Draw and identify each of the curves defined by the following parametrizations.

• [math]P(t) = (t, t^2)[/math], \quad [math]-\infty \lt t \lt \infty[/math].
• [math]P(t) = (t-1, t^2)[/math], \quad [math]-\infty \lt t \lt \infty[/math].
• [math]P(t) = (t^2-1, t+1)[/math], \quad [math]-\infty \lt t \lt \infty[/math].
• [math]P(t) = (2t^{\frac13}, 3t^{\frac13})[/math] \quad [math]-\infty \lt t \lt \infty[/math].
• [math]P(t) = (t-1, t^3)[/math], \quad [math]-\infty \lt t \lt \infty[/math].
• [math]P(t) = (3\cos t, 3\sin t)[/math], \quad [math]0 \leq t \leq \pi[/math].
• [math]P(s) = (\sin s, 2)[/math], \quad [math]-\infty \lt s \lt \infty[/math].
• [math]Q(r) = (2\sin r, 3\cos r)[/math], \quad [math]-\infty \lt r \lt \infty[/math].

Draw and identify each of the following parametrized curves.

• [math]\dilemma{x=t-1,} {y=2t+3, & -\infty \lt t \lt \infty.}[/math]
• [math]\dilemma{x=t^2,} {y=t-3, & -\infty \lt t \lt \infty.}[/math]
• [math]\dilemma{2 \cos t,} {y=\cos t, & -\infty \lt t \lt \infty.}[/math]
• [math]\dilemma{x=3\sec\theta,} {y=2\tan\theta, & -\frac{\pi}2 \lt t \lt \frac{\pi}2.}[/math]

Each of the following parametrized curves is a function [math]f[/math] of [math]x[/math]. [To put it another way, each is the graph of an equation [math]y = f(x)[/math].] Find [math]f(x)[/math].

• [math]P(t) = (t-1, t^2+1)[/math], \quad [math]-\infty \lt t \lt \infty[/math].
• [math]\dilemma{x=t,}{y=e^{t^2}, & -\infty \lt t \lt \infty.}[/math]
• [math]\dilemma{x=2\cos t}{y=3\sin t, & 0 \leq t \leq \pi.}[/math]
• [math]P(t) = (e^t, t)[/math], \quad [math]-\infty \lt t \lt \infty[/math].

For each of the following parametrization, find an equation [math]F(x,y) = c[/math] whose graph is the parametrized curve

• [math]P(t) = (t^2,t)[/math], \quad [math]-\infty \lt t \lt \infty[/math].
• lab{10.1.4b} [math]\dilemma{x=e^{3t},}{y=e^t, & -\infty \lt t \lt \infty.}[/math]
• lab{10.1.4c} [math]\dilemma{x=e^t+e^{-t},} {y=e^t-e^{-t}, & -\infty \lt t \lt \infty.}[/math]

[For \ref{ex10.1.4b} and \ref{ex10.1.4c}, you will need in addition to the equation [math]F(x,y) = c[/math], the inequality [math]x \gt 0[/math].]

For the ellipse in Example \ref{exam 10.1.2} parametrized by the equations [math]x=4 \cos t[/math] and [math]y=3 \sin t[/math], interpret [math]t[/math] geometrically. (Hint: See Figure.)

Sketch the curve defined by the parametrization

This curve is traced by a point, fixed on a radius of a wheel of radius [math]a[/math] at a distance [math]b[/math] from the center, as the wheel rolls along a straight line. There are two cases.

• The curtate cycloid, where [math]a \gt b[/math]. (Think of a point on the spoke of a wheel.)
• The prolate cycloid, where [math]a \lt b[/math]. (Think of a point on the flange of a railway wheel.)

For each of the following parametrized curves, assume that [math]y[/math] is defined as a differentiable function of [math]x[/math] in a neighborhood of the points indicated, and find [math]\dydx[/math] at the point.

• [math]P(t) = (2t+1, t^2)[/math], when [math]t=2[/math].
• [math]\dilemma{x=5 \cos s,} {y=3 \sin s, \mbox{when} \: s = \frac{\pi}4.}[/math]
• [math]\dilemma{x=e^t,} {y=t, \mbox{when} \: t=0.}[/math]
• [math]\dilemma{x=e^t,} {y=t, \mbox{when} \: t = \ln 5.}[/math]

Find the slope of each of the following parametrized curves at the point indicated.

• [math]P(t) = (t-1, t^3-3t^2+3t-1)[/math], at [math]P(1)[/math].
• [math]\dilemma{x=3 \cos t,} {y=3 \sin t, \mbox{when} \: t = \frac{\pi}4.}[/math]
• [math]\dilemma{x = t^3 - t + 1,} {y = t^2 + t +1, \mbox{at} \: (1, 3).}[/math]
• [math]Q(t) = (t^2 - t + 1, e^t + 1)[/math], at [math]Q(0)[/math].

• Assuming the necessary differentiability conditions on the parametrization [math]P(t) = (x(t), y(t))[/math], derive a formula for [math]\deriv2y[/math]
• For the curve parametrized by the equations [math]x = t^2 + t + 1[/math] and [math]y = t^3 + 3t[/math], find [math]\deriv2y[/math] when [math]t=1[/math].