Math 2210-1
Homework 1
Due Wednesday May 19
Show all work. Please box
√ your answers. Be sure to write in complete sentences when appropriate. Also,
I prefer exact answers like 2 instead of 1.414. Note that a symbol indicates that graph paper might be
useful for that problem.
Plane Curves: Parametric Representation
1.
The following curves are given parametrically. Graph the curve, state if the curve is closed or simple,
and obtain the Cartesian equation of the curve by eliminating the parameter.
(a) x = 3t, y = 2t; t ∈ R
√
(b) x = t − 3, y = 2t; 0 ≤ t ≤ 8
(c) x = 3 sin r, y = −2 cos r; 0 ≤ r ≤ 2π
(d) x = 2 cos2 r, y = 3 sin2 r; 0 ≤ r ≤ 2π
(e) x = sin θ, y = 2 cos2 2θ; θ ∈ R
2.
What curves do the following sets of parametric equations trace out?
Find an implicit or explicit
equation for each curve.
(a) x = 2 + cos t, y = 2 − sin t
(b) x = 2 + cos t, y = 2 − cos t
(c) x = 2 + cos t, y = cos2 t
3.
If t is allowed to take on all real values, the parametric equations
x = 2 + 3t, y = 4 + 7t
describe a line in the plane.
(a) What part of the line is obtained by restricting t to nonnegative numbers?
(b) What part of the line is obtained if t is restricted to −1 ≤ t ≤ 0?
(c) How should t be restricted to give the part of the line to the left of the y-axis?
4. Two particles are traveling through the plane. At time t the first particle is at the point (−1 + t, 4 − t)
and the second is at (−7 + 2t, −6 + 2t).
(a) Describe the two paths.
(b) Will the two particles collide? If so, when and where?
(c) Do the paths of the two particles cross, and if so, where?
5. Each of the three circles A, B and C of the figure below can be parameterized by equations of the form
x = a + k cos t, y = b + k sin t, 0 ≤ t ≤ 2π.
What can you say about the values of a, b and k for each of these circles?
y
10
B
C
x
10
A
-10
6. Find
d2 y
dy
without eliminating the parameter.
and
dx
dx2
(a) x = 6t2 , y = −2t3 ; t 6= 0
(b) x = 3 tan θ − 1, y = 5 sec θ + 2; t 6=
7.
(2n + 1)π
2
Given the curve x = 2et, y = 13 e−t, find the equation of the tangent to the given curve at the point
t = 0 without eliminating the parameter. Make a sketch.
8.
Find the area of the region between the curve x = e2t, y = e−t and the x-axis from t = 0 to t = ln 5.
Make a sketch.