The cycloid has parametric equations x = a(t + sin t), y = a(1 -
cos...
The cycloid has parametric equations x = a(t + sin t), y = a(1 -
cos t). Find
the length of the arc from t = 0 to t = pi. [ Hint: 1 + cosA = 2
cos2 A/2 ]. and the arc length of a
parametric
Consider the parametric equation of a curve:
x=cos(t), y= 1- sin(t), 0 ≤ t ≤ π
Part (a): Find the Cartesian equation of the
curve by eliminating the parameter. Also, graph the curve and
indicate with an arrow the direction in which the curve is traced
as the parameter increases. Label any x and y intercepts.
Part(b): Find the point (x,y) on the curve with
tangent slope 1 and write the equation of the tangent line.
Consider the curve traced out by the parametric equations: { x =
1 + cos(t) y = t + sin(t) for 0 ≤ t ≤ 4π.
1. Show that that dy dx = − 1 + cos(t)/sin(t) = − csc(t) −
cot(t).
2. Make a Sign Diagram for dy dx to find the intervals of t over
which C is increasing or decreasing.
• C is increasing on: • C is decreasing on:
3. Show that d2y/dx2 = − csc2...
a. 1 1 cos(x)cos(y) = -cos(x-y) + -cos(x + y) 1 l
sin(x)sin(y) = -cos(x-y)--cos(x+ y) 1 l sin(x)cos(y) =—sin(x-y)
+-sin(x + y) A DSB-FC (double sideband-full carrier) signal s(t) is
given by, s(t) = n cos(2rr/cf)+ cos(2«-/mt)cos(2«-fct) What is the
numeric value for the AM index of modulation, m, fors(f) ?
3. Consider the parametric curve x = sin 2t, y = − cos 2t for
−π/4 ≤ t ≤ π/4.
(a) (2 pts) Find the Cartesian form of the curve.
(b) (3 pts) Sketch the curve. Label the starting point and
ending point, and draw an
arrow on the curve to indicate the direction of travel.
(c) (5 pts) Find an equation for the curve’s tangent line at the
point
√2/2, −√2/2
.
The plane curve represented by x(t) = t − sin(t), y(t) = 7 −
cos(t), is a cycloid.
(a) Find the slope of the tangent line to the cycloid for 0 <
t < 2π.
dy
dx
(b) Find an equation of the tangent line to the cycloid at t
=
π
3
(c) Find the length of the cycloid from t = 0 to t =
π
2
Find the distance between the skew lines with parametric
equations x = 1 + t, y = 3 +
6t, z = 2t, and
x = 1 + 2s, y = 6 + 15s, z
= −2 + 6s.
Find the equation of the line that passes through the points on
the two lines where the shortest distance is measured.
Consider the parametric curve given by the equations
x = tsin(t) and y = tcos(t) for 0 ≤ t ≤ 1
Find the slope of a tangent line to this curve when t = 1.
Find the arclength of this curve (make sure to do it by
integration by parts if you find yourself integrating powers of
sec(θ))
Consider the line L with parametric equations x = 5t − 2, y = −t
+ 4, z= 2t + 5. Consider the plane P given by the equation
x+3y−z=6.
Find the distance from L to P .