Inquiry-Based Learning
- There is no requirement that you get everything correct, nor finish an entire problem. Try everything.
- Come with questions, present periodically in class when you have a complete idea, and enjoy the learning process.
Brain Gains
- 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 $t=2$. In other words, give a vector equation of a line that passes through $\vec r(2) = (6,-2)$ and is parallel to $\frac{d\vec r}{dt}(2) = (4,-3)$.
Answer:
Passing through $(6,-2)$ and parallel to $(4,-3)$ means an equation is $$(x,y) = (4,-3)t+(6,-2)\quad \text{or}\quad \vec r(t) = (4,-3)t+(6,-2).$$ This is the same as $$(x,y) = (4t+6,-3t-2).$$
- The curve above represents the path of an object. Find the object's velocity and speed at $t=1$.
Answer:
The derivative $\frac{d\vec r}{dt} =(2t,-3)$ gives us the velocity at any time $t$. So at $t=1$ we have the velocity as $$\vec v = (2,-3).$$ The speed is the magnitude of the velocity, which gives the speed as $$v=|\vec v|=||\vec v|| = \sqrt{13}.$$ Note that we may use the same letter, without a vector symbol, to represent the magnitude of the corresponding vector, and you'll find that some people prefer double bars $||\vec v||$ instead of single bards $|\vec v|$.
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 (simplify your answer to get a speed of 3). What is the difference between velocity and speed?
- Draw $\vec r(t) = (3 \cos 2t, 3 \sin 2t)$. (Suggestion - use multiples of $\pi/4$ for a table, rather than $\pi/2$. Why?) What is the speed of this curve? (Simplify the speed to get 6.)
- Draw $\vec r(t) = (7 \cos 5t, 7 \sin 5t)$. What is the speed of this curve? (Did you get 35?)
- 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.
- Use Mathematica and the ParametricPlot[] command to draw your parametrization.
- Another hurricane has diameter $d$ miles, with sustained winds of $w$ mi/hr, and is moving in the direction of $\vec v$. Give a parametrization, similar to the one above.
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