   Chapter 14.4, Problem 7E

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.7. z = x2 + xy + 3y2, (1, 1, 5)

To determine

To graph: The surface z=x2+xy+3y2 and the tangent plane at the point (1,1,5).

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 z=f(x,y)=x2+xy+3y2 . (1)

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

fx(x,y)=2x+(1)y=2x+y

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

fy(x,y)=x+6y

Obtain the partial derivative of x at the given point (1,1,5).

fx(x0,y0)=fx(1,1)=2(1)+1=3

Thus, fx(x0,y0)=3 .

The partial derivative of y at the given point (1,1,5).

fy(x0,y0)=fy(1,1)=1+6(1)=7

Thus, fy(x0,y0)=7

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