Brain Gains
- Compute the derivatives:
- $f(x) = \ln (2x+1)$
- $g(x) = \cos 3x$
- $p(x) = e^{2x} \sin x$
Answer:
- $\ds \frac{df}{dx} = \left(\frac{1}{ (2x+1) }\right)(2)=\frac{2}{2x+1}$
- $g'(x) = -3\sin 3x$
- $p'(x) = 2e^{2x} \sin x + e^{2x}\cos x$
- 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.
Answer:
The solution is an ellipse centered at the origin that goes left and right 4 units, while it goes up and down 3 units. Draw a box around this ellipse, and then connect opposing corners with lines to get the hyperbola that opens left and right.
- Sketch the curve $\vec r(t)=\left<t^2+2,-2t+1\right>$ for $-1\leq t\leq 2$.
Answer:
The graph is a small portion of a parabola. The vertex is at $ (2,1) $. It opens towards the right. At time $t=0$, it passes through the point $ (3,3) $, then through $(2,1)$, then through $ (3,-1)$, and then finishes at $t=2$ at $ (6,-3) $.
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.
- Draw the parametric curve $x=4+5\cos t$, $y=3+7\sin t$. Give a Cartesian equation of the curve.
- 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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