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3 years ago

6 times 7 is 42 6 times 5 is 30 if that's any help please help me out.

Mathematics

1 answer:

alexgriva [62]3 years ago

6 0

6 * 5 = 30

42 - 30 = 12

? * 2 = 12

? = 6

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Consider the line integral Z C (sin x dx + cos y dy), where C consists of the top part of the circle x 2 + y 2 = 1 from (1, 0) t

pashok25 [27]

Direct computation:

Parameterize the top part of the circle $x^2+y^2=1$ by

$\vec r(t)=(x(t),y(t))=(\cos t,\sin t)$

with $0\le t\le\pi$ , and the line segment by

$\vec s(t)=(1-t)(-1,0)+t(2,-\pi)=(3t-1,-\pi t)$

with $0\le t\le1$ . Then

$\displaystyle\int_C(\sin x\,\mathrm dx+\cos y\,\mathrm dy)$

$=\displaystyle\int_0^\pi(-\sin t\sin(\cos t)+\cos t\cos(\sin t)\,\mathrm dt+\int_0^1(3\sin(3t-1)-\pi\cos(-\pi t))\,\mathrm dt$

$=0+(\cos1-\cos2)=\boxed{\cos1-\cos2}$

Using the fundamental theorem of calculus:

The integral can be written as

$\displaystyle\int_C(\sin x\,\mathrm dx+\cos y\,\mathrm dy)=\int_C\underbrace{(\sin x,\cos y)}_{\vec F}\cdot\underbrace{(\mathrm dx,\mathrm dy)}_{\vec r}$

If there happens to be a scalar function $f$ such that $\vec F=\nabla f$ , then $\vec F$ is conservative and the integral is path-independent, so we only need to worry about the value of $f$ at the path's endpoints.

This requires

$\dfrac{\partial f}{\partial x}=\sin x\implies f(x,y)=-\cos x+g(y)$

$\dfrac{\partial f}{\partial y}=\cos y=\dfrac{\mathrm dg}{\mathrm dy}\implies g(y)=\sin y+C$

So we have

$f(x,y)=-\cos x+\sin y+C$

which means $\vec F$ is indeed conservative. By the fundamental theorem, we have

$\displaystyle\int_C(\sin x\,\mathrm dx+\cos y\,\mathrm dy)=f(2,-\pi)-f(1,0)=-\cos2-(-\cos1)=\boxed{\cos1-\cos2}$

3 0

3 years ago

Need help with number 3 inequalities with variables on both sides

Volgvan

Answer:

x ≤ 2

Step-by-step explanation:

We are given the inequality:

$\displaystyle{2x-3 \leq \dfrac{x}{2}}$

First, get rid of the denominator by multiplying both sides by 2:

$\displaystyle{2x\cdot 2-3\cdot 2 \leq \dfrac{x}{2}\cdot 2}\\\\\displaystyle{4x-6 \leq x}$

Add both sides by 6 then subtract both sides by x:

$\displaystyle{4x-6+6 \leq x+6}\\\\\displaystyle{4x \leq x+6}\\\\\displaystyle{4x-x \leq x+6-x}\\\\\displaystyle{4x-x \leq 6}\\\\\displaystyle{3x \leq 6}$