Find the planes tangent to the following surfaces at the

Jaden Easton 2021-08-13 Answered
Find the planes tangent to the following surfaces at the indicated points:
a) x2+2y2+3xz=10, at the point (1,2,13)
b) y2x2=3, at the point (1,2,8)
c) xyz=1, at the point (1,1,1)
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Expert Answer

Malena
Answered 2021-08-14 Author has 83 answers

Let f:RnR be a differentiable function. Recal that the tangent plane of the surface consisting of points such that f(x)=k for some constant k at the point x0 is given with:
f(x0)(xx0)=0 (1)
a) Here we set f(x,y,z)=x2+2y2+3xz,x0=(1,2,13) and k=10. Note that f(x0)=k.
Now we can calculate:
f(x)=(2x+3z,4y,3x)f(1,2,13)=(3,8,3)
Using (1) we easily get:
0=(3,8,3)(x1,y2,z13)=3(x1)+8(y2)+3(z13)
Thus, the tangent plane equation is:
3x+8y+3z=20
b) Here we set f(x,y,z)=y2x2,x0=(1,2,8) and k=3. Note that f(x0)=k
Now we can calculate:
f(x)=(2x,2y,0)f(1,2,8)=(2,4,0)
Using (1) we easily get:
0=(2,4,0)(x1,y2,z8)=2(x1)+4(y2)+0(z8)
Thus, the tangent plane equation is:
c) Here we set f(x,y,z)=xyz,x0=(1,1,1) and k=1. Note that f(x0)=k. Now we can calculate:
f(x)=(yz,xz,xy)f(1,1,1)=(1,1,1)
Using (1) we easily get:
0=(1,1,1)(x1,y1,z1)=1(x1)+1(y1)+1(z1)
Thus, the tangent plane equation is:

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