   Chapter 14, Problem 26RE

Chapter
Section
Textbook Problem

Find equations of (a) the tangent plane and (b) the normal line to the given surface at the specified point.26. z = ex cos y, (0, 0, 1)

(a)

To determine

To find: The equation of the tangent planeto the surface z=excosy at the point (0,0,1).

Explanation

Given:

Thesurface is, z=excosy.

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)=zexcosy. (1)

The equation of the tangent plane to the given surface at the point (0,0,1) is defined by,

Fx(0,0,1)(x0)+Fy(0,0,1)(y0)+Fz(0,0,1)(z1)=0. (2)

Take partial derivative with respect to x at the point (0,0,1) in the equation (1),

Fx(x,y,z)=x(zexcosy)=0excosy=excosy

The value of Fx(x,y,z) at the point (0,0,1) is,

Fx(0,0,1)=e0cos(0)=1

Thus, the value of Fx(0,0,1)=1

(b)

To determine

To find: The equation of the normal line to the surface z=excosy at the point (0,0,1).

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