- As a class, we'll look at this Mathematica Notebook as a wrap up from our difference quotient discussion yesterday.
Rapid Recall
- Draw the parametric curve $x=4+3\cos t$, $y=5+2\sin t$.
Answer:
The graph is an ellipse centered at $(4,5)$ that opens left/right 3, and up/down 2.
- Give a Cartesian equation of the curve above.
Answer:
We know $\cos^2t+\sin^2t = 1$. Solving the equations above for $\cos t$ and $\sin t$ gives $\cos t =\frac{x-4}{3}$ and $\sin t = \frac{y-5}{2}$. This means, upon substitution, that $$\left(\frac{x-4}{3}\right)^2+\left(\frac{y-5}{2}\right)^2 = 1.$$
- For the curve $\vec r(t) = (t^2+2, -3t+4)$, note that $\frac{d\vec r}{dt} =(2t,-3)$. Give a vector equation of the tangent line to $\vec r(t)$ at $t=1$. In other words, give a vector equation of a line that passes through $\vec r(1) = (3,1)$ and is parallel to $\frac{d\vec r}{dt}(1) = (2,-3)$.
Answer:
Passing through $(3,1)$ and parallel to $(2,-3)$ means an equation is $$(x,y) = (2,-3)t+(3,1)\quad \text{or}\quad \vec r(t) = (2,-3)t+(3,1).$$ This is the same as $$(x,y) = (2t+3,-3t+1).$$
Group problems
- Draw $\vec r(t) = (3 \cos t, 3 \sin t)$.
- The curve above represents the position of an object. Compute the velocity of the object, so $\vec v(t) = \frac{d\vec r}{dt}$.
- State the speed of the object above.
- Draw $\vec r(t) = (3 \cos 2t, 3 \sin 2t)$. What is the speed of this curve?
- Draw $\vec r(t) = (7 \cos 5t, 7 \sin 5t)$. What is the speed of this curve?
- Hurricane Matthew has a diameter of 28 miles. Assuming the eye is at the origin $(0,0)$, give a parametrization of the exterior edge of the hurricane.
- Sustained winds are 128 mi/hr. Modify your parametrization above so that the speed is 128 mi/hr.
- The eye of the hurricane is moving north west at a speed of 12 mi/hr. Modify your parametrization so that the center moves north west at 12 mi/hr.
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