Rapid Recall
- Compute the derivatives:
- $f(x) = \ln 2x$
- $g(x) = \cos 3x$
- $p(x) = e^{2x} \sin x$
Answer:
- $\ds \frac{df}{dx} = \left(\frac{1}{2x}\right)(2)=\frac{1}{x}$
- $g'(x) = -3\sin 3x$
- $p'(x) = 2e^{2x} \sin x + e^{2x}\cos x$
- Draw $\ds \frac{x^2}{16}+\frac{y^2}{9}=1$ and $\ds \frac{x^2}{16}-\frac{y^2}{9}=1$. Use the same set of axes if you'd like.
- Sketch the curve $\vec r(t)=\left<t^2+2,-2t+1\right>$ for $-1\leq t\leq 2$.
Group problems
- Draw $\ds \frac{(x+2)^2}{9}+\frac{(y-4)^2}{25}=1$ and then draw $\ds -\frac{(x+2)^2}{9}+\frac{(y-4)^2}{25}=1$.
- Draw the parametric curve $x=2+3\cos t$, $y=4+5\sin t$. Make a $t,x,y$ table of points, and then graph the $(x,y)$ coordinates.
- 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 $\lim_{h\to 0}\frac{\vec r(t+h)-\vec r(t)}{h}$.
- As a class, we'll look at this Mathematica Notebook.
- Draw $\vec r(t) = (3 \cos t, 3 \sin t)$.
- Find the velocity of an object parametrized by the curve above. Then state the speed. [Hint: derivatives will help.]
- Draw $\vec r(t) = (3 \cos 2t, 3 \sin 2t)$. What is the speed of this curve?
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