(a) Show that a differentiable function f decreases most rapidly at x in the direction opposite the gradient vector, that is, in the direction of −∇f(x). Let θ be the angle between ∇f(x) and unit vector u. Then Du f = |∇f| cos θ _________ . Since the minimum value of cos θ _________ is -1 ________ occurring, for 0 ≤ θ < 2π, when θ = ________ , the minimum value of Du f is −|∇f|, occurring when the direction of u is the opposite of _______ the direction of ∇f (assuming ∇f is not zero). (b) Use the result of part (a) to find the direction in which the function f(x, y) = x4y − x2y3 decreases fastest at the point (2, −5)._________

Answers

Answer 1

Answer:

Step-by-step explanation:

[tex]\text{Show that a differentiable function f decreases most rapidly at x in the }[/tex]

[tex]\text{direction opposite the gradient vector, that is, in the direction of}[/tex] [tex]-\bigtriangledown f(x)[/tex][tex]\text{. Let}\ \theta \ \text{be the angle between} \bigtriangledown f(x) \ \text{and unit vector u. Then } D_u f = \mathbf{|\bigtriangledown f| \ cos \ \theta }}[/tex]

[tex]\text{Since the minimum value of} \ \ \mathbf{cos \ \theta} \ \ is \mathbf{-1} \ \text{occuring \ for \ 0} \le \ \theta \ < 2x, \\ \\ when \ \theta = \mathbf{\pi} , \text{the mnimum value of} \ D_uf \ is} -|\bigtriangledown f|, \text{occuring when the direction of u is } \\ \\ \ \mathbf{the \ opposite \ of} \ \text{the direction of } \ \bigtriangledown f (assuming \ \bigtriangledown f\ is \ not \ zero)[/tex]

b) [tex]\text{From part A:}[/tex]

[tex]If \ f(x,y) = x^4y -x^2y^2 \ \ decreases \ fastest \ at \ the \point \ (2,-5)\\ \\ F(x,y) = x^4y -x^2y^3 \\ \\ f_x = \dfrac{df}{dx}= \dfrac{d}{dx}(x^4y-x^2y^3) \\ \\ f_x = \dfrac{df}{dx}= y4x^3 -2y^3x \\ \\ For(2,-5) \\ \\ f_x = (-5)4(2)^3 -2(-5)^3(2) \\ \\ \mathbf{ f_x = 340}[/tex]

[tex]However; f_y = \dfrac{df}{dy} = \dfrac{d}{dy}(x^4y - x^2y^3) \\ \\ f_y = x^4 -3x^2y^2 \\ \\ Now, for (2, -5)\\ \\f_y = (2)^4 -3(2)^2(-5)^2 \\ \\ f_y = -284[/tex]

[tex]So; \bigtriangledown = < 340,-284> \text{this is the direction of fastest decrease}[/tex]


Related Questions

What is m<4? Pls help me

Answers

Answer:

65°

Step-by-step explanation:

180-115=65

-7 2/3+(-5 1/2) +8 3/4=

Answers

Exact Form : -53/12
Decimal Form : -4.416
Mixed numbers : -4 5/12

Step-by-step explanation:

Please refer to the attachment

Help pleaseee is for nowwww

Answers

Answer:

your bo**bs are so far wooooooooooooooooow!!!!

Write 11/8 as a mixed fraction

Answers

Answer:

1 3/8

Step-by-step explanation:

Since 11 is over 8 that means you already have a whole number so you just add the 3 and you get 1 3/8

To check just in case you just multiply the whole number and denominator and then add the numerator.

a profit of 24% was made wheb a book was sold for birr 34% find the selling price that would have given a profit 28%​

Answers

Answer:

$35.20

Step-by-step explanation:

Given that the profit of 24% was made on $34.10,

And Profit = Selling price - cost price,

Where profit is 24%, and selling price is $34.10, cost price = ?

Then, to find the cost price (z):

Assume the cost price be a 100% and the selling price of $34.10 be 124%

Hence, we have

$34.10 = 124%

z = 100%

Substitue

=> z / 34.10 = 100% / 124 %

=> 124z = 34.10 (100)

124z = 3410

Divide both sides by the coefficient of z which is 124

z = $27.50

Therefore, to find the selling price for a profit of 28%

We have 28% of 27.5 = 7.70

With equation of Profit = Selling price - cost price

We have 7.70 = Selling price - 27.50

Removing 7.7 from both sides

We have, Selling price = 27.50 + 7.70

=> Selling price = $35.20

Hence, the selling prices for a profit of 28% is $35.20

Determine the mean of given data
-7, 6, -6,5, -5, -11

Answers

Answer: The mean is -11

Answer:

mean= average  -3

Step-by-step explanation:  Add all numbers (6+5-5-11-6-7) you will get -18.  Divide by the number of numbers you have.  You have 6 numbers so divide by 6 and get -3 the average.

The population of a village is in the following ratios.
women: children = 13:4
men: children = 7:3
a) Find the ratio women : men.
Give your answer in its simplest form.
women : men

Answers

Answer:

13:7

Step-by-step explanation:

We see that the ratio from women:children is 13:4.

Therefore, the number of women is 13.

In the ratio comparing men to children we see that it is 7:3.

Therefore, the number of men is 7.

Now we need to compare the number of women to the number of men which is 13:7.

If AB is parallel to CD and the slope of AB is -3, what is the slope of CD?

Answers

The slow would have to be the same, so it’s also -3

If a Tv costs $910 and the sales tax is $80, how much will the TV cost in total?

Answers

Answer:

$1,638.00

Step-by-step explanation:

Before Tax Price: $910.00

Sale Tax: 80.00% or $728.00

After Tax Price: $1,638.00

There Are The Before And After :D!!

Please Help me!!! with this problem

Answers

9514 1404 393

Answer:

  D  120

Step-by-step explanation:

We are given the relation ...

  48 = 0.40 × sales

Dividing by 0.40 gives the sales for the day.

  sales = 48/0.40 = 120

120 milkshakes were sold that day.

A quality report for a production line shows a score of 59 points on a 65-point inspection test. A passing score is 95% compliance. Did the production line pass inspection?​

Answers

Answer:

the production line did not pass

Step-by-step explanation:

8*10^4 is how many times as large as 4*10^-5

Answers

answer: D
80 000 > 0,00004
80 000 : 0,00004 = 2 000 000 000
2 000 000 000 = 2*10^9

What is the approximate length of the base of the parallelogram in the diagram?

Answers

Answer:

2piR

Step-by-step explanation:

The base of the parallelogram is equal to the circumference.

The length of the base of the parallelogram will be πr.

What is Parallelogram? What is triangle? What is a expression? What is a mathematical equation?In Euclidean geometry, a parallelogram is a simple quadrilateral with two pairs of parallel sides. The opposite or facing sides of a parallelogram are of equal length and the opposite angles of a parallelogram are of equal measure.A triangle is a polygon with three edges and three vertices. The sum of all the angles of a triangle is 180 degrees. Mathematically - ∠x + ∠y + ∠z = 180°There are different types of triangles such as - equilateral triangle , scalene triangle , isosceles triangle etc.A mathematical expression is made up of terms (constants and variables) separated by mathematical operators. A mathematical equation is used to equate two expressions.

We have a circle converted into a parallelogram as shown.

The angle subtended by each arc at the center =

Ф = (360°/8) = 45°

Length of each arc will be -

L = (Ф/360°) x 2πr

L = (45/360) x 2πr

L = (5/40) x 2πr

L = 1/8 x 2πr

L(A) = πr/4

Length of the base will be -

L(B) = 4 x L(A) = 4 x πr/4 = πr

Therefore, the length of the base of the parallelogram will be πr.

To solve more questions on 2 -D figures, visit the link below-

brainly.com/question/25950519

#SPJ2

Use the Distributive Property to
REWRITE 11(x + 5)

Answers

Answer:

11X+55

Step-by-step explanation:

Multiplicacion

To draw a line that is parallel to 9x+12y=16, what should the slope of the line be?(1 point)

A. 4/3
B. -3/4
C. -4/3
D. 3/4

Answers

Answer:

Hi how are you doing today Jasmine

HELP PLEASE DUE IN 4 MINUTES

Answers

The answer would be B. Hope this helps.

need help geometry question central angles and arcs plsss

Answers

Measure of angle HFG is 250°

Keisha is babysitting at $8 per hour to earn money for a car. So far she has saved $1300. The car that Keisha wants to buy costs at least $5440. Write and solve an inequality to determine the amount Keisha still needs to earn for the car.

Answers

Answer:

$4100

Step-by-step explanation:

For Kaisha to get this car we have these information available

Amount saved = $1300

The cost of the car = $5400

Let The amount left to complete the car = k

We have this equation

K + $1300 = $5400

K = 5400 - 1300

K = 4100

Therefore Kaisha still needs to earn an additional amount of $4100 for the total amount of the car to be completed.

What is the value of x?



Enter your answer in the box.

x =

Answers

10x - 20 + 6x + 8 = 180

16x - 12 = 180

16x = 192

x = 12

hope this helps...!!!

Step-by-step explanation:

The value of x is 12.

please answer this correctly without making mistakes.I want ace,experts and genius people to answer this correctly without making mistakes


add q and 5, add 2 to the result, then add what you have to 4

Answers

Answer:

q+11

Step-by-step explanation:

you have q and you add 5

q+5

they want you to now add 2

q+5 +2

5+2=7

q+7

now add 4

q+7+4

7+4=11

the final answer is q+11

8) How many terms of the arithmetic sequence (2, 4, 6, 8, ...} will give a sum of 600?

PLZ ANSWER

Answers

Answer:

I believe the answer is 300

Step-by-step explanation:

Shade 50% of the figure

Answers

Answer:

All you have to so is shade in 2 mini squares and your good

Two swim clubs are having a membership sale. Aqua Plus is offering a two-year membership for $479.76. Water
World is offering an 18-month membership for $350.00. Which describes the swim club that has the lower-priced
offer?
O Aqua Plus has the lower cost of $239.88 per year
O Water World has the lower cost of $239.88 per year.
O Aqua Plus has the lower cost of $19.44 per month
Water World has the lower cost of $19.44 per month

Answers

Answer:

option d is correct

Step-by-step explanation:

hope this helped

Answer:

D

Step-by-step explanation:

Edge 2019

Assume that a simple random sample has been selected and test the given claim.Claim: The average age of actresses when they win an acting award is more than 34 years oldTo test this claim a simple random sample (SRS) of 27 actresses is used. This sample of 27 actresses had a sample mean of 31.4 years and a standard deviation = 11.2 years. Assume the the distribution of actresses ages is approximately normal.Select all TRUE statements from those given below..For this hypothesis test; H0: ? 34 H1: ?>342. The correct decision is to "support" (fail to reject) the null hypothesis H0.3. We decide to reject the claim that the average age of actresses when they win an acting award is more than 34 years old.4. For this hypothesis test, the P-value = .8815. For this hypothesis test, the test statistic is t = -1.21

Answers

Answer:

how big is your noodle

Step-by-step explanation:

Quincy measured a picnic area near the river and made a scale drawing. He used the scale 1 inch : 10 yards. In the drawing, the picnic area is 17 inches long. What is the length of the actual picnic area?

Answers

Answer:

170 yards

Step-by-step explanation:

1:10

2:20

3:30

4:40

5:50

6:60

7:70

8:80

9:90

10:100

11:110

12:120

13:130

14:140

15:150

16:160

17:170

or you could just use fractions or find the formula

the formula for this equation: number of yards = number of inches times ten

c2 - 6c-5
Evaluate the expression when c = -2

Answers

-2(2)-6(-2)-5
-4+12-5
Answer is 3

If I lost 4% of 100 pencils an 6% of 200 pens how many pencils do I have left and how many pens do I have left?

Answers

To find 4% you multiply 100 by .04
100•.04=4
Then subtract
100-4=96 *you have 96 pencils left*

To find 6% of 200, multiply 200 by .06
200•.06=12
Then subtract
200-12=188 *you have 188 pens left*

I need helpppppppp plssssssssss

Answers

Just put the x in for the equation like 4+2 8+2 10+2 and the outputs would be 6, 10, and 12
4 division sign 2= 2
8 division sign 2= 4
10division sign 2 = 5

im sure you put the output as what you got like 2,4,5 Hope this helps and makes sense!

Hello! I was trying to Do 69x24 but it is sadly not giving helpful tips so I was wondering if you can help me step by step?​

Answers

the answer is 1,656 .

Round 796,154 to nearest 100,000

Answers

Answer:

800,000

Step-by-step explanation:

The 9 makes the 7 round up making it 800,000.

Answer:

I'm pretty sure it's 800,000!

Step-by-step explanation:

Since 9 is larger than 5, you're rounding up. This would cause 7 to be rounded up to 8, making it 800,000!

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