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Evgesh-ka [11]
3 years ago
9

Supposed soda is on sale for $0.50 can and you have a coupon for $0.80 off your total purchase write a function rule for the cos

t of an cans of soda
Mathematics
2 answers:
Fed [463]3 years ago
7 0
If C is the cost and x the amount of cans.
C = 0.5x - 0.8
shepuryov [24]3 years ago
6 0

As per the problem

Soda is on sale for $0.50.

and you have a coupon for $0.80 off your total purchase.

we are supposed to write a function rule for the cost of an can of soda.

If you buy x cans of soda then the cost function can be written as

C(x)=0.5x-0.80\\ \\ \text{we can replace x by the number of cans, here we have x=1 }\\ \\ C(1)=0.50*1-0.80

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Use the Fundamental Theorem for Line Integrals to find Z C y cos(xy)dx + (x cos(xy) − zeyz)dy − yeyzdz, where C is the curve giv
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The Line integral is π/2.

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We have to find a funtion f such that its gradient is (ycos(xy), x(cos(xy)-ze^(yz), -ye^(yz)). In other words:

f_x = ycos(xy)

f_y = xcos(xy) - ze^{yz}

f_z = -ye^{yz}

we can find the value of f using integration over each separate, variable. For example, if we integrate ycos(x,y) over the x variable (assuming y and z as constants), we should obtain any function like f plus a function h(y,z). We will use the substitution method. We call u(x) = xy. The derivate of u (in respect to x) is y, hence

\int{ycos(xy)} \, dx = \int cos(u) \, du = sen(u) + C = sen(xy) + C(y,z)  

(Remember that c is treated like a constant just for the x-variable).

This means that f(x,y,z) = sen(x,y)+C(y,z). The derivate of f respect to the y-variable is xcos(xy) + d/dy (C(y,z)) = xcos(x,y) - ye^{yz}. Then, the derivate of C respect to y is -ze^{yz}. To obtain C, we can integrate that expression over the y-variable using again the substitution method, this time calling u(y) = yz, and du = zdy.

\int {-ye^{yz}} \, dy = \int {-e^{u} \, dy} = -e^u +K = -e^{yz} + K(z)

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As a result,

f(x,y,z) = cos(xy) - e^{yz} + K(z)

if we derivate f over z, we obtain

f_z(x,y,z) = -ye^{yz} + d/dz K(z)

That should be equal to -ye^(yz), hence the derivate of K(z) is 0 and, as a consecuence, K can be any constant. We can take K = 0. We obtain, therefore, that f(x,y,z) = cos(xy) - e^(yz)

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3 0
3 years ago
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