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brilliants [131]
3 years ago
13

Determine whether or not F is a conservative vector field. If it is, find a function f such that F = ∇f. (If the vector field is

not conservative, enter DNE.) F(x, y) = (y2 − 4x)i + 2xyj
Mathematics
1 answer:
Andru [333]3 years ago
7 0

Answer:

f = xy²-2x² satisfies that  F = ∇f.

Step-by-step explanation:

F(x,y) = (y² - 4x) i + 2xy j

We want f(x,y) such that

f_x(x,y) = y^2 -4x\\f_y(x,y) = 2xy

Lets find a primitive of y²-4x respect to the variable x. We need to think y (and y²) as constants here, so a primitive of y² would be xy² the same way that a primitive of k is xk (because we treat y² as constant). A primitive of x is x²/2, thus a primitive of 4x is 2x². Thus, a primitive of y²-4x is xy² - 2x². We can obtain any other primitive by summing a constant, however since we treated y as constant, then we have that

f(x,y) = xy^2 - 2x^2 + c(y)

where c(y) only depends on y (thus, it is constant repsect with x).

We will derivate the expression in terms of y to obtain information about c(y)

2xy = f_y(x,y) = \frac{d}{dy} (xy^2 - 2x^2 + c(y) ) = 2xy -0 + \frac{d}{dy} c(y) = 2xy + \frac{d}{dy} c(y)

Thus, \frac{d}{dy} is constant. We can take f(x,y) = xy²-2x². This function f satisfies that  F = ∇f.

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The perimeter of any shape is a measure of the length of the peripheral of that shape.

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green and blue.

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3 years ago
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Answer:

<u>Oliver and Lenny reached the top of the hill at the same time</u>

<u>The proportion of time Oliver spent hiking to that of Lenny is 1:1</u>

Step-by-step explanation:

1. Let's review the information given to us to answer the question correctly:

Oliver's speed = 5 km/h during the first half, then decreased his speed to 4 km/h during the second half.

Lenny's speed = 4 km/h during the first half, then increased his speed to 5 km/h in the second half.

2. Who reached the top of the hill first? What is the proportion of time he spent hiking to that of his friend?

Let's assume the distance from the bottom to the top of the hill is 10 kilometers for calculating the time it will take Oliver and Lenny to reach the top of the hill:

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Lenny = Time of hiking first 5 km + Time of hiking second 5 km

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Lenny = 5/4 + 5/5

Lenny = 5/4 + 1 = 2 1/4 hours

<u>Oliver and Lenny reached the top of the hill at the same time</u>

<u>The proportion of time Oliver spent hiking to that of Lenny is 1:1</u>

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