Math 215 - Multivariate Calculus - Schedule

Brain Gains

Draw the parametric curve $x=3t$, $y=t^2$ for $-1\leq t\leq 3$. Mark the point $t=2$ on your graph. Below is a table you can use to help you (you'll need to fill in the missing 2 points). $$\begin{array}{c|c} t&(x,y)\\\hline -1&(-3,1)\\ 0&(0,0)\\ 1&\\ 2&\\ 3&(9,9) \end{array}$$
A rover follows the path given by $\vec r(t) = (3t,t^2)$. Find the velocity of the rover at $t=2$.
Give a vector equation of the tangent line to the rover's path at $t=2$, so $$\begin{pmatrix}x\\y\end{pmatrix} = \begin{pmatrix}?\\?\end{pmatrix} t+\begin{pmatrix}?\\?\end{pmatrix}.$$
Give the rover's speed at any time $t$.

Consider the parametric curve $\vec r(t) = (3-2t^2,4t+5)$ for $-1\leq t\leq 2$.
- Draw the curve.
- Find $\frac{d\vec r}{dt}$. Then state $\frac{dx}{dt}$, $\frac{dy}{dt}$, and $\frac{dy}{dx}$, all in terms of $t$.
- At $t=1$, state the point on the curve and a vector tangent to the curve. Then give a vector equation of the tangent line to the curve at $t=1$.
An object travels along straight lines. Its velocity is $ (0,3,4) $ m/s for 2 seconds, and then turns so its velocity is $ (1,2,-2) $ m/s for 5 seconds. Show the total distance (arc length) traveled by the object is 25m. [Hint: find the speed from the velocity.]
Find the arc length of the curve $\vec r(t) = (t^2, t^3)$ for $0\leq t\leq 2$. Recall the arc length formula is $$\int_C ds = \int_a^b\sqrt{\left(\frac{dx}{dt}\right)^2+\left(\frac{dy}{dt}\right)^2}dt.$$ Actually compute any integrals you encounter.
Consider again the curve $C$ parametrized by $\vec r(t) = (3-2t^2,4t+5)$ for $-1\leq t\leq 2$.
- Set up a formula to compute the length of the curve.
- A wire lies along the curve $C$. The density (mass per length) of the wire at a point $(x,y)$ on the curve is given by $\delta(x,y) = y+2$. Set up an integral formula that gives the total mass of the wire.
- The wire contains charged particles. The charge density (charge per length) at a point $(x,y)$ on the curve is given by the product $q(x,y)=xy$. Set up an integral formula that gives the total charge on the wire.