1.) Find the equation of the tangent plane to the surface z = e^(-1x/17) ln(3y)
ID: 2832814 • Letter: 1
Question
1.) Find the equation of the tangent plane to the surface z = e^(-1x/17) ln(3y)
at the point (-3, 2, 2.138)?
2.) Consider the ellipsoid 2x^2 + 5y^2 + z^2 = 26
a.) The implicit form of the tangent plane to this ellipsoid at (1, -2, -2) is?
b.) The parametric form of the line through this point that is perpendicular to that tangent plane is L(t) =?
3.) Suppose z = x^2 siny, x = 3s^2 + 2t^2 , y = 4st
a.) Find the numerical values of [(Dz)/(Ds)] and [(Dz)/(Dt)] when (s , t) = (2, 1)
i.)[(Dz)/(Ds)] (2, 1).............................................?
ii)[(Dz)/(Dt)] (2, 1).......................................?
4.) Suppose z = x^2 sin y, x = 2s^2 - 4t^2 , y = 2st.
a.) Find the numerical values of [(Dz)/(Ds)] and [(Dz)/(Dt)] when (s, t) = (-2, -3)
i)[(Dz)/(Ds)] (-2, -3)...........................................?
ii)[(Dz)/(Dt)] (-2, -3).....................................?
5.) Use the chain rule to find [(Dz)/ (Dt) , where
z = x^2 y + xy^2 , x = -4 + t^4 , y = 2 - t^4
a.) End result (in terms of just t):
Explanation / Answer
1)
z = e^(-x/17) * ln(3y)
differentiate w.r.t x
zx = (-1/17) * e^(-x/17) * ln(3y)
zx(-3, 2) = (-1/17) * e^(3/17) * ln(6) = -0.12574
and
differentiate w.r.t y
zy = e^(-x/17) * 1/y
zy(-3, 2) = e^(3/17) * 1/2 = 0.5965
Tangent eq. is
z = 2.138 - 0.12574*(x + 3) + 0.5965*(y - 2)
=>
z = -0.12574x + 0.5965y + 0.56778
2)
2x^2 + 5y^2 + z^2 = 26
=>
z = sqrt[26 - 2x^2 - 5y^2]
differentiate w.r.t x
zx = -2x/[sqrt[26 - 2x^2 - 5y^2]]
zx(1, -2) = -2/[sqrt[26 - 2 - 20]] = -1
and
z = sqrt[26 - 2x^2 - 5y^2]
differentiate w.r.t y
zy = -5y/[sqrt[26 - 2x^2 - 5y^2]]
zy(1, -2) = 10/[sqrt[26 - 2 - 20]] = 5
tangent is
z = -2 - 1*(x- 1) + 5*(y + 2)
=>
z = -x + 5y + 9
3) z = x^2 siny, x = 3s^2 + 2t^2 , y = 4st
z = x^2 siny
dz/dx = 2x siny
and
dz/dy = x^2 cosy
x = 3s^2 + 2t^2
dx/ds = 6s
and
dx/dt = 4t
y = 4st
dy/ds = 4t
and
dy/dt = 4s
now
i)
dz/ds = dz/dx * dx/ds + dz/dy * dy/ds
= 2xsiny * 6s + x^2 cosy * 4t
= 2*[3s^2 + 2t^2] * sin(4st) * 6s + (3s^2 + 2t^2)^2 * cos(4st) * 4t
now put s = 2, t = 1
[Dz/Ds](2, 1) = 2*[12 + 2] * sin(8) * 12 + (12 + 2)^2 * cos(8) * 4 = 218.352
ii)
dz/dt = dz/dx * dx/dt + dz/dy * dy/dt
= 2xsiny * 4t + x^2 cosy * 4s
= 2*[3s^2 + 2t^2] * sin(4st) * 4t + (3s^2 + 2t^2)^2 * cos(4st) * 4s
now put s = 2, t = 1
[Dz/Dt](2, 1) = 2*[12 + 2] * sin(8) * 4 + (12 + 2)^2 * cos(8) * 8
= -117.336
4) z = x^2 sin y, x = 2s^2 - 4t^2 , y = 2st
z = x^2 siny
dz/dx = 2x siny
and
dz/dy = x^2 cosy
x = 2s^2 - 4t^2
dx/ds = 4s
and
dx/dt = -8t
y = 2st
dy/ds = 2t
and
dy/dt = 2s
now
i)
dz/ds = dz/dx * dx/ds + dz/dy * dy/ds
= 2xsiny * 4s + x^2 cosy * 2t
= 2*[2s^2 - 4t^2] * sin(2st) * 4s + (2s^2 - 4t^2)^2 * cos(2st) * 2t
now put s = -2, t = -3
[Dz/Ds](-2, -3) = 2*[8 - 4*9] * sin(12) * -8 + (8 - 4*9)^2 * cos(12) * -6
= -4209.875
ii)
dz/dt = dz/dx * dx/dt + dz/dy * dy/dt
= 2xsiny * -8t + x^2 cosy * 2s
= 2*[2s^2 - 4t^2] * sin(2st) * -8t + (2s^2 - 4t^2)^2 * cos(2st) * 2s
now put s = -2, t = -3
[Dz/Dt](-2, -3) = 2*[8 - 4*9] * sin(12) * 24 + (8 - 4*9)^2 * cos(12) * -4
= -1925.172
5)
z = x^2 y + xy^2 , x = -4 + t^4 , y = 2 - t^4
z = x^2 y + xy^2
dz/dx = 2xy + y^2
and
dz/dy = x^2 + 2xy
x = -4 + t^4
dx/dt = 4t^3
y = -4 + t^4
dy/dt = -4t^3
So,
Dz/Dt = dz/dx * dx/dt + dz/dy * dy/dt
= (2xy + y^2) * 4t^3 + (x^2 + 2xy) * (-4t^3)
= (2*(-4 + t^4 )(-4 + t^4) + (
1)
z = e^(-x/17) * ln(3y)
differentiate w.r.t x
zx = (-1/17) * e^(-x/17) * ln(3y)
zx(-3, 2) = (-1/17) * e^(3/17) * ln(6) = -0.12574
and
differentiate w.r.t y
zy = e^(-x/17) * 1/y
zy(-3, 2) = e^(3/17) * 1/2 = 0.5965
Tangent eq. is
z = 2.138 - 0.12574*(x + 3) + 0.5965*(y - 2)
=>
z = -0.12574x + 0.5965y + 0.56778
2)
2x^2 + 5y^2 + z^2 = 26
=>
z = sqrt[26 - 2x^2 - 5y^2]
differentiate w.r.t x
zx = -2x/[sqrt[26 - 2x^2 - 5y^2]]
zx(1, -2) = -2/[sqrt[26 - 2 - 20]] = -1
and
z = sqrt[26 - 2x^2 - 5y^2]
differentiate w.r.t y
zy = -5y/[sqrt[26 - 2x^2 - 5y^2]]
zy(1, -2) = 10/[sqrt[26 - 2 - 20]] = 5
tangent is
z = -2 - 1*(x- 1) + 5*(y + 2)
=>
z = -x + 5y + 9
3) z = x^2 siny, x = 3s^2 + 2t^2 , y = 4st
z = x^2 siny
dz/dx = 2x siny
and
dz/dy = x^2 cosy
x = 3s^2 + 2t^2
dx/ds = 6s
and
dx/dt = 4t
y = 4st
dy/ds = 4t
and
dy/dt = 4s
now
i)
dz/ds = dz/dx * dx/ds + dz/dy * dy/ds
= 2xsiny * 6s + x^2 cosy * 4t
= 2*[3s^2 + 2t^2] * sin(4st) * 6s + (3s^2 + 2t^2)^2 * cos(4st) * 4t
now put s = 2, t = 1
[Dz/Ds](2, 1) = 2*[12 + 2] * sin(8) * 12 + (12 + 2)^2 * cos(8) * 4 = 218.352
ii)
dz/dt = dz/dx * dx/dt + dz/dy * dy/dt
= 2xsiny * 4t + x^2 cosy * 4s
= 2*[3s^2 + 2t^2] * sin(4st) * 4t + (3s^2 + 2t^2)^2 * cos(4st) * 4s
now put s = 2, t = 1
[Dz/Dt](2, 1) = 2*[12 + 2] * sin(8) * 4 + (12 + 2)^2 * cos(8) * 8
= -117.336
4) z = x^2 sin y, x = 2s^2 - 4t^2 , y = 2st
z = x^2 siny
dz/dx = 2x siny
and
dz/dy = x^2 cosy
x = 2s^2 - 4t^2
dx/ds = 4s
and
dx/dt = -8t
y = 2st
dy/ds = 2t
and
dy/dt = 2s
now
i)
dz/ds = dz/dx * dx/ds + dz/dy * dy/ds
= 2xsiny * 4s + x^2 cosy * 2t
= 2*[2s^2 - 4t^2] * sin(2st) * 4s + (2s^2 - 4t^2)^2 * cos(2st) * 2t
now put s = -2, t = -3
[Dz/Ds](-2, -3) = 2*[8 - 4*9] * sin(12) * -8 + (8 - 4*9)^2 * cos(12) * -6
= -4209.875
ii)
dz/dt = dz/dx * dx/dt + dz/dy * dy/dt
= 2xsiny * -8t + x^2 cosy * 2s
= 2*[2s^2 - 4t^2] * sin(2st) * -8t + (2s^2 - 4t^2)^2 * cos(2st) * 2s
now put s = -2, t = -3
[Dz/Dt](-2, -3) = 2*[8 - 4*9] * sin(12) * 24 + (8 - 4*9)^2 * cos(12) * -4
= -1925.172
5)
z = x^2 y + xy^2 , x = -4 + t^4 , y = 2 - t^4
z = x^2 y + xy^2
dz/dx = 2xy + y^2
and
dz/dy = x^2 + 2xy
x = -4 + t^4
dx/dt = 4t^3
y = 2 - t^4
dy/dt = -4t^3
So,
Dz/Dt = dz/dx * dx/dt + dz/dy * dy/dt
= (2xy + y^2) * 4t^3 + (x^2 + 2xy) * (-4t^3)
= (2*(-4 + t^4 )(2 - t^4) + (2 - t^4 )^2) * 4t^3 + ((-4 + t^4)^2 + 2*(-4 + t^4)*(2 - t^4)) * (-4t^3)
simplify it
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