Extended Problems
Problem Set 3
This problem set is an assessment of your understanding of the material so far, so it is very important that
the work you submit is your own. This problem set paper should be submitted in class by Friday, October
21, and an accompanying Mathematica notebook should be submitted via http://d2l.ship.edu by midnight
on Wednesday, October 19.
1. In the space below give parametric equations of one line that is perpendicular to the plane
3x + y − 2z = 6 and two non-parallel lines that are in the plane. In Mathematica produce a
picture that shows the plane and all three of your lines on the same coordinate axis.
2. In the space below show how to find the equation of the line determined by the intersection
of the planes 3x + y − 2z = 1 and 2x − y − 3z = 4. In Mathematica show both planes along
with a highlighted (i.e., thick) line of intersection on a single 3D plot.
3. In the space below, use algebra to find the exact value of x for which y = ex has maximum
curvature. (Feel free to use Mathematica to check your steps.) In Mathematica plot y = ex
and the curvature function on the same axes.
4. The Cornu spiral has vector equation
Z t
Z t
u2
u2
r(t) =
sin du,
cos du
2
2
0
0
In the space below, use algebra to find a formula for the curvature of the spiral. (Feel free to
use Mathematica to check your steps.) In Mathematica plot r(t) and r0 (t) on the same axes.
5. For a curve r(t) = hx(t), y(t)i define θ(t) to be the angle between the tangent line and
the x-axis at any time t. (θ(t) is called the angle of inclination of the curve.) Draw a
pitcture that illustrates that T(t) = hcos θ(t), sin θ(t)i, and then explain why it follows that
|θ0 (t)| = kT0 (x)k.