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NISA [10]
2 years ago
10

If z=(x+y)e^y and x=5t and y=1−t^2, find the following derivative using the chain rule. Enter your answer as a function of t.

Mathematics
1 answer:
Vitek1552 [10]2 years ago
4 0

By the chain rule, if z=f(x,y)=f(x(t),y(t)), then

\dfrac{\mathrm dz}{\mathrm dt} = \dfrac{\partial f}{\partial x}\dfrac{\mathrm dx}{\mathrm dt} + \dfrac{\partial f}{\partial y}\dfrac{\mathrm dy}{\mathrm dt}

We have f(x,y)=(x+y)e^y, for which

• \dfrac{\partial f}{\partial x} = e^y

• \dfrac{\partial f}{\partial y} = e^y + (x+y)e^y = (x+y+1)e^y

and x(t)=5t and y(t)=1-t^2, so that

• \dfrac{\mathrm dx}{\mathrm dt} = 5

• \dfrac{\mathrm dy}{\mathrm dt} = -2t

Putting everything together, we get

\dfrac{\mathrm dz}{\mathrm dt} = 5e^y - 2t(x+y+1)e^y \\\\ \dfrac{\mathrm dz}{\mathrm dt} = 5e^{1-t^2} - 2t(5t-t^2+2)e^{1-t^2} = \boxed{\left(2t^3-10t^2-4t+5\right)e^{1-t^2}}

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Answer:

E-F and E-D

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Step-by-step explanation:

The circle and triangle are as shown.  The options are:

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By drawing radius lines from the center of the circle to the tangent points B, D, and F, we can divide the triangle into 3 kites.  Therefore, only segments that are legs of the same kite are congruent.  So the answer must be E-F and E-D, and C-B and C-D.

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zloy xaker [14]

The two congruent base angles tell us that this is an isosceles triangle, meaning the triangle has two congruent sides. Therefore, we can set these two expressions equal to each other and solve from there.

15x + 7 = 23x - 17

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4 0
3 years ago
A card is drawn from a​ well-shuffled deck of 52 cards. Find the probability of drawing a diamond or a spade.
AveGali [126]

Answer:

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Step-by-step explanation:

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Total spade = 4

Total diamond = 4

pr(drawing a spade) = number of spade/total cards

pr(drawing a spade) = 4/52 = 1/13

pr(drawing a diamond)  = 4/52 = 1/13

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8 0
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Answer:

y = -1/2x - 1/2

Step-by-step explanation:

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m - slope

b - y-intercept

Step 1: Define

y = -1/2x + 3/7

Point (-1, 0)

Step 2: Find parallel line

<em>Parallel lines have the same slope as the original but different y-intercepts.</em>

<em>m</em> = -1/2

y = -1/2x + b

0 = -1/2(-1) + b

0 = 1/2 + b

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Step 3: Write parallel line

y = -1/2x - 1/2

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