- 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) $ for 2 seconds, and then turns so its velocity is $ (1,2,-2) $ for 5 seconds. What is the distance traveled by the object.
- 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 $$\int_C ds = \int_a^b\sqrt{\left(\frac{dx}{dt}\right)^2+\left(\frac{dy}{dt}\right)^2}dt.$$
- Consider the curve $C$ parametrized by $\vec r(t) = (3-2t^2,4t+5)$ for $-1\leq t\leq 2$.
- Draw the curve $C$.
- Give a vector equation of the tangent line to the curve at $t=1$.
- 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.
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