   Chapter 14.4, Problem 8E

Chapter
Section
Textbook Problem

Graph the surface and the tangent plane at the given point. (Choose the domain and viewpoint so that you get a good view of both the surface and the tangent plane.) Then zoom in until the surface and the tangent plane become indistinguishable.8. z = 9 + x 2 y 2 , (2, 2, 5)

To determine

To graph: The surface z=9+x2y2 and the tangent plane at the point (2,2,5).

Explanation

Result used:

“Suppose f has continuous partial derivatives, the equation of the tangent plane to the surface z=f(x,y) at the point P(x0,y0,z0) is zz0=fx(x0,y0)(xx0)+fy(x0,y0)(yy0) ”.

Calculation:

The given surface is, z=9+x2y2 . (1)

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

fx(x,y)=12(9+x2y2)12(2xy2)=xy29+x2y2

Take partial derivative with respect to y in the equation (1),

fy(x,y)=12(9+x2y2)12(2xy2)=x2y9+x2y2

Obtain the partial derivative of x at the given point (2,2,5).

fx(x0,y0)=fx(2,2)=2(22)9+2222=2(4)9+16=85

Obtain the partial derivative of y at the given point (2,2,5)

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