SUNY Geneseo Department of Mathematics
Math 223 01
Fall 2022
Prof. Doug Baldwin
Complete by Thursday, October 6
Grade by Friday, October 14
This exercise focuses on developing your understanding of arc length and curvature of vector-valued functions. While doing that, it also reinforces your understanding of differentiation and integration for vector-valued functions, and of ways of using Mathematica in connection with such functions. It thus contributes to the following learning outcomes for this course:
This exercise is mainly based on material in sections 2.2 and 2.3 in our textbook. We talked about those sections in classes between September 26 and 28. On September 28 we experienced using Mathematica to solve multi-step problems, something that may be helpful on this problem set.
For several questions, you will want to find numeric values of integrals in
Mathematica. There are two ways to do this: First, use the NIntegrate
function, which works exactly the same way Integrate does for definite
integrals, but guarantees to give its answer as a decimal number instead of possibly
giving a symbolic answer. A slightly more general way to do this is to apply
Mathematica’s N function to the result of ordinary
Integrate. For example…
N[ Integrate[ ..., {t,...} ] ]
You can use the N function on any value in Mathematica to force
Mathematica to show you that value as a decimal number, so it’s a bit more
general (but a bit more combersome) than NIntegrate, which only gives you
a numeric value of a definite integral.
Solve each of the following problems.
Evaluate the following integrals by hand, and then confirm your answers by evaluating them with Mathematica.
\[\int \langle t^2 + 3t, e^t \cos(e^t), \sqrt{t+1} \rangle\,dt\] \[\int_0^1 \langle 1, t, t^2, t^3 \rangle\,dt\]When we started talking about vector-valued functions, we experimented with creating one that would look like a line drawing of the computer animated snake. My version of that function is
\[s(t) = \langle t, \frac{t \sin(4 \pi t)}{2}, \frac{t^{30}}{4} \rangle, 0 \le t \le 1\]Use Mathematica to calculate what the total length of this snake would be if it were stretched out in a straight line from tail to nose.
Find the length of one turn of the 4-dimensional helix \(\vec{r}(t) = \langle 2\sin t, \sqrt{3}t, 2\cos t, 2t \rangle\). Do not use Mathematica on this problem.
Find a unit vector that points in the direction the curve \(\vec{r}(t) = \langle \cos (e^t), \sin (e^t), 0 \rangle\) is turning when \(t = \ln \pi\). Also find the curvature of \(\vec{r}(t)\). Use Mathematica to carry out and organize the calculations.
I will grade this exercise during an individual meeting with you. That meeting should happen on or before the “Grade By” date above. During the meeting I will look at your solution, ask you any questions I have about it, answer questions you have, etc. Sign up for the meeting via Google calendar. Please have a written solution to the exercise ready to share with me during your meeting, as that will speed the process along.