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1 year ago

3. If dy = x^2y^2 , then what is d2y/dx2

Mathematics

1 answer:

lakkis [162]1 year ago

5 0

$\text{Given that,}\\\\\dfrac{dy}{dx} = x^2 y^2\\\\ \implies \dfrac{d^2 y}{dx^2} = x^2\cdot 2y \dfrac{dy}{dx}+ y^2 \cdot 2x\\\\\implies \dfrac{d^2 y}{dx^2} = x^2 \cdot 2y \cdot x^2 y^2 +2xy^2\\\\\implies \dfrac{d^2 y}{dx^2} =2x^4y^3 + 2xy^2 = 2xy^2(x^3y+1)$

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Which angles are coterminal with −6π5 ? Select each correct answer. 14π/5 −16π/5 −π/5 −11π/5

wariber [46]

ANSWER

$- \frac{16 \pi}{5} \: and \: \frac{14\pi}{5}$

EXPLANATION

Coterminal angles are angles in standard position that have the same terminal side.

To find an angle in standard position that is coterminal with

$- \frac{6 \pi}{5}$

We add or subtract multiples of
$2 \pi$

The first addition gives,

$- \frac{6 \pi}{5} + 2 \pi = \frac{4 \pi}{5}$

We add the second multiple to get,

$- \frac{6 \pi}{5} +2( 2 \pi )= \frac{14 \pi}{5}$

Since this is the maximum value among the options, we end the addition here.

Let us now subtract the first multiple to get,

$- \frac{6 \pi}{5} - 2 \pi = - \frac{16 \pi}{5}$

We end the subtraction here because this is the least value among the options.

Therefore the angles that are coterminal with
$- \frac{6 \pi}{5}$
are

$- \frac{16 \pi}{5} \: and \: \frac{14\pi}{5}$

8 0

2 years ago

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Two brothers shared the coins in a piggy bank. Alan received 140 coins. When he gave 48 coins to his brother, the number of his

Musya8 [376]

Assume Alan's brother has x coins before Alan gave him 48 coins
x+48=1.4x (increased by 40%)
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2 years ago

Cesar has read 89 pages of a book for his

slega [8]

Answer:

X=8

Step-by-step explanation:

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s2008m [1.1K]

Answer:

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2 years ago

Urgent, It is a Calculus question and I’ll appreciate your help. Thanks

BaLLatris [955]

Answer:

4733

Step-by-step explanation:

Please refer to the attached diagram.

Point A can be assigned x-coordinate "p". Then its y-coordinate is 6p^2. The slope at that point is y'(p) = 12p.

Point B can be assigned x-coordinate "r". Then its y-coordinate is 6r^2. The slope at that point is y'(r) = 12r.

We want the slopes at those points to have a product of -1 (so the tangents are perpendicular). This means ...

(12p)(12r) = -1

r = -1/(144p)

The slope of line AB in the diagram is the ratio of the differences of y- and x-coordinates:

slope AB = (ry -py)/(rx -px) = (6r^2 -6p^2)/(r -p) = 6(r+p) . . . . simplified

The slope of AB is also the tangent of the sum of these angles: the angle AC makes with the x-axis and angle CAB. The tangent of a sum of angles is given by ...

tan(α+β) = (tan(α) +tan(β))/1 -tan(α)·tan(β))

Of course the slope of a line is equal to the tangent of the angle it makes with the x-axis. The tangent of angle CAB is 2 (because the aspect ratio of the rectangle is 2). This means we can write ...

slope AB = ((slope AC) +2)/(1 -(slope AC)(2))

$6(p+r)=\dfrac{12p+2}{1-(12p)(2)}\\\\3(p+r)(1-24p)=6p+1\qquad\text{multiply by $1-24p$}\\\\3\left(p-\dfrac{1}{144p}\right)(1-24p)=6p+1\qquad\text{use the value for r}\\\\3(144p^2-1)(1-24p)=144p(6p+1)\qquad\text{multiply by 144p}\\\\ 3456 p^3+ 144 p^2+ 24 p+1 =0\qquad\text{put in standard form}\\\\144p^2(24p+1)+(24p+1)=0\qquad\text{factor by pairs}\\\\(144p^2+1)(24p+1)=0\qquad\text{finish factoring}\\\\p=-\dfrac{1}{24}\qquad\text{only real solution}\\\\r=\dfrac{-1}{144p}=\dfrac{1}{6}$

So, now we can figure the coordinates of points A and B, and the distance between them. That distance is given by the Pythagorean theorem as ...

d^2 = (6r^2 -6p^2)^2 +(r -p)^2

d^2 = (6(1/6)^2 -6(-1/24)^2)^2 +(1/6 +1/24)^2 = 25/1024 +25/576 = 625/9216

Because of the aspect ratio of the rectangle, the area is 2/5 of this value, so we have ...

Rectangle Area = (2/5)(625/9216) = 125/4608 = a/b

Then a+b = 125 +4608 = 4733.

_____

<em>Comment on the solution</em>

The point of intersection of the tangent lines is a fairly messy expression, and that propagates through any distance formulas used to find rectangle side lengths. This seemed much cleaner, though maybe not so obvious at first.

6 0

2 years ago