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M427L Midterm Exam
Name
NO NOTES. NO CALCULATORS.
1. Find and classify the critical points of
f
(
x,y,z
) =
x
2
+
y
2
+
z
2
+
yz
+
xz.
2. Find the length of the curve
c
(
t
) = (
t
cos
t,t
sin
t,
2
√
2
3
t
3
/
2
)
,
0
≤
t
≤
π.
3. Let
F
(
x,y,z
) = (
x
2

y,
4
z,x
2
). Find div
F
and curl
F
.
4. Express the equation
x
2
+
y
2
= 1 in spherical coordinates.
5. Find the point where the lines
c
1
(
t
) = (1

t,
2
t,

3+2
t
) and
c
2
(
t
) = (2
t
+2
,
3
t
+19
,
4
t
+19) intersect. Then ﬁnd
the equation of the plain containing both lines.
6. Let
f
(
x,y
) =
√
xy
+ 1. Write the derivative and Hessian of
f
at the point (2
,
4). Then use a quadratic
approximation to estimate
f
(2
.
03
,
3
.
99).
7. Suppose
u
=
f
(
x,y
) and
x
=
r
cos
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Unformatted text preview: θ and y = r sin θ. Show that ± ∂u ∂x ¶ 2 + ± ∂u ∂y ¶ 2 = ± ∂u ∂r ¶ 2 + 1 r 2 ± ∂u ∂θ ¶ 2 . (This is a chain rule problem.) 8. Let f ( x,y ) = 3 x 2y 2 . Find all unit vectors v such that D v f (1 , 2) = 0 . 9. Let F : R 3 → R 3 be a C 2 vector ﬁeld. Show that ∇ • ( ∇ × F ) = 0 . 10. Find T , N , B ,κ, and τ for the curve c ( t ) = (3sin t, 4 t, 3cos t ) at the point where t = π/ 2. Sketch the curve and show these quantities on your sketch....
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This note was uploaded on 05/02/2011 for the course M 427L taught by Professor Keel during the Spring '07 term at University of Texas at Austin.
 Spring '07
 Keel
 Critical Point, Matrices

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