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))
Given the curve C in parametric form :
C : x = 2cos t , y = 2sin t , z = 2t ; 0≤ t ≤ 2pi
a) the velocity v(t)
b) the speed ds/dt
c) the acceleration a(t)
d) the unit tangent vector T(t)
e) The curvature k and the normal vector N(t)
f) the binormal vector B(t)
g) The tangential and normal components of accelertation
3. Given the parametric equations x = 3t /1 + t^3 and y = 3t^2
/1 + t^3 . a) Show that the curve produced also satisfies the
equation x^3 + y^3 = 3xy. b) Compute the limits of x and y as t
approaches −1: i. from the left ii. from the right c) To avoid the
issue in part b), graph the curve twice in the same command, once
for −5 < t < −1.5 and once for...
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...
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 .
Find all horizontal and vertical tangent lines for the
parametric curve defined by x(t) = t^3 - 3t +1, y(t) = 4t^2 +5.
then write our the equations for the tangent lines
We have a parametric curve x = - 6t + t^3 + 1, y = -2t − t^2
a) Sketch a graph of the curve (use technology to help)
b) Find the equation of the tangent line at t = –1
c) Find the value(s) of t where the tangent is horizontal.
d) Find the value(s) of t where the tangent is vertical.
Find the derivative of the parametric curve x = 6t - t^2, y =
lnt where t > 0
A. Find all values of t where the tangent line is horizontal. After
you find the values of t where the tangent lines are horizontal,
find the corresponding x and y values giving your answers as
ordered pairs.
B. Find all values of t where the tangent line is vertical.
After you find the values of t where the tangent lines...
Sketch the curve given by the parametric equation x= tan(t) , y=sec
(t) for -pi/2 < t < pi/2. Eliminate the parameter "t" and
find the Cartesian form of this curve. What type of curve is this?
What curve would be good if "t" belong to the interval (pi/2,
3pi/2)?