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Tangent plane and normal line to a given surface
125206 Tangent plane and normal line to a given surface Find an equation for the tangent plane and parametric equations for the normal line to the surface at the point P: z = x e^(-y) ; P(1,0,1) . The following is the text part of the solution.
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Multiple Variable Calculus
Solution:
Here f(x,y) = 3x2 + 2y2 - 3
fx(x,y) = 3*2x = 6x
fx(1,1) = 6*1 = 6
fy(x,y) = 2*2y = 4y
fy(1,1) = 4*1 = 4
An equation of the tangent plane to the surface z = f(x, y) at the point P ( is:
z - 2 = 6(x-1) + 4(y-1)
z-2 = 6x-6 +
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Parametric Equations of a Line through a Surface
As the normal vector is n=(2, 18, 18, an equation for the tangent plane through P is
2(x-1)+18(y-1)+18(z-1)=0
i.e.,
x-1+9y-9+9z-9=0
So,
x+9y+9z-19=0
i.e.,
z=(19-x-19y)/9 Parametric Equations of
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Integrals, moment of inertia, and Stokes theorem
x = 0, y = 0, z = 0, y = 1 - x2, and 4x + 3y + 2z = 12, assume _(x,y,z)_1.
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Evaluate ∫CP(x,y)dx+Q(x,y)dy
Given: P(x,y)=y2, Q(x,y)= 3x; C is the part of the graph of y=3x2 from (-1,3) to (2,12).
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Use the divergence theorem to evaluate
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Parametric Equations and Equation of an Ellipsoid
128104 Parametric Equations and Equation of an Ellipsoid (1) Find parametric equations for the line through the point (0, 1,2) that is parallele to the plane x+y+z = 2 and perpendicular to the line x = 1 +t, y = 1 ?t,z = 2t.
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Find the equations of the tangent line and the normal line
52781 Find the equations of the tangent line and the normal line Please explain how to find the equations of the tangent line and the normal line to the graph of the equation at the indicated point and achieve the specified answer.
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Normal gradient vector
13786 Normal Gradient Vector Use the normal gradient vector to write an equation of the line (or plane) tangent to the given curve (or surface) at the given point
P: x^(1/3) + y^(1/3) + z^(1/3) = 1; P(1, -1, 1).
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10 Calculus Problems
equation of plane:
5x-2y+3z = 4
equation of line:
x = 2t+1; y = 3t-1; z = 1-t
At the intersection point the line equation will satisfy the plane equation.
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Equation of a Tangent Plane
115067 Equation of a Tangent Plane 1) Find the equation of the tangent plane to the graph of the function f(x, y) = (x3 + siny) / (y^2+1) at the point (2, 0, 8).
2) Let g(x, y, z) = x2 - y3 + z4.