2 Week 15

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2.1 HW12.4 - SCalcCC5 1.6.010.

Consider the parametric equations below. \[ x = t^2,\quad y = t^5 \]

2.1.1 (a)

Sketch the curve by using the parametric equations to plot points. Indicate with an arrow the direction in which the curve is traced as t increases.

To help with sketching, you might want to observe that the tangent line at a point on the parametric curve where \(t=t_0\) is determined by its tangent vector, i.e. a line passing through the points \(\big(x(t_0),y(t_0)\big)\) and \(\big(x(t_0)+x'(t_0),y(t_0)+y'(t_0)\big)\).

2.1.1.1 Solution

The left plot is animated.

Figure 2.1: Plot of algebraic curve given by \(x=t^2\) and \(y=t^5\) from \(t=-1\) to \(t=1\)

2.1.2 (b)

Eliminate the parameter to find a Cartesian equation of the curve.

2.1.2.1 Solution

\[ \begin{aligned} &y=t^5 \Rightarrow t=\sqrt[5]{y}=y^{1/5}\\ \Rightarrow\,&x=t^2=(y^{1/5})^2=y^{2 / 5} \end{aligned} \]

You might (or might not…) want to know that the singularity at \((0,0)\) is called a cusp, or even read a bit more on algebraic curves.

2.2 Recall: Curves Defined with Polar Coordinates

2.2.1 Change of Coordinate System

From what you already know about coordinates, one can make a change of coordinate for curves from polar to Cartesian by \[ \left\{\begin{aligned} x(t)&=r(t)\cdot\cos\theta(t)\\ y(t)&=r(t)\cdot\sin\theta(t); \end{aligned}\right.\quad \left\{\begin{aligned} r(t)&=\sqrt{x(t)^2+y(t)^2}\\ \theta(t)&=\arctan\left(\frac{y(t)}{x(t)}\right). \end{aligned}\right. \]

For a curve defined by \(r=f(\theta)\), one may parameterized it as \(r=f(t),\theta(t)=t\).

However, since we are considering negative \(r\) sometimes, e.g. HW 13.9 - SCalcCC5 A.H.1.035., you might wish to segment the curve when it passes through the origin.

If the curve is given by an equation, you might wish to try to manipulate the equation symbolically first \[ f(r)=g(\theta)\quad\leadsto\quad f\left(\sqrt{x(t)^2+y(t)^2}\right)=g\left(\frac{y(t)}{x(t)}\right), \] which might make the process easier.

2.2.2 Arc Length and Area inside Curve

\[ \begin{aligned} A&=\int_a^b \frac{1}{2} r^2\,\mathrm{d}\theta,\\ L&=\int_a^b \sqrt{r^2+\left(\frac{d r}{d \theta}\right)^2}\,\mathrm{d}\theta. \end{aligned} \]

From Prof. Rico’s notes.

2.3 HW 13.5 - SCalcCC5 A.H.1.016. [FIXME]

Find a Cartesian equation for the curve and identify it. \[ r=4\tan\theta\sec\theta \]

Please be advised of an ambiguity in this problem.

Figure 2.2: Contour plot of the components of \(x^4-16y^2=0\) over \(\mathbb{Q}\)

One may thus observe that the curve is a parabola, “re-traced” over itself.