20240926

• I-Learn, Class Pictures, Learning Targets, Text Book Practice
• Prep Tasks: Unit 1 - Motion, Unit 2 - Derivatives, Unit 3 - Integration, Unit 4 - Vector Calculus

Prep

We're in Unit 1 - Motion. Your homework assignment each day is to spend 1-2 hours working on the next 4 tasks from the current unit's prep.

Brain Gains (Rapid Recall, Jivin' Generation)

• Set up a work integral that will give the work done by $\vec F(x,y) = (x+y, -x^2)$ on an object that follows the path $\vec r(t) = (t^2, t^3)$ for $t\in [-1,2] $. The work integral is written using any of $$\ds\int_C Mdx+Ndy=\ds\int_C \vec F\cdot d\vec r=\ds\int_{a}^{b}\vec F(\vec r(t))\frac{d\vec r}{dt}dt.$$

ClearAll[F, r, tB, xB, yB]
r[t_] := {t^2, t^3}
tB = {t, -1, 2}
F[x_, y_] := {x + y, -x^2}

F @@ r[t]
dr = r'[t]
dW = (F @@ r[t]) . r'[t]
Integrate[dW, tB]

xB = {x, 0, 4}
yB = {y, -2, 10}
Show[
 ParametricPlot[r[t], Evaluate[tB]],
 VectorPlot[F[x, y], xB, yB, VectorScaling -> Automatic, VectorStyle -> Automatic]
 ]

Self Directed Learning Projects

1. Have a specific plan that helps you develop deeper understanding related to something from the unit. Think about Bloom's Taxonomy (see 1 and 2), and focus your efforts towards the highest levels.
2. Carry out the plan, making modifications as needed (follow new leads, keep in the time constraints, etc.).
3. Create something based on what you learned (Bloom's taxonomy).
4. Share your work publicly.
5. Complete a short steward report to reflect on your learning process.

Group Problems

1. A force given by $\vec F = (2y,-x+y)$ acts on an object as it moves along the curve $\vec r(t) =(t^2,3t+1)$ for $-1\leq t\leq 2$. Compute $\ds \int_C\vec F\cdot \frac{d\vec r}{dt}dt$ (the work done by the force along the curve).
2. Draw the parametric curve $\vec r(t) = (3\cos t, t/\pi, 4\sin t)$ for $0\leq t\leq 6\pi$.
   • If an object follows this parametrization, state the velocity, speed, and acceleration of that object.
   • Give a vector equation of the tangent line to the curve above at $t=\pi$.
   • Set up an integral that gives the length of this curve. Just set it up.
   • At time $t=2\pi$, compute the component of the acceleration that is parallel to the velocity, and the component of the acceleration that is orthogonal to the velocity.
3. Let $\vec F = (a,b,c)$ and $\vec d = (x,y,z)$. Compute the component of $\vec F$ that parallel to $\vec d$ and the component of $\vec F$ that is orthogonal to $\vec d$. [Feel free to pick specific values for $a, b, c, x, y, z$, or do this problem symbolically.]

Today

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