Suppose that an imaginary string of unlimited length is attached to a point on a circle and pulled taut so that it is tangent to the circle. Keeping the string pulled tight, wind (or unwind) the string around the circle. The path traced out by the end of the string as it moves is called an involute of the circle (shown in blue). Given a circle of radius r with parametric equations x = r cos θ , y = r sin θ , show that the parametric equations of the involute of the circle are x = r cos θ + θ sin θ and y = r sin θ − θ cos θ .
Suppose that an imaginary string of unlimited length is attached to a point on a circle and pulled taut so that it is tangent to the circle. Keeping the string pulled tight, wind (or unwind) the string around the circle. The path traced out by the end of the string as it moves is called an involute of the circle (shown in blue). Given a circle of radius r with parametric equations x = r cos θ , y = r sin θ , show that the parametric equations of the involute of the circle are x = r cos θ + θ sin θ and y = r sin θ − θ cos θ .
Solution Summary: The author illustrates how an imaginary string of unlimited length is attached to a point on the circle of radius r with parametric equations.
Suppose that an imaginary string of unlimited length is attached to a point on a circle and pulled taut so that it is tangent to the circle. Keeping the string pulled tight, wind (or unwind) the string around the circle. The path traced out by the end of the string as it moves is called an involute of the circle (shown in blue).
Given a circle of radius
r
with parametric equations
x
=
r
cos
θ
,
y
=
r
sin
θ
,
show that the parametric equations of the involute of the circle are
x
=
r
cos
θ
+
θ
sin
θ
and
y
=
r
sin
θ
−
θ
cos
θ
.
1. A bicyclist is riding their bike along the Chicago Lakefront Trail. The velocity (in
feet per second) of the bicyclist is recorded below. Use (a) Simpson's Rule, and (b)
the Trapezoidal Rule to estimate the total distance the bicyclist traveled during the
8-second period.
t
0 2
4 6 8
V
10 15
12 10 16
2. Find the midpoint rule approximation for
(a) n = 4
+5
x²dx using n subintervals.
1° 2
(b) n = 8
36
32
28
36
32
28
24
24
20
20
16
16
12
8-
4
1
2
3
4
5
6
12
8
4
1
2
3
4
5
6
=
5 37
A 4 8 0.5
06
9
Consider the following system of equations, Ax=b :
x+2y+3z - w = 2
2x4z2w = 3
-x+6y+17z7w = 0
-9x-2y+13z7w = -14
a. Find the solution to the system. Write it as a parametric equation. You can use a
computer to do the row reduction.
b. What is a geometric description of the solution? Explain how you know.
c. Write the solution in vector form?
d. What is the solution to the homogeneous system, Ax=0?
Elementary Statistics: Picturing the World (7th Edition)
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Fundamental Theorem of Calculus 1 | Geometric Idea + Chain Rule Example; Author: Dr. Trefor Bazett;https://www.youtube.com/watch?v=hAfpl8jLFOs;License: Standard YouTube License, CC-BY