MATHEMATICS COLLEGE

Suppose f(x,y)=xy, P=(−4,−4) and v=2i+3j. A. Find the gradient of f. ∇f= i+ j Note: Your answers should be expressions of x and y; e.g. "3x - 4y" B. Find the gradient of f at the point P. (∇f)(P)= i+ j Note: Your answers should be numbers C. Find the directional derivative of f at P in the direction of v. Duf= Note: Your answer should be a number D. Find the maximum rate of change of f at P. Note: Your answer should be a number E. Find the (unit) direction vector in which the maximum rate of change occurs at P. u= i+ j Note: Your answers should be numbers

Answers

Answer 1

Answer:

Answers:

Gradient of f: $\nabla f = y\hat{i} + x\hat{j}$

Gradient of f at point p: $\nabla f = -4\hat{i} -4\hat{j}$

Directional derivative of f and P in direction of v: $\nabla f(P)v = -20\n$

The maximum rate of change of f at P: $| \nabla f(P)| = 4√(2)$

The (unit) direction vector in which the maximum rate of change occurs at P is: $v = -(1)/(√(2))\hat{i}-(1)/(√(2))\hat{j}$

Step by step solutions:

Given that:

$f(x,y) = xy$

$P = (-4,4)\n$

$v = 2i + 3j$

A: Gradient of f

$\nabla f = ((\partial f)/(\partial x), (\partial f)/(\partial y)) = (y,x) = y\hat{i} + x\hat{j}$

B: Gradient of f at point P:

Just put the coordinates of p in above formula:

$\nabla f = -4\hat{i} -4\hat{j}$

C: The directional derivative of f and P in direction of v:

The directional derivative is found by dot product of $\nabla f(P) \: \rm and \: \rm v$ :

$\nabla f(P)v = [-4,4][2,3]^T = -20\n$

D: The maximum rate of change of f at P is calculated by evaluating the magnitude of gradient vector at P:

$| \nabla f(P)| = √((-4)^2 + (-4)^2) = 4√(2)$

E: The (unit) direction vector in which the maximum rate of change occurs at P is:

$v = ((-4)/(4√(2)), (-4)/(4√(2))) = -(1)/(√(2))\hat{i}-(1)/(√(2))\hat{j}$

That vector v is the needed unit vector in this case.

we divided by $4√(2)$ to make that vector as of unit length.

Learn more about vectors here:

brainly.com/question/12969462

Answer 2

Answer:

a) The gradient of a function is the vector of partial derivatives. Then

$\nabla f=((\partial f)/(\partial x), (\partial f)/(\partial y))=(y,x)=y\hat{i} + x\hat{j}$

b) It's enough evaluate P in the gradient.

$\nabla f(P)=(-4,-4)=-4\hat{i} - 4 \hat{j}$

c) The directional derivative of f at P in direction of V is the dot produtc of $\nabla f(P)$ and v.

$\nabla f(P) v=(-4,-4)\left[\begin{array}{ccc}2\n3\end{array}\right] =(-4)2+(-4)3=-20$

d) The maximum rate of change of f at P is the magnitude of the gradient vector at P.

$||\nabla f(P)||=√((-4)^2+(-4)^2)=√(32)=4√(2)$

e) The maximum rate of change occurs in the direction of the gradient. Then

$v=(1)/(4√(2))(-4,-4)=((-1)/(√(2)),(-1)/(√(2)))= (-1)/(√(2))\hat{i}-(1)/(√(2))\hat{j}$

is the direction vector in which the maximum rate of change occurs at P.

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What does the point (1, 18) represent

will give points please hurry

Answers

A graph is the plot of a function or a proportion thus (1,18) represents the number of cookies made by using 1 cup of flour is 18.

What is a graph?

A graph is a diametrical representation of any function between the dependent and independent variables.

For example y = x² form a parabola now by looking at only the graph we can predict that it has only a positive value irrespective of the interval of x.

As per the given graph of the number of cookies and cups of flour.

The coordinate of a point is (x,y) and here x-axis is the cups of flour and the y-axis is the number of cookies.

Since x = 1 so cups of flour are 1.

Since y = 18 so the number of cookies is 18.

Hence "A graph is the plot of a function or a proportion thus (1,18) represents the number of cookies made by using 1 cup of flour is 18".

To learn more about graphs,

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#SPJ2

1 cup of flour makes 18 cookies

Global trade provides consumers withоооо
more options and lower prices.
fewer options and lower prices.
more options and higher prices.
fewer options and higher prices.

Answers

The major advantage of participating in global trade is the fact that global trade offers the consumer more options and lower prices.

What is global trade?

The term global trade is the kind of trade that involves people from different countries of the world who trade on goods and services.

The major advantage of participating in global trade is the fact that global trade offers the consumer more options and lower prices.

Learn more about global trade: brainly.com/question/20400015

Answer:

A) More option and lower prices

Step-by-step explanation:

ON ENG 2021

Doris and Miguel are saving money weekly but at different rates. Doris and Miguel both write and graph equations to represent their savings. In both equations y represents the amount in their savings account after a numberof weeks. When the equations are graphed, the lines intersect at the point (6, 114). Which statement best explains the point of intersection?
o Miguel will have $114 more than Doris after 6 weeks
O Doris will have $114 more than Miguel after 6 weeks
O Doris and Miguel will have the same amount after 6 weeks
Doris and Miguel will have the same amount after 114 weeks

Answers

$\large\mathfrak\purple{answer}$

DorisandMiguelwillhavethesameamountafter6weeks.

(hopethisistherightanswerandmarkmebrainliestIhave2moretogo)

X + y + w = b2x + 3y + z + 5w = 6

z + w = 4

2y + 2z + aw = 1

For what values a, b (constants) is the system:

(a) inconsistent?

(b) consistent w/ a unique sol'n?

(c) consistent w/ infinitely-many sol'ns?