Parametric Equations Assignment.sagews

Parametric Equations Assignment

Question 0

Watch the lecture video here.

Did you watch the video? [Type yes or no.]

Question 1

Graph the parametric equations $x(t)=\sin(t+\sin(t))$ and $y(t)=\cos(t+\cos(t))$ for $t=0$ to $t=2\pi$.

Question 2

Consider the parametric equations $x(t)=\sin(2t)$ and $y(t)=\sin(3t)$.

Part a

Graph these equations from $t=0$ to $t=2\pi$.

Part b

What values of $t$ result in the point $\left(\frac{\sqrt{3}}{2},0\right)$? (see Example 5)

Part c

Find the derivative $\displaystyle\frac{dy}{dx}$ (this will be a function of $t$).

Part d

Find the slopes for the values of $t$ you found in part (b). [There are two answers.]

Part e

Find equations for the tangent lines at $\left(\frac{\sqrt{3}}{2},0\right)$. [There are two tangent lines.]

Part f

Add the tangent lines to the graph above (For the tangent line plots, use $xmin=-1,\ xmax=1.5,\ ymin=-1,\ ymax=1$).

Question 3

Return to Example 2 and try some values of a and b to get an interesting picture.

This material was developed by Aaron Tresham at the University of Hawaii at Hilo and is
licensed under a Creative Commons Attribution-ShareAlike 4.0 International License.