Use the Chain Rule to find the indicated partial derivatives. w = xy + yz + zx, x = r cos(θ), y = r sin(θ), z = rθ; âw âr , âw âθ when r = 4, θ = Ï2

δw/δr = 4π

δw/δθ = -8π

Step-by-step explanation:

Given the following functions

w = xy + yz + zx, x = r cosθ y = r sinθ, z = rθ where r = 4 and θ = π/2

We are to find δw/δr and δw/δθ

δw/δr = δw/δx•δx/δr + δw/δy•δy/δr + δw/δz•δz/δr

δw/δx = y+z

δx/δr = cosθ

δw/δy = x+z

δy/δr = sinθ

δw/δz = y+x

δz/δr = θ

Substitute the given values into the formula

δw/δr = (y+z)cosθ+(x+z)sinθ+(y+x)θ

Substitute the value of x, y and z in terms of theta into the resulting function

δw/δr = (y+z)cosθ+(x+z)sinθ+(y+x)θ

δw/δr = (rsinθ+rθ)cosθ+(rcosθ+rθ)sinθ+(rsinθ+rcosθ)θ

Substitute r = 4 and θ = π/2

δw/δr = (4sinπ/2+4π/2)cosπ/2+(4cosπ/2+4π/2)sinπ/2+(4sinπ/2+4cosπ/2)π/2

Note that cos π/2 = 0 and sinπ/2 = 1

δw/δr = (4+2π)(0)+(0+2π)(1)+(4(1)+4(0))π/2

δw/δr = 0+2π+4π/2

δw/δr = 2π+2π

δw/δr = 4π

For δw/δθ

δw/δθ = δw/δx•δx/δθ + δw/δy•δy/δθ + δw/δz•δz/δθ

δw/δx = y+z

δx/δθ = -rsinθ

δw/δy = x+z

δy/δθ =rcosθ

δw/δz = y+x

δz/δθ = r

Substitute the given values into the formula

δw/δθ = (y+z)-rsinθ+(x+z)rcosθ+(y+x)r

Substitute the value of x, y and z in terms of theta into the resulting function

δw/δθ = (y+z)-rsinθ+(x+z)rcosθ+(y+x)r

δw/δθ = (rsinθ+rθ)-rsinθ+(rcosθ+rθ)rcosθ+(rsinθ+rcosθ)r

Substitute r = 4 and θ = π/2

δw/δθ = (4sinπ/2+4π/2)-4sinπ/2+(4cosπ/2+4π/2)4cosπ/2+(4sinπ/2+4cosπ/2)(4)

Note that cos π/2 = 0 and sinπ/2 = 1

δw/δθ = (4+2π)(-4)+(0+2π)(0)+(4(1)+4(0))(4)

δw/δθ = -16-8π+0+4(4)

δw/δθ = -16+16-8π

δw/δθ = -8π

a. turk and mcalister

b. howard, mission, and market

c. yes. since streets end some do not cross yet are not parallel to others. example: mcalister and 5th

step-by-step explanation:

parallel lines are lines which do not intersect ever and look a lot like railroad tracks. turk, mcalister and golden gate are examples of these.

intersect are streets which cross each other and a car on one can reach the other. howard, mission, and market intersect with 6th.

skew lines are lines which neither intersect nor are parallel. these lines exist here because some streets end and never intersect another like mcalister and 5th.

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step-by-step explanation: