30 1. ADVANCED DERIVATIVES
Problem 1.56 Find, in the form
ax
C
by
C
cz
D
d;
the tangent planes to the following curves at the indicated points.
1. f .x; y/ D x
2
C y
2
1 at .2; 2/
2. g.x; y/ D 3x
2
C 2xy C4
2
1 at .1; 1/
3. h.x; y/ D .x C y C 1/
2
at .0; 4/
4. r.x; y/ D y ln.x
2
C 1/ at .3; 1/
5. s.x; y/ D e
x
2
Cy
2
at .0; 1/
6. q.x; y/ D sin.x/ cos.y/ at .=3; =6/
Problem 1.57 Suppose that
x
2
C y
2
C z
2
D 12:
Find the tangent plane at each of the following points.
1. p D .2; 2; 2/
2. q D .2; 2; 2/
3. r D .2; 2; 2/
4. u D .1; 1;
p
10/
5. v D .
p
7; 1; 2/
6. w D .1=2; 2:5;
p
22=2/
Problem 1.58 Find all tangent planes to
x
2
C y
2
C z
2
D 75
that are at right angles to the vector .2; 4 ; 6/.
Problem 1.59 Find the tangent plane at .1; =3/ to
g.x; y/ D x cos.y/
1.3. TANGENT PLANES 31
Problem 1.60 Find the tangent plane to each of the following surfaces at the indicated point.
1. Surface x C y C z
2
D 6 at (1,1,2)
2. Surface x
2
y
3
C 5z D 19 at (4,-2,-1)
3. Surface 3x C y
2
C z
2
D 8 at (2,1,1)
4. Surface .x y/
3
C 2z D 14 at (2,0,3)
5. Surface xyz Cx C y C z D 4 at (1,1,1)
6. Surface xy C yz C xz D 0 at (-1,1,-2)
7. Surface x
2
C y
2
C z
3
D 6 at (2,1,1)
8. Surface xy C xz C yz D 7 at (1,1,3)
Problem 1.61 Find the tangent plane at .0; 0/ to
g.x; y/ D e
.x
2
Cy
2
/
Problem 1.62 Find the tangent plane at .=4; =4; =4/ to
cos.xyz/ D 0
Problem 1.63 If
x
2
C y
2
C z
2
D r
2
is a sphere of radius r centered at the origin .0; 0; 0/ (it is), show that a sphere has a tangent
plane at right angles to any non-zero vector.
Problem 1.64 If you want to find the tangent plane to a point on a sphere, what is the simplest
method? Explain.
Problem 1.65 If P and Q are planes that are both at right angles to a vector Ev, and they are
not equal, what can be said about the intersection of P and Q?
Problem 1.66 Suppose that Eu and Ev are vectors so that Ev Eu D 0. If P is a plane at right angles
to Eu, and Q is a plane at right angles to Ev in three-dimensional space, then what is the most that
can be said about the intersection of the planes?
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