91Ó°ÊÓ

Find the slope of the tangent line to the curve at the point corresponding to the value of the parameter. $$ x=e^{2 t}, \quad y=\ln t ; \quad t=1 $$

Plot the point with the rectangular coordinates. Then find the polar coordinates of the point taking \(r>0\) and \(0 \leq \theta<2 \pi\). \((5,-12)\)

Find the foci and vertices of the ellipse, and sketch its graph. $$ 4 x^{2}+9 y^{2}=36 $$

The parametric equations of the astroid \(x^{2 / 3}+y^{2 / 3}=a^{2 / 3}\) are \(x=a \cos ^{3} t\) and \(y=a \sin ^{3} t\). (Verify this!) Find an expression for the slope of the tangent line to the astroid in terms of \(t\). At what points on the astroid is the slope of the tangent line equal to \(-1 ?\) Equal to 1 ?

Show that a conic with focus at the origin, eccentricity \(e\), and directrix \(x=d\) has polar equation $$ r=\frac{e d}{1+e \cos \theta} $$

Find the length of the cardioid with parametric equations \(x=a(2 \cos t-\cos 2 t) \quad\) and \(\quad y=a(2 \sin t-\sin 2 t)\)

Let \(P\) be a point located a distance \(d\) from the center of a circle of radius \(r\). The curve traced out by \(P\) as the circle rolls without slipping along a straight line is called a trochoid. (The cycloid is the special case of a trochoid with \(d=r .\) ) Suppose that the circle rolls along the \(x\) -axis in the positive direction with \(\theta=0\) when the point \(P\) is at one of the lowest points on the trochoid. Show that the parametric equations of the trochoid are $$ x=r \theta-d \sin \theta \quad \text { and } \quad y=r-d \cos \theta $$ where \(\theta\) is the same parameter as that for the cycloid. Sketch the trochoid for the cases in which \(dr\).

The cornu spiral is a curve defined by the parametric equations \(x=C(t)=\int_{0}^{t} \cos \left(\pi u^{2} / 2\right) d u \quad y=S(t)=\int_{0}^{t} \sin \left(\pi u^{2} / 2\right) d u\) where \(C\) and \(S\) are called Fresnel integrals. They are used to explain the phenomenon of light diffraction. a. Plot the spiral. Describe the behavior of the curve as \(t \rightarrow \infty\) and as \(t \rightarrow-\infty\). b. Find the length of the spiral from \(t=0\) to \(t=a\).

The butterfly catastrophe curve, which is described by the parametric equations \(x=c\left(8 a t^{3}+24 t^{5}\right) \quad\) and \(\quad y=c\left(-6 a t^{2}-15 t^{4}\right)\) occurs in the study of catastrophe theory. Plot the curve with \(a=-7\) and \(c=0.03\) for \(t\) in the parameter interval \([-1.629,1.629] .\)

The Lissajous curves, also known as Bowditch curves, have applications in physics, astronomy, and other sciences. They are described by the parametric equations \(x=\sin (a t+b \pi), \quad a\) a rational number, and \(y=\sin t\) Plot the curve with \(a=0.75\) and \(b=0\) for \(t\) in the parameter interval \([0,8 \pi]\).