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Multivariable Calculus

8th Edition
James Stewart
ISBN: 9781305266643

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BuyFindarrow_forward

Multivariable Calculus

8th Edition
James Stewart
ISBN: 9781305266643
Textbook Problem

Find equations of (a) the tangent plane and (b) the normal line to the given surface at the specified point.

27. x2 + 2y2 − 3z2 = 3, (2, −1, 1)

(a)

To determine

To find: The equation of the tangent planeto the surface x2+2y23z2=3 at the point (2,1,1) .

Explanation

Given:

Thesurface is, x2+2y23z2=3 .

Result used:

“The tangent plane to the level surface at the point P(x0,y0,z0) is defined as Fx(x0,y0,z0)(xx0)+Fy(x0,y0,z0)(yy0)+Fz(x0,y0,z0)(zz0)=0

Calculation:

Let the surface function be, F(x,y,z)=x2+2y23z23 (1)

The equation of the tangent plane to the given surface at the point (2,1,1) is defined by,

Fx(2,1,1)(x2)+Fy(2,1,1)(y+1)+Fz(2,1,1)(z1)=0 (2)

Take partial derivative with respect to x at the point (2,1,1) in the equation (1),

Fx(x,y,z)=x(x2+2y23z23)=2x+000=2xFx(2,1,1)=4

Thus, the value of Fx(2,1,1)=4

(b)

To determine

To find: The equation of the normal line to the surface x2+2y23z2=3 at the point (2,1,1) .

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