Designer curves
Custom curvature. Let (s) 0 be a real-valued function. We construct a
unit-speed curve f (s) with curvature function (s). Let
s
x(s) =
(t)dt d,
cos
0
0
s
y(s) =
(t)dt d
sin
0
0
Let f (s) = (x(s), y(s).
(1) Compute velocity vector f (s) and
MATH 381 (FALL 2013): LABS
1. Mommy, where do parametrizations come from?
The only general method I know of for parametrizing a curve is to parametrize
it in terms of arc length. The existence of this uniform method for parametrizing
curves is very useful
MATH 381 FALL 2013 TEST 1 SOLUTIONS
(1) By the diagram,
r(t) = OS + SR
= (3 cos t, 3 sin t) + (cos( + 3t), sin( + 3t)
= (3 cos t, 3 sin t) + (cos cos 3t sin sin 3t, sin cos 3t + cos sin 3t)
= (3 cos t cos 3t, 3 sin t sin 3t).
(2)
ds(t) =
=
x (t)2 + y (t
MATH 381 (FALL 2013)
The textbook is Adams and Essex, Calculus, a complete course, 8th edition. If you
dont have a copy, you can borrow one of mine so you can photocopy the exercises.
Week 1
Textbook problems: 8.2 1-28, 8.3 1-25, 8.4 1-9, 15-231
Please su
From the textbook, for you own practice: 12.1 1-36, 12.2 #1-20, 12.3
#1-40, 12.4 #1-27 12.5 #1-40 12.6 #1-23, 12.7 #1-37
HW2 to hand in: 12.2 #13, 20, 12.3 #12, 23, 31, 12.5 #23
(1) [HW2 hand in.] Let U be an open interval in R and suppose that f is an
n-
am
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MATH 381 FALL 2013 PROBLEMS
(1) Let
F(x, y) =
Find a function f such that
x2
x
y
,
2 x2 + y 2
+y
.
f = F.
(2) For (x, y) R2 , (x, y) = (0, 0), let (x, y) (, ] be the angle between the positive
x-axis and the ray from (0, 0) through (x, y).
(a) For which (