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Morgarella [4.7K]
2 years ago
6

Somebody help me out

Mathematics
1 answer:
Alekssandra [29.7K]2 years ago
3 0

Given Function : f(x) = -3x² + 3x + 7

To find out the value of f(8) : Substitute 8 in the function in place of x

⇒   f(8) = -3(8)² + 3(8) + 7

⇒   f(8) = -3(64) + 24 + 7

⇒   f(8) = -192 + 24 + 7

⇒   f(8) = -161

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Show that the line integral is independent of path by finding a function f such that ?f = f. c 2xe?ydx (2y ? x2e?ydy, c is any p
Juli2301 [7.4K]
I'm reading this as

\displaystyle\int_C2xe^{-y}\,\mathrm dx+(2y-x^2e^{-y})\,\mathrm dy

with \nabla f=(2xe^{-y},2y-x^2e^{-y}).

The value of the integral will be independent of the path if we can find a function f(x,y) that satisfies the gradient equation above.

You have

\begin{cases}\dfrac{\partial f}{\partial x}=2xe^{-y}\\\\\dfrac{\partial f}{\partial y}=2y-x^2e^{-y}\end{cases}

Integrate \dfrac{\partial f}{\partial x} with respect to x. You get

\displaystyle\int\dfrac{\partial f}{\partial x}\,\mathrm dx=\int2xe^{-y}\,\mathrm dx
f=x^2e^{-y}+g(y)

Differentiate with respect to y. You get

\dfrac{\partial f}{\partial y}=\dfrac{\partial}{\partial y}[x^2e^{-y}+g(y)]
2y-x^2e^{-y}=-x^2e^{-y}+g'(y)
2y=g'(y)

Integrate both sides with respect to y to arrive at

\displaystyle\int2y\,\mathrm dy=\int g'(y)\,\mathrm dy
y^2=g(y)+C
g(y)=y^2+C

So you have

f(x,y)=x^2e^{-y}+y^2+C

The gradient is continuous for all x,y, so the fundamental theorem of calculus applies, and so the value of the integral, regardless of the path taken, is

\displaystyle\int_C2xe^{-y}\,\mathrm dx+(2y-x^2e^{-y})\,\mathrm dy=f(4,1)-f(1,0)=\frac9e
8 0
3 years ago
Use the square root property of equality to solve<br> (x - 3)2 = -4.<br> The solutions are
Volgvan

Answer:

3+2i

Step-by-step explanation:

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larisa [96]
Formula is π * r^2

π = about 3.14 and the radius is half of the diameter so 28/2 = 14

3.14 * 14^2 = 615.44

B

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3 years ago
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Which term below correctly completes the following sentence?
Norma-Jean [14]

Answer:

c

Step-by-step explanation:

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Is there another way to convert fractions to decimals (explanation pls ❤️)
AlekseyPX

Answer:

The easiest way to convert a fraction to a decimal is to divide the numerator (the top of the fraction) by the denominator (the bottom of the fraction) by using a calculator. The resulting answer will be the value of the fraction expressed as a decimal number.

Step-by-step explanation:

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