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-BARSIC- [3]
3 years ago
15

Find the midpoint: H(-1, 2) and K(-7,-4)

Advanced Placement (AP)
1 answer:
UNO [17]3 years ago
7 0

Answer:

(-4,-1  )

Explanation:

Midpoint Formula: (\frac{x_1+x_2}{2},\frac{y_1+y_2}{2}  )

Simply plug in your 2 coordinates into the formula to find midpoint:

(\frac{-1-7}{2},\frac{2-4}{2}  )

(\frac{-8}{2},\frac{-2}{2}  )

(-4,-1  )

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2. Find the solution of each differential equation. (a) y/2y-8y = 0 (b) 25y/- 20y + 4y = 0 (c) y + 2y + 2y = 0 2. Find the solut
jek_recluse [69]

Each of these ODEs is linear and homogeneous with constant coefficients, so we only need to find the roots to their respective characteristic equations.

(a) The characteristic equation for

y'' - 2y' - 8y = 0

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The characteristic roots are r=4 and r=-2. Then the general solution is

\boxed{y = C_1 e^{4x} + C_2 e^{-2x}}

where C_1,C_2 are arbitrary constants.

(b) The characteristic equation here is

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\boxed{y = C_1 e^{2/5\,x} + C_2 x e^{2/5\,x}}

(c) The characteristic equation is

r^2 + 2r + 2 = (r + 1)^2 + 1 = 0

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y = C_1 e^{(-1+i)x} + C_2 e^{(-1-i)x}

Recall Euler's identity,

e^{ix} = \cos(x) + i \sin(x)

Then we can rewrite the solution as

y = C_1 e^{-x} (\cos(x) + i \sin(x)) + C_2 e^{-x} (\cos(x) - i \sin(x))

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When Isabel wakes up, she realizes to her horror that she's slept late—and Ruth is nowhere to be found. When she confronts Becky about Ruth's whereabouts, Becky dances around the question until Isabel finally demands an answer. It is as she fears: Madam sold Ruth the night before and has sent her to the West Indies. Becky's theory is that the milk had a sedative mixed in with it to knock Isabel out and keep her from interfering. Ugh.

Isabel confronts Madam about the news. In her anger, Madam grabs a painting off the wall and smashes it over Isabel's head, so Isabel runs out of the house and into the street, not caring about how badly it looks to be a slave running down the street as Madam Lockton chases you.  

In her mind, there's only one clear solution to all this: Go to Colonel Regan and demand that he make good on his promise. She goes to his headquarters and shouts the ad astra code until someone lets her in and takes her to the Colonel. Before she even gets a chance to open her mouth, though, Madam barges in and demands to know what's going on.

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In one last attempt at freedom, Isabel runs for an open window and almost makes it out before being pulled back in.

Explanation:

all based on research

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