Page 132
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Tangent Planes and Normal Lines
1.
Find a unit nonnal vector to the swface at the indicated point:
(a) x
+ y + z= 4, P(2,0,2)
(b) z =
Jx
2 + y2 , P(3,4,5)
(c) x 2 y4 - z = 0, P(l,2,16)
(d) z - x siny= 4, P(6,1l /6,7)
2.
Find an equation of the tangent plane to the surface at the indicated point:
(a) z = 25- x 2 - y2, P(3,1,15)
(b) f(x,y)
=y
(c) g(x,y)
=x 2
-
/ x, P(1,2,2)
y2, P(5,4,9)
(d) z = eX (siny+ 1), P(0,1l /2,2)
•
(e) h(x,y) = lnJx 2 + y2 , P(3,4,ln5)
3.
(f) x 2 + 4y2 + z2
=36, P(2,- 2,4)
(g) xy2 + 3x- z2
=4, P(2,1,- 1)
Find the equation of the nonnalline to the surface at the indicated point:
(a) x 2
+ y2 + Z =9, P(l,2,4)
(b) xy- Z = 0, P(-2,- 3,6)
(c) XYZ= 10, P(l,2,5)
•
4.
Show that any tangent plane to the cone Z2 = (axl + (byl passes through the
origin.
*5.
Suppose that f is any differentiable function of a single variable and suppose that
a surface is defined by z = xf(y/x). Show that the tangent plane at every point of
this surface passes through the origin.
185
Page 133
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Page 134
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Page 135
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Page 136
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