by-step explanation:
6x - 2(x + 2) > 2 - 3(x+3)
Solve the inequality
plz help
Answers
Answer:
x > -3/7
Step-by-step explanation:
6x - 2(x + 2) > 2 - 3(x + 3)
Distribute the two and three inside the parenthesis.
6x - 2x - 4 > 2 - 3x - 9
Combine like terms.
4x - 4 > -3x - 7
Add 4 to both sides.
4x > -3x - 3
Add 3x to both sides.
7x > -3
Divide both sides by 7.
x > -3/7
Answer:
x>-3/7
Step-by-step explanation:
6x-2x-4>2-3x-9
4x-4>-3x-7
7x-4>-7
7x>-3
x>-3/7
Data where the difference between data values has meaning but there is no defined starting point
Answers
Examples of such data include stock prices, economic indicators, and meteorological data such as temperature, wind speed, and barometric pressure.
These types of data often have a trend or pattern, but the difference between the data points has meaning and does not necessarily have a defined starting point.
1. Data where the difference between data values has meaning but there is no defined starting point can be defined as data that is not linearly dependent.
2. Examples of such data include stock prices, economic indicators, and meteorological data. These types of data often have a trend or pattern, but the difference between the data points has meaning and does not necessarily have a defined starting point.
3. Stock prices, economic indicators, and meteorological data all have different scales and can move in different directions, making it difficult to track the exact difference between data points.
4. Despite this, these types of data still allow for meaningful analysis and can be used to make predictions and draw conclusions.
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a number increased by 7 given 20
Answers
Answer:
13
Step-by-step explanation:
Which number is closest to 63(square root)?
A. 8.4
B. 7.2
C. 7.9
D. 6.7
Answers
Answer:
C-7.9
Step-by-step explanation:
Because the square root is actually 7.93
Answer:
C. 7.9
Step-by-step explanation:
Do the square root of 63.
\(\sqrt{63} = 7.93725\)
Your answer isn't perfect, but it's asking for which number is the closest to the \(\sqrt{63}\).
\(\sqrt{63}\) ≈ \(7.9\)
solve 16x = 64x + 4 please
Answers
Answer:-1/12
Step-by-step explanation:
16x=64x+4
16x-64x-4
-48x=4
x=4/-48
Ans:-1/12
Answer:
\(\huge\boxed{\sf x = -1/12}\)
Step-by-step explanation:
\(\sf 16x = 64x+4\\\\Subtract \ 64x \ to \ both \ sides\\\\16x-64x=4\\\\-48x = 4\\\\Divide \ -48 \ to \ both \ sides\\\\x= 4 / -48\\\\x = - 1 / 12\\\\\rule[225]{225}{2}\)
Hope this helped!
~AH1807Which is a quadratic function?
A y = X-9
B y = x² +9
C y = x + 2x -9
Answers
Answer:
B y = x² +9
Step-by-step explanation:
A quadratic function must have the highest power of x being x^2 and must have an x^2 term
A y = X-9 - linear
B y = x² +9 quadratic
C y = x + 2x -9 - linear
∠xangle, x and ∠ � ∠yangle, y are supplementary angles. ∠ � ∠yangle, y measures 10 8 ∘ 108 ∘ 108, degrees. What is the measure of ∠ � ∠xangle, x?
Answers
The measure of ∠x is 72 degrees.
If two angles are supplementary, it means that their measures add up to 180 degrees. Let's denote the measure of ∠x as x degrees. Supplementary angles are a pair of angles that, when combined, add up to 180 degrees. In other words, if you have two angles that are supplementary, the sum of their measures will always be 180 degrees.
Given that ∠y measures 108 degrees, we can set up the equation:
∠x + ∠y = 180
Substituting the known value for ∠y, we have:
x + 108 = 180
To find the measure of ∠x, we need to isolate x on one side of the equation. Subtracting 108 from both sides, we get:
x = 180 - 108
x = 72
Therefore, the measure of ∠x is 72 degrees.
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I'm just asking for other people's opinions but isn't this correct? I can still change it so feel free to tell me if you think you know the answer
Answers
Consider the transformation Y = g(X) = 2 −3X, for the random variable X, where the cdf for X is<
FX (x) = 0 , x > 0
1 −e^−4x , x >0
Find the cdf for Y.
Answers
The CDF of Y is: FY(y) = FX((2 - y)/3)
= {1 - \(e^{(-4(2-y)/3)}\), for y ∈ (-∞, 2]
{0, for y < -∞
To find the CDF of Y, we need to first find the range of Y:
Y = g(X) = 2 - 3X
When X = 0, Y = 2
As X approaches infinity, Y approaches negative infinity
Thus, the range of Y is (-∞, 2].
Now, let's find the CDF of Y for y ∈ (-∞, 2]:
FY(y) = P(Y ≤ y) = P(2 - 3X ≤ y)
Solving for X:
X = (2 - y)/3
Since FX(x) = P(X ≤ x), we can substitute (2 - y)/3 for x in FX(x) to get:
FX((2 - y)/3) = P(X ≤ (2 - y)/3)
= 1 - \(e^{(-4(2-y)/3)}\), for y ∈ (-∞, 2]
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An experimenter studies vision in low light by
having participants sit in a darkened room for
either 5, 15, or 25 minutes and then testing their
ability to correctly identify 20 objects. (a) What is
the independent variable here? (b) What are the
conditions? (c) What is the dependent variable?
(d) You would use the scores from which variable
to create a frequency distribution?
Answers
Answer:
Independent variable- time spent in the dark room
Dependent variable - number of objects correctly identified
Conditions of the experiment- low light environment and lighted environments
Scores from the independent variable are used to construct the frequency distribution
Step-by-step explanation:
The independent variable is the factor that is being manipulated in an experiment and this time it is the time spent in a darkened room.
The dependent variable is the response variable, in this case, the number of objects correctly identified.
The experimental conditions involved bringing people from a lighted area into a darkened room.
an exponential function is expressed in the form y ab x the relation represents a growth when
Answers
Answer:
b > 1
Step-by-step explanation:
You want to know the conditions on an exponential function that represents growth.
Growth factorThe value of 'b' in the exponential function y = a·b^x is called the "growth factor." Each time x increases by 1 unit, the value of y is multiplied by 'b'. If that product is increasing, the value of 'b' must be greater than 1.
The relation represents growth when b > 1.
An exponential function in the form \(y = ab^x\) represents growth when the base (b) is greater than 1.
What is exponential function?In an exponential function of the form y = ab^x, the base (b) is a crucial component. The behavior of the function depends on the value of the base.
When the base (b) is greater than 1, it means that b is a positive number larger than 1. In this scenario, as the value of x increases, the value of \(b^x\) also increases exponentially. This results in the function \(y = ab^x\) exhibiting growth.
To better understand this growth behavior, let's consider an example. Suppose we have an exponential function \(y = 2^x\). As x increases from 0, the values of \(2^x\) will be as follows:
For x = 0, \(2^0\) = 1
For x = 1, \(2^1\) = 2
For x = 2, \(2^2\) = 4
For x = 3, \(2^3\) = 8
For x = 4, \(2^4\) = 16
As you can see, as x increases, the values of \(2^x\) grow exponentially. This demonstrates the growth behavior of exponential functions when the base is greater than 1.
It's important to note that when the base (b) is between 0 and 1 (exclusive), the exponential function will exhibit decay or decreasing behavior rather than growth.
In summary, an exponential function of the form \(y = ab^x\) represents growth when the base (b) is greater than 1. As x increases, the function values increase exponentially, indicating a growth pattern.
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I NEED HELP FAST PLEASE I DON'T UNDERSTAND THIS
Answers
The variable associated with the interior and exterior angles in the triangle is equal to 56.
The value of the exterior angle is equal to 116°.
How to determine the value of a variable associated with interior and exterior angles in a triangle
This question presents the case of a geometric system formed by a triangle and a semirray that includes two interior angles and a exterior angle. Please notice that the sum of the measures of interior angles in a triangle equals 180° and the sum of two supplementary angles equals 180°.
First, derive the equation of the missing interior angles in triangle PQR:
m ∠ R = 180° - 60° - x
m ∠ R = 120° - x
Second, obtain the equation for the two supplementary angles:
(120° - x) + (2 · x + 4°) = 180°
124° + x = 180°
x = 56
Third, determine the value of the exterior angle:
θ = 2 · x + 4
θ = 2 · 56 + 4
θ = 116°
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HELP PLEASE!!!!!!
Write the missing improper fraction in each box. Express the answers in simplest form.
Answers
Answer:13/8 and 7/4. 2/1 then 5/2 and 21/8
Step-by-step explanation: If you continue the pattern, you get 13/8 which is already simplified because 13 is a primw number. Then 14/8 which simplifies to 7/4 if divided by 2. Similar process for other ones
How many milligrams are equal to 17. 9 grams?
A. 0. 0179 mg
B. 179 mg
C. 17,900 mg
DX. 179,000 mg
Answers
Answer:
17,900 mg
Step-by-step explanation:
I took the k12 test today
The concept that a message gives different meanings to different objects is called _____.
a. ​encapsulation
b. ​polymorphism
c. ​linear addressing
d. ​dynamic addressing
Answers
The concept that a message gives different meanings to different objects is called option (b) polymorphism.
Polymorphism is a fundamental concept in object-oriented programming (OOP) that allows different objects to respond to the same message or method invocation in different ways. In other words, it allows objects of different classes to be treated as if they were of the same class, as long as they implement the same method or message.
This can make code more flexible, reusable, and easier to maintain. Polymorphism is achieved through inheritance, interfaces, or overloading methods. For example, a "draw" method could be implemented differently for different shapes, such as circles, rectangles, or triangles.
Therefore, the correct option is (b) polymorphism
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What is the answer to The sum of 2 numbers is 19 and their difference is 3? And thank you brainly for helping. I need the answers correct because I play prodigy, a math game. Thank you!
Answers
Answer:
11 and 8
Step-by-step explanation:
11+8=19
11-8=3
Answer:
8 and 11
Step-by-step explanation:
Peter had 4 bags which had equal number of mangoes.He sold 8 mangoes and remained with 24 mangoes.How many mangoes were in each bag?
Answers
Answer: 8 mangoes
Step-by-step explanation:
1. To find the total number of mangoes Peter had, add 8 and 24 which gives you 32. 8 + 24 = 32.
2. If Peter had an equal number of mangoes in each bag, and a total number of 32 mangoes, then you must divide the total number of mangoes by the number of bags, which is 4. 32 ÷ 4 = 8. Therefore, there were 8 mangoes in each bag.
Answer:
The answer is 8 mangoes
What is the midpoint of PQ
Answers
Drag the tiles to the correct boxes to complete the pairs. Not all tiles will be used.
The school band director has a budget $4,428.00 to spend on the band room, instruments, music, and uniforms. There are four sections in the band: drum corps, woodwinds, horns, and color guard. She has received three offers.
Option 1. Repair and replace uniforms at $564.00 per section and spend $182.00 on music.
Option 2. Repair and replace uniforms at $593.00 per section and spend $78.00 on music.
Option 3. Repair uniforms at $287.00 per section, spend $856.00 for improving the acoustics in the band room, and spend $422.00 on music and repairing instruments.
The remainder of the money will be split evenly among the four sections for additional costs. Match each option with the amount of money each section will receive for additional costs.
Answers
Answer:
The amount of money left for each section to spend from the budget in additional costs if they follow exact of the
options is
Option 1-$497.5
Option 2 - $494.5
Option 3 - $500.5
The image to show the amounts to match to isn't attagged to the question, but these are the answers and they should
be matched accordingly.
Step-by-step explanation:
We first calculate the amount left per section after following each of the options.
Budget $4,428.00
Option 1.
Repair and replace uniforms at $564.00 per section and spend $182.00 on music.
There are 4 sections, hence, total amount spent in this option:
4(564) + 182 $2438
Amount of money left from the budget after spending= 4428-2438 = $1990
Amount left per section = (1990/4) =
$497.5
Option 2
ធ
Repair and replace uniforms at $593.00 per section and spend $78.00 on music.
There are 4 sections, hence, total amount spent in this option:
4(593) + 78 $2450
Amount of money left from the budget after spending= 4428 - 2450= $1978
Amount left per section = (1978/4)=
$494.5
Option 3
Repair uniforms at $287.00 per
section, spend $856.00 for improving the acoustics in the band room, and spend $422.00 on music and repairing instruments.
There are 4 sections, hence, total amount spent in this option:
4(287) + 856 +422 = $2426
Amount of money left from the budget
after spending = 4428-2426 = $2002
Amount left per section = (2002/4)= $500.5
Factor completely
18x3 + 81x - 27
Answers
Answer:
108x
Step-by-step explanation:
Answer:
108
Step-by-step explanation:
18x3 + 81x - 27
(18x3) + (81x - 27)
54 + 54
108
What is the value of x in the equation 8 + 4 = 2(x - 1)?
a
13/2
b
5
c
7
d
11/2
Answers
Answer:
Step-by-step explanation:
Find the value of x and found to the nearest degree help ASAP
Answers
Answer:
22
Step-by-step explanation:
sin(X) = perpendicular/base
sin(X) = 7/19
X=arcsin(7/19)=22