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3 years ago

One third divided by eight

Mathematics

2 answers:

pshichka [43]3 years ago

5 0

Answer:

1/24

Step-by-step explanation:

stellarik [79]3 years ago

3 0

Answer:

1/24

Step-by-step explanation:

1/3÷ 8

Copy dot flip

1/3 * 1/8

1/24

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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

What is the slope and y-Intercept of the equation 3(y – 2) + 6(x + 1) - 2 = 0?

Ne4ueva [31]

Answer:

A. Slope = -2, y-intercept = 2/3

Step-by-step explanation:

<u>Convert the given equation into slope-intercept form:</u>

3(y - 2) + 6(x + 1) - 2 = 0
3y - 6 + 6x + 6 - 2 = 0
3y + 6x - 2 = 0
3y = - 6x + 2
y = - 2x + 2/3

The slope is -2, the y-intercept is 2/3

Correct choice is A

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2 years ago

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At a soccer tournament 121212 teams are wearing red shirts, 666 teams are wearing blue shirts, 444 teams are wearing orange shir

Furkat [3]

Answer:

innappropiate question.

Step-by-step explanation:

8 0

3 years ago

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If demand is D1, what is the lowest price that Stromnord can charge so that it will not run out of this model of shoe in the mon

Molodets [167]

D1 = 60 for a price of $80. Charging $80 will ensure supply exceeds demand.
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The store apparently could charge a price slightly lower than $80, but we cannot tell from the chart how much lower.

3 0

4 years ago

Can someone answer this question please answer it correctly if it’s corect I will mark you brainliest

Licemer1 [7]

Answer:

$43.95

Step-by-step explanation:

15/100 × 293

= 43.95

7 0

3 years ago

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