- Draw the parametric curve $x=2+3\cos t$, $y=5+2\sin t$. Make a $t,x,y$ table of points, and then graph the $(x,y)$ coordinates. Finish by giving a Cartesian equation of this curve.
- Draw the ellipse $x^2+6x+4y^2+8y=12$.
- Draw the hyperbola $x^2+6x-4y^2+8y=-1$.
- Draw the parametric curve $\vec r(t) = (2+t^2,3t-4)$ for $-2\leq t\leq 3$.
- Give $\frac{d\vec r}{dt}$. Then state $\frac{dx}{dt}$, $\frac{dy}{dt}$, and $\frac{dy}{dx}$.
- Give a vector equation of the tangent line to the curve at $t=2$.
- Consider the curve $\vec r(t) = (2+4t,5-2t)$. Our goal is to visualize the difference quotient $\frac{\vec r(t+h)-\vec r(t)}{h}$.
- Draw the parametric curve $\vec r(t) = (2+4t,5-2t)$ for $t\in[-2,3]$.
- When $t=0$ and $h=1$, add to a graph of the curve the vectors $\vec r(t+h)$, $\vec r(t)$, and the difference $\vec r(t+h)-\vec r(t)$.
- When $t=0$ and $h=1/2$, add to a graph of the curve the vectors $\vec r(t+h)$, $\vec r(t)$, and the difference $\vec r(t+h)-\vec r(t)$. How does division by $h$ affect the difference?
- When $t=0$ and $h=1/4$, add to a graph of the curve the vectors $\vec r(t+h)$, $\vec r(t)$, and the difference $\vec r(t+h)-\vec r(t)$. How does division by $h$ affect the difference?
- Visually, what vector do you obtain by computing $\lim_{h\to 0}\frac{\vec r(t+h)-\vec r(t)}{h}$.
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