Calculus: Analytic Geometry and Calculus, with Vectors

570 Partial derivatives

10 When

u = x2 + Y2 - z2,

the graph of the equation u = 0 is a cone having its vertex at the origin. If

Po(xo,yo,zo) is a point on the cone which is not the origin, show that the equation

of the plane tangent to the cone at Po is

zox + YoY - zoz = 0.

Note that the gradient at the origin is 0 and that we have no information about

planes (if any) that are tangent to a cone at its vertex.

11 The graph of the equation

(1) z YZ x2

b2 a2

is a hyperbolic paraboloid or saddle surface more or less like that shown in Figure

6.672. The sections in planes parallel to the xz and yz planes are parabolas

while other sections are hyperbolas. Letting

x2 y2

i P,

find the gradient of u at the point Po(xo,yo,zo) on the graph of (1) and show that

the equation of the plane tangent to the graph at Po is

(3) x0x-yoY -f- z + Z0 = 0.

a2 b2^2

12 Letting u be the left member of the equation

(1) auxx + a12xy + a13xz + bi(x + x) + a21Yx + a22YY + a2syz + b2(Y + y)

+ aalzx + as2zy + asazz + bs(z + z) = c,

where a21 = a12, a31 = ais, and a82 = a23, show that

(2) Vu = 2[a11x + a12y + aiaz + b1]i + 2[a21x + a22y + a2az + b2]7

+ 2[aslx + as2Y + assz + b3]k.

Supposing that the graph of (1) is a quadric surface S containing a pointPo(xo,yo,zo)

at which Vu 0 0, write the equation of the plane tangent to S at Po. Put the

equation in the form obtained from (1) by the following ritual. Wherever the prod -

uct or sum of two variables appears, award a subscript to the first factor or term.

13 Prove that each plane tangent to the surface having the equation

x3i + yi4 + z% = a%

intersects the coordinate axes in points that are projections upon these axes of a

point on the sphere having the equation

x2+y2+z2 = a2.

14 Find the gradient of the function F for which

F(x,y,z) = xyz - 1,

find equations of the line normal to the surface having the equation

xyz=1