Find the maximum rate of change of the function f (x, y) = x^2y^2 + xy −10 at the
point (1, 1), and the direction in which it occurs.

Find The Maximum Rate Of Change Of The Function F (x, Y) = X^2y^2 + Xy 10 At Thepoint (1, 1), And The

Answers

Answer 1

Answer:

The maximum rate of change of the given function is equal to 3√2.

The direction at which the vector points is given by the unit vector [tex]\displaystyle \frac{\sqrt{2}}{2} \hat{\i} + \frac{\sqrt{2}}{2} \hat{\j}[/tex].

General Formulas and Concepts:
Pre-Calculus

Vectors

Unit Vector:
[tex]\displaystyle \frac{\overline{v}}{|\overline{v}|}[/tex]

Calculus

Differentiation

DerivativesDerivative Notation

Derivative Property [Addition/Subtraction]:
[tex]\displaystyle (u + v)' = u' + v'[/tex]

Derivative Rule [Basic Power Rule]:

f(x) = cxⁿf’(x) = c·nxⁿ⁻¹

Multivariable Calculus

Partial Derivatives

Gradient:
[tex]\displaystyle \nabla f(x, y, z) = \frac{\partial f}{\partial x} \hat{\i} + \frac{\partial f}{\partial y} \hat{\j} + \frac{\partial f}{\partial z} \hat{\text{k}}[/tex]

Gradient Property [Addition/Subtraction]:
[tex]\displaystyle \nabla \big[ f(x) + g(x) \big] = \nabla f(x) + \nabla g(x)[/tex]

Step-by-step explanation:

Step 1: Define

Identify given.

[tex]\displaystyle f(x, y) = x^2y^2 + xy - 10[/tex]

[tex]\displaystyle \text{Point} \ (1, 1)[/tex]

Step 2: Find Maximum Rate of Change

[Function] Take gradient:
[tex]\displaystyle \nabla f(x, y) = \frac{\partial}{\partial x} \bigg[ x^2y^2 + xy - 10 \bigg] \hat{\i} + \frac{\partial}{\partial y} \bigg[ x^2y^2 + xy - 10 \bigg] \hat{\j}[/tex][Gradient] Rewrite [Gradient Property - Addition/Subtraction]:
[tex]\displaystyle \nabla f(x, y) = \bigg[ \frac{\partial}{\partial x} x^2y^2 + \frac{\partial}{\partial x} xy - \frac{\partial}{\partial x} 10 \bigg] \hat{\i} + \bigg[ \frac{\partial}{\partial y} x^2y^2 + \frac{\partial}{\partial y} xy - \frac{\partial}{\partial y} 10 \bigg] \hat{\j}[/tex][Gradient] Differentiate [Derivative Rule - Basic Power Rule]:
[tex]\displaystyle \nabla f(x, y) = \bigg( 2xy^2 + y \bigg) \hat{\i} + \bigg( 2yx^2 + x \bigg) \hat{\j}[/tex][Gradient] Substitute in point:
[tex]\displaystyle \nabla f(1, 1) = 3 \hat{\i} + 3 \hat{\j}[/tex][Gradient] Take magnitude:
[tex]\displaystyle \bigg| \nabla f(1, 1) \bigg| = \boxed{3 \sqrt{2}}[/tex]

∴ the maximum rate of change of the given function f(x, y) is equal to 3√2.

Step 3: Find Direction

[Gradient] Convert to unit vector:
[tex]\displaystyle \text{Direction} = \boxed{ \frac{\sqrt{2}}{2} \hat{\i} + \frac{\sqrt{2}}{2} \hat{\j} }[/tex]

∴ we have found the direction of the gradient.

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Topic: Multivariable Calculus

Unit: Directional Derivatives


Related Questions

where can I get study guide help for algebra 2 thank you ​

Answers

Answer:

Stay after school more often with your math teacher or come early to school to practice math or to see what your stuggling at.

Please help me answer this! Thank you! No links!

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Answer:

139

Step-by-step explanation:

Q. What are the solutions to this equation? (x-3)(x+2)=0
answer choices
Сх
Cx=3,2
A x=3,-2
B x=2,3
x
D x=3,2
X=

Answers

Answer:

x=3,-2

Step-by-step explanation:

take the first part ;(x+2)=0. x=-2 x_3=0 so x=3

8. The expression below is evaluated when x = 9, y = 3,
and z = 2. What is the value of the expression?
8x - 72 + 2y

Answers

Answer:

64

Step-by-step explanation:

First of all substitute the values to where ever you see their corresponding variables to make them multiply.

x = 9

y = 3

z = 2

so we get

8x as 8(9),

7z as 7(2)

2y as 2(3)

then the it's will be

8(9) - 7(2) + 2(3)

which is

72 - 14 + 6

finally we get

64

Solve the inequality.

2(4+2x)≥5x+

Answers

I think it might be 3

A new LED TV currently cost 1,300 when paid in full. There is also a payment plan available where the customer must pay a monthly payment of $72 for 2 years. How much more will a customer who pays in monthly installments end up paying for the LED TV than a customer who pays the 1,300 in full?

Answers

Answer:

$428

Step-by-step explanation:

you can do 72*24= the 2 year period and then you get 1728

you then do 1728-1300 and you will get $428 more

help me with this pls​

Answers

Answer:

sure I'll help u with this as this question required to have a and asuch a being used to review with an associate with you in the comment of the item that you used in the comment is the only one that was a good one to review the product and details about the refund

Please help me with this math question. Thank you.

Answers

Answer:

D

Step-by-step explanation:

We want to find the values of x such that:

[tex]\displaystyle -x^2 + 7x + 5 = 0[/tex]

We can use the quadratic formula. Recall that:
[tex]\displaystyle x = \frac{-b\pm\sqrt{b^2-4ac}}{2a}[/tex]

In this case, a = -1, b = 7, and c = 5. Substitute and evaluate:
[tex]\displaystyle \begin{aligned} x & = \frac{-(7)\pm\sqrt{(7)^2-4(-1)(5)}}{2(-1)}\\ \\ & = \frac{-7\pm\sqrt{49+20}}{-2} \\ \\ & = \frac{-7\pm\sqrt{69}}{-2}\end{aligned}[/tex]

In conclusion, the answer is D.

What is a rewritten form of the radical?
A.
-8816
B.
C.
D.
-90.75
-98616
-2.904

Answers

the only the answer is D

what is the greatest prime number less than 50?

Answers

Answer:

The greatest prime number less than 50 is 47

Two angles are supplementary. The larger angle is
150

larger than the smaller angle. Find the measure of both angles, and separate your answers with a comma.

Answers

Answer:

x = 165, y = 15

Step-by-step explanation:

1. Well, what does supplementary mean? - The angles add up to 180°.

2. Let's form an equation: x + y = 180°

3. Let's say x is the larger angle and y is the smaller angle.

4. Let's form another equation: x = 150 + y

5. Put the two equations next to each other:

x + y = 180

x = 150 + y

6. We know what x is equaled to, so let's plug it into the first equation!

150 + y + y = 180

150 + 2y = 180

-150           -150

2y = 30

y = 15

7. Now, we know what y is, let's find x!

x + 15 = 180

   -15     -15

x = 165

The measure of both angles are  15 degrees and 165 degrees.

What is Angle?

An angle is the figure formed by two rays, called the sides of the angle, sharing a common endpoint, called the vertex of the angle

Given that the two angles are supplementary.

we know that supplementary angles are the angles whose sum is added to 180 degrees.

Let the smaller angle is x.

The larger angle is 150∘ larger than the smaller angle.

y=x+150

Now add both the angles which makes 180 degrees

x+x+150=180

2x+150=180

Subtract 150 on both sides

2x=30

Divide both sides by 2

x=15

So the smaller angle is 15 and the larger angle is 165 degrees

Hence,  the measure of both angles are  15 degrees and 165 degrees.

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Find the missing dimension of the triangle.
Area = 30 mm squared
6 mm = height
b = mm

Answers

Answer:

b = 10 mm

Step-by-step explanation:

the area (A) of a triangle is calculated as

A = [tex]\frac{1}{2}[/tex]bh ( b is the base and h the height )

given A = 30 and h = 6 , then

[tex]\frac{1}{2}[/tex]b × 6 = 30 , that is

3b = 30 ( divide both sides by 3 )

b = 10

why does thsi always happen

Answers

Answer:

It always happens because you asked the question

Step-by-step explanation:

⦁ Martina created this box plot to represent the number of inches of snow that fell during the winter in several different cities.

⦁ What was the least amount of snowfall in any of the cities?
⦁ In which quarter is the data most concentrated? Explain how you know.
⦁ In which quarter is the data most spread out? Explain how you know.

Answers

Using the box plot, it is found that:

The least amount of snowfall in any of the cities was of 5 inches.The data was most concentraded in the third quarter, betwen 35 and 40.The data was most spread out in the second quarter, betwen 13 and 35.

What is a box plot?

It is a plot that focuses the interquartile range of the population, that is, it gives:

The smallest value.The first quartile.The median.The third quartile.The highest value.

In this problem, the box plot starts at 5, hence the least amount of snowfall in any of the cities was of 5 inches.

The data was most concentraded in the third quarter, betwen 35 and 40, as it is the smallest quarter in the graph.

The data was most spread out in the second quarter, betwen 13 and 35, as it is the greater quarter in the graph.

More can be learned about box plots at https://brainly.com/question/27353162

Answer:

a) 5

b) 35 - 40 this is the smallest part in the purple bar

c) 13 - 35, this is the biggest part of the purple bar

Step-by-step explanation:

(this is off my test!)

By looking at the plot, you can see where everything is, and since it asks how you know, there you go.

Please help on math.​

Answers

Answer:

The median is 45

Step-by-step explanation:

The median is also referred to as "the middle" value, so you want to take all the values in order and find the middle number. that will be the median. It's the same either box plots.

A quicker way to find it with box plot is to look at the line inside the of the box (line is not always in center of box) on the plot. that is where you'll find the median on questions like these.

hope this helps! :)

Find the missing side length.
Assume that all intersecting sides meet at right angles.
Be sure to include the correct unit in your answer.

Answers

Answer:

the answer is in the picture

What are all possible values of x if 10x + 7 ≥ 67

Answers

[tex]~~~~10x+7\geq 67\\\\\implies 10x \geq 67 -7\\\\\implies 10x \geq 60\\\\\implies x \geq \dfrac{60}{10}\\\\\implies x \geq 6[/tex]

What is the answer to3/8 + 3/4=?

Answers

9/8 or 9 1/8

You multiply 3/4 with 2/2 to give you 6/8. You then add 3 to 6 and you get 9.

Can someone pls help me I’m stuck on this problem and this is due tomorrow!

Answers

Answer:

I'm not sure if you are allowed to use improper fractions but that would be the only way to have an answer be greater than both of its factors if one of the factors has to be a fraction.

I would do something simple like:

[tex]\frac{2}{1} * 2 = 4[/tex]

Step-by-step explanation:

Since both factors are greater than 1 the product will be greater than both the factors.

Answer:

Equation is below. See model attached.

6/4 x 3 = 18/4

The Jurassic Zoo charges ​$7 for each adult admission and ​$5 for each child. The total bill for the 200 people from a school trip was ​$1124. How many adults and how many children went to the​ zoo?

Answers

Answer:

There were 29 adult admissions; and 72 child's.


The angle of elevation to the top of a Building in New York is found to be 3 degrees from the ground at a
distance of 2 miles from the base of the building. Using this information, find the height of the building.
Your answer is in feet

Answers

Answer:

The height of the building is about 553.4 feet.

Step-by-step explanation:

tan (3°) = h/2

h = 2 tan (3°) = .10481 miles = 553.4 feet

Pls help me worth 50 points + brainliest
(i only can give brainliest if 2 PEOPLE answer)

Answers

box one
box three
box four

box two is incorrect because 0.80/100 x 400 is 3.2
and we are looking for the common denominator of 320 :)

how do i know the common denominator is 320? it's is 80% of 400!

i hope this helps

Answer:

first box, second box and third box.

Step-by-step explanation:

This is because they shared the same answer which is 320.

hope it helps =)

Which inequality is equivalent to |X-4| <9

a) -9>x-4<9

b) -9
c) x-4<-9 or -4<9

d) x-4>-9 or x-4<9

Answers

Answer:

D. x-4>-9 or x-4<9

Step-by-step explanation:

To solve this inequality, you have to isolate the term of x from one side of the equation.

|x-4|<9

-9<x-4<9

x-4>-9

First, add by 4 from both sides.

x-4+4>-9+4

Solve.

-9+4=-5

x>-5

x-4<9

Add by 4 from both sides.

x-4+4<9+4

Solve.

Add the numbers from left to right.

9+4=13

x<13

Therefore, the correct answer is D. "x-4>-9 or x-4<9".

I hope this helps. Let me know if you have any questions.

OLZ HELP ILL MARK AS BRAINLEST
Find the slope of the line that goes through the following points​

Answers

Answer:

B. 1 :)

Step-by-step explanation:

Every time the x value gets bigger, the y laue does too. Positive numbers are larger than negative ones so every time we get closer to 0, our values get bigger :)

Have an amazing day!!

Please rate and mark as brainliest!!

write an inequality for this situation
Embiid scored at least 31 points
A, E > 31
B, E < 31

Answers

Answer:  [tex]E \ge 31[/tex]

Reason:

"at least 31 points" means "31 or more".

So either E = 31 or E > 31 (meaning Embiid scored 31 points or he scored more than 31 points).

Choice A looks like the closest match assuming the inequality sign is [tex]\ge[/tex]

Saying E > 31 instead of [tex]E \ge 31[/tex] would be false because it leaves out the case E = 31.

1. Find the volume of the figure.
4 cm
2 cm
10 cm
10 cm
1 cm
3 cm
1 cm 1 cm

Answers

Put an attachment so we can help you, please.

Maya purchased a boat for $18,340. It's value depreciated by 15% in the first year she owned it. What was her boat worth at the end of this first year?

Answers

The worth of the boat of Maya which goes for 15% deprication this year at the end of the first year of purchase, with the considered purchase price, is $15,589

How to find the percentage from the total value?

Suppose the value of which a thing is expressed in percentage is "a'

Suppose the percent that considered thing is of "a" is b%

Then since percent shows per 100 (since cent means 100), thus we will first divide the whole part in 100 parts and then we multiply it with b so that we collect b items per 100 items(that is exactly what b per cent means).

Thus, that thing in number is

[tex]\dfrac{a}{100} \times b[/tex]

Deprecated price = Initial price- amount of deprication

We have:

Initial price = $18,340

And the amount of deprication = 15% of $18,340

= [tex]\dfrac{18340}{100} \times 15 = 2751 \: \rm dollars[/tex]

(as deprication on first year was by 15% of the original price, and we want to know the deprication for the first year only)

Thus, we get:

Depricated price of Maya's boat = $18,340 - $2,751 = $15,589

Thus, the worth of the boat of Maya which goes for 15% deprication this year at the end of the first year of purchase, with the considered purchase price, is $15,589

Learn more about percent here:

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Mike invests $1,778 in a retirement
account with a fixed annual interest rate of
3% compounded 2 times
per year.
What
will the account balance be after 15 years?

Answers

Answer:

A = $2,779.16

Step-by-step explanation:

A = $2,779.16

A = P + I where

P (principal) = $1,778.00

I (interest) = $1,001.16

First, convert R as a percent to r as a decimal

r = R/100

r = 3/100

r = 0.03 rate per year,

Then solve the equation for A

A = P(1 + r/n)^nt

A = 1,778.00(1 + 0.03/2)^(2)(15)

A = 1,778.00(1 + 0.015)^(30)

A = $2,779.16

Summary:

The total amount accrued, principal plus interest, with compound interest on a principal of $1,778.00 at a rate of 3% per year compounded 2 times per year over 15 years is $2,779.16.

A table of values for the quadratic function f is shown below. If 3 is one solution to f(x) = 0, what is the value of the other solution?
Record your answer and fill in the bubbles on your answer document.

Answers

Answer:The answer to this oroblem would be -7

Step-by-step explanation:We can see this by when f(x) is 0 right next to it is -7

The value of the other solution for f(x)=0 is -7. This is obtained by taking the general form of the quadratic equation f(x)=y.

Quadratic function:A function is represented by an equation f(x)=ax²+bx+c where a≠0 and b, c are any real numbers.This can be generalized into f(x)=y where x is an independent variable and y is a dependent variable.

Given data:

In the table, x values range from -8 to 1

So, we can write using the general form f(x)=y for these values

For x=-8, f(-8)=-2.75

x=-7, f(-7)=0

x=-6, f(-6)=2.25

x=-5, f(-5)=4

x=-4, f(-4)=5.25

x=-3, f(-3)=6

x=-2, f(-2)=6.25

x=-1, f(-1)=6

x=0, f(0)=5.25

x=1, f(1)=4

And 3 is one of the solution to f(x)=0.

Calculating solution for f(x)=0:

It is given that f(x)=0 when x=3

Thus,

f(3)=0

From the table,

f(-7)=0

So, the other solution for the function f(x)=0 is -7.

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Use the function rule f(x)= x• |-1|^x. Find the output f/(2).

Answers

-1^2 = 1
so f(2) = 2*1
= 2. (by multiplication)
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