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agasfer [191]
3 years ago
8

Find the product: ( x-1/x ) ( x+1/x ) ( x2+1/x2 ) ( x4+1/x4 )

Mathematics
1 answer:
erma4kov [3.2K]3 years ago
3 0

Use the difference of 2 squares (a - b)(a + b) = a^2 - b^2:-

(x - 1/x)(x + 1/x) = x^2 - (1/x)^2

= x^2 - 1/x^2

(x ^2 - 1/x^2)(x^2 + 1 /x^2)

= x^4 - 1/x^4

(x^4 - 1/x^4)(x^4 + 1/x^4)

= x^8 - 1 /x^8 Answer

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Find the work done by F= (x^2+y)i + (y^2+x)j +(ze^z)k over the following path from (4,0,0) to (4,0,4)
babunello [35]

\vec F(x,y,z)=(x^2+y)\,\vec\imath+(y^2+x)\,\vec\jmath+ze^z\,\vec k

We want to find f(x,y,z) such that \nabla f=\vec F. This means

\dfrac{\partial f}{\partial x}=x^2+y

\dfrac{\partial f}{\partial y}=y^2+x

\dfrac{\partial f}{\partial z}=ze^z

Integrating both sides of the latter equation with respect to z tells us

f(x,y,z)=e^z(z-1)+g(x,y)

and differentiating with respect to x gives

x^2+y=\dfrac{\partial g}{\partial x}

Integrating both sides with respect to x gives

g(x,y)=\dfrac{x^3}3+xy+h(y)

Then

f(x,y,z)=e^z(z-1)+\dfrac{x^3}3+xy+h(y)

and differentiating both sides with respect to y gives

y^2+x=x+\dfrac{\mathrm dh}{\mathrm dy}\implies\dfrac{\mathrm dh}{\mathrm dy}=y^2\implies h(y)=\dfrac{y^3}3+C

So the scalar potential function is

\boxed{f(x,y,z)=e^z(z-1)+\dfrac{x^3}3+xy+\dfrac{y^3}3+C}

By the fundamental theorem of calculus, the work done by \vec F along any path depends only on the endpoints of that path. In particular, the work done over the line segment (call it L) in part (a) is

\displaystyle\int_L\vec F\cdot\mathrm d\vec r=f(4,0,4)-f(4,0,0)=\boxed{1+3e^4}

and \vec F does the same amount of work over both of the other paths.

In part (b), I don't know what is meant by "df/dt for F"...

In part (c), you're asked to find the work over the 2 parts (call them L_1 and L_2) of the given path. Using the fundamental theorem makes this trivial:

\displaystyle\int_{L_1}\vec F\cdot\mathrm d\vec r=f(0,0,0)-f(4,0,0)=-\frac{64}3

\displaystyle\int_{L_2}\vec F\cdot\mathrm d\vec r=f(4,0,4)-f(0,0,0)=\frac{67}3+3e^4

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2 years ago
Hillary earns $60 per week at her job, plus $2 for each item she sells. She wants to
AnnZ [28]

Answer:

60+2x≥300

Step-by-step explanation:

So we know that she makes a base of $60 a week and gets $2 extra for every item she sells. First, we set up an equation for how many items she would have to sell to make exactly $300 in a week. 60+2x=300. Now, all we do is replace the equal sign with a ≥ sign because it says at least which means more than or equal to.

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Mrs. Siebenaller bought a bus for 25,000 with a 7% interest rate mrs s gets a loan payoff of 60 months how much interest would s
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Answer:

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Step-by-step explanation:

Mrs. Siebenaller bought a bus for 25,000 with a 7% interest rate and she gets a loan payoff of 60 months,

We know that,

\text{PV of annuity}=P\left[\dfrac{1-(1+r)^{-n}}{r}\right]

Where,

PV = Present value of annuity = 25000,

r = rate of interest of each period = \dfrac{7}{12}% monthly

n = number of periods = 60 months,

Putting the values,

\Rightarrow 25000=P\left[\dfrac{1-(1+\frac{0.07}{12})^{-60}}{\frac{0.07}{12}}\right]

\Rightarrow P=\dfrac{25000}{\left[\dfrac{1-(1+\frac{0.07}{12})^{-60}}{\frac{0.07}{12}}\right]}

\Rightarrow P=\$495.03

Hence total amount paid is,

=495.03\times 60=\$29,701.80

Therefore interest amount is,

=29,701.80-25,000=\$4701.80


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