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Digiron [165]
1 year ago
10

Calculus 283 Chapter 16​

Mathematics
2 answers:
ycow [4]1 year ago
8 0

(a) The vector field \vec F is conservative if it is a gradient field. So we try to find a scalar function f such that

\nabla f(x,y) = \dfrac{\partial f}{\partial x} \, \vec\imath + \dfrac{\partial f}{\partial y} = \vec F(x,y)

Given the field

\vec F(x,y) = y\,\vec\imath+x\,\vec\jmath

we need to have

\dfrac{\partial f}{\partial x} = y

\dfrac{\partial f}{\partial y} = x

Integrate both sides of the first equation with respect to x.

\displaystyle \int \frac{\partial f}{\partial x} \, dx = \int y \, dx \implies f(x,y) = xy + g(y)

Differentiate both sides with respect to y.

\dfrac{\partial f}{\partial y} = x + \dfrac{dg}{dy} = x \implies \dfrac{dg}{dy} = 0 \implies g(y) = C

So we have

f(x,y) = xy + C \implies \nabla f = \vec F

is indeed conservative.

(b) By the gradient theorem, the integral is path-independent and

\displaystyle \int_C \vec F \cdot d\vec r = f(5,4) - f(0,2) = \boxed{20}

for <em>any</em> path C that starts at (0, 2) and ends at (5, 4).

Lisa [10]1 year ago
7 0

Answer:

¿Quieres respuestas personalizadas verificadas por un tutor? ¡Pruebe Brainly Tutor y obtenga ayuda de un profesional en minutos!

Step-by-step explanation:

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Nutka1998 [239]
<h3>Answer: n = -11</h3>

=========================================================

Explanation:

Since x-2 is a factor of f(x), this means f(2) = 0.

More generally, if x-k is a factor of p(x), then p(k) = 0. This is a special case of the remainder theorem.

So if we plugged x = 2 into f(x), we'd get

f(x) = x^3+x^2+nx+10

f(2) = 2^3+2^2+n(2)+10

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f(2) = 2n+22

Set this equal to 0 and solve for n

2n+22 = 0

2n = -22

n = -22/2

n = -11 is the answer

Therefore, x-2 is a factor of f(x) = x^3+x^2-11x+10

Plug x = 2 into that updated f(x) function to find....

f(x) = x^3+x^2-11x+10

f(2) = 2^3+2^2-11(2)+10

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3 years ago
Consider this function.
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Answer:

3. Since the range of the original function is limited to y> 6, the domain of the inverse function is x ≥ 6.

Step-by-step explanation:

The domain of a function is the range of its inverse, and vice versa. The only answer choice that expresses this relationship is choice 3.

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Comment on the answer choice:

The slope of the function is undefined at x=4, so restricting the function domain to the portion with positive slope means the domain restriction of the function is x > 4. That also means the range restriction of the function is y > 6. The domain restriction of the inverse function is the same: x > 6, not x ≥ 6. The answer choice has an error.

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y=125x .... (2)

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It means for 4 hours, the total cost of both the DJ's is same at $500.

b. Substitution Method.

Plug y = 125x in equation (1)

125x= 200+75x

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For x= 6, equation (2)

y=125\times 6=750

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