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3 years ago

9

If you drew a horizontal line on the coordinate plane, what could you say about the y-coordinate for every point on the line.

1 answer:

Paha777 [63]3 years ago

7 0

It's the second number that you use for coordinate plane

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Suppose that \nabla f(x,y,z) = 2xyze^{x^2}\mathbf{i} + ze^{x^2}\mathbf{j} + ye^{x^2}\mathbf{k}. if f(0,0,0) = 2, find f(1,1,1).

lesya [120]

The simplest path from (0, 0, 0) to (1, 1, 1) is a straight line, denoted $C$ , which we can parameterize by the vector-valued function,

$\mathbf r(t)=(1-t)(\mathbf i+\mathbf j+\mathbf k)$

for $0\le t\le1$ , which has differential

$\mathrm d\mathbf r=-(\mathbf i+\mathbf j+\mathbf k)\,\mathrm dt$

Then with $x(t)=y(t)=z(t)=1-t$ , we have

$\displaystyle\int_{\mathcal C}\nabla f(x,y,z)\cdot\mathrm d\mathbf r=\int_{t=0}^{t=1}\nabla f(x(t),y(t),z(t))\cdot\mathrm d\mathbf r$

$=\displaystyle\int_{t=0}^{t=1}\left(2(1-t)^3e^{(1-t)^2}\,\mathbf i+(1-t)e^{(1-t)^2}\,\mathbf j+(1-t)e^{(1-t)^2}\,\mathbf k\right)\cdot-(\mathbf i+\mathbf j+\mathbf k)\,\mathrm dt$

$\displaystyle=-2\int_{t=0}^{t=1}e^{(1-t)^2}(1-t)(t^2-2t+2)\,\mathrm dt$

Complete the square in the quadratic term of the integrand: $t^2-2t+2=(t-1)^2+1=(1-t)^2+1$ , then in the integral we substitute $u=1-t$ :

$\displaystyle=-2\int_{t=0}^{t=1}e^{(1-t)^2}(1-t)((1-t)^2+1)\,\mathrm dt$

$\displaystyle=-2\int_{u=0}^{u=1}e^{u^2}u(u^2+1)\,\mathrm du$

Make another substitution of $v=u^2$ :

$\displaystyle=-\int_{v=0}^{v=1}e^v(v+1)\,\mathrm dv$

Integrate by parts, taking

$r=v+1\implies\mathrm dr=\mathrm dv$

$\mathrm ds=e^v\,\mathrm dv\implies s=e^v$

$\displaystyle=-e^v(v+1)\bigg|_{v=0}^{v=1}+\int_{v=0}^{v=1}e^v\,\mathrm dv$

$\displaystyle=-(2e-1)+(e-1)=-e$

So, we have by the fundamental theorem of calculus that

$\displaystyle\int_C\nabla f(x,y,z)\cdot\mathrm d\mathbf r=f(1,1,1)-f(0,0,0)$

$\implies-e=f(1,1,1)-2$

$\implies f(1,1,1)=2-e$

3 0

3 years ago

M is directionly proptional to r squared

sergejj [24]

9514 1404 393

Answer:

A) 504

B) 8

Step-by-step explanation:

The statement of proportionality can be written as the equation ...

m = kr²

Then we can find k from ...

k = m/r² = 14/2² = 7/2

That is, ...

m = (7/2)r²

__

Using this relation, we can find m for r=12:

m = (7/2)·12² = 7·72

m = 504 . . . for r = 12

__

And, we can use the same equation to solve for r when m=224.

224 = 7/2·r²

r² = 64

r = √64

r = 8 . . . for m = 224

4 0

2 years ago

What is the value of m? 12+(7+m)=30<br><br> 11<br> 19<br> 25<br> 35<br> 49

Marat540 [252]

The answer is 11
hope that helps

4 0

3 years ago

Read 2 more answers

-3x + y = -2 IF x=-3 x=-1 x=2 what does y equal?

nikdorinn [45]

-11, -1, -8 that is all of what y would =

5 0

2 years ago

Read 2 more answers

5+8^-38 what is the answer please

Eddi Din [679]

Well if the 8 is squared the answer should be 31

3 0

3 years ago