[math] \newcommand{\ex}[1]{\item } \newcommand{\sx}{\item} \newcommand{\x}{\sx} \newcommand{\sxlab}[1]{} \newcommand{\xlab}{\sxlab} \newcommand{\prov}[1] {\quad #1} \newcommand{\provx}[1] {\quad \mbox{#1}} \newcommand{\intext}[1]{\quad \mbox{#1} \quad} \newcommand{\R}{\mathrm{\bf R}} \newcommand{\Q}{\mathrm{\bf Q}} \newcommand{\Z}{\mathrm{\bf Z}} \newcommand{\C}{\mathrm{\bf C}} \newcommand{\dt}{\textbf} \newcommand{\goesto}{\rightarrow} \newcommand{\ddxof}[1]{\frac{d #1}{d x}} \newcommand{\ddx}{\frac{d}{dx}} \newcommand{\ddt}{\frac{d}{dt}} \newcommand{\dydx}{\ddxof y} \newcommand{\nxder}[3]{\frac{d^{#1}{#2}}{d{#3}^{#1}}} \newcommand{\deriv}[2]{\frac{d^{#1}{#2}}{dx^{#1}}} \newcommand{\dist}{\mathrm{distance}} \newcommand{\arccot}{\mathrm{arccot\:}} \newcommand{\arccsc}{\mathrm{arccsc\:}} \newcommand{\arcsec}{\mathrm{arcsec\:}} \newcommand{\arctanh}{\mathrm{arctanh\:}} \newcommand{\arcsinh}{\mathrm{arcsinh\:}} \newcommand{\arccosh}{\mathrm{arccosh\:}} \newcommand{\sech}{\mathrm{sech\:}} \newcommand{\csch}{\mathrm{csch\:}} \newcommand{\conj}[1]{\overline{#1}} \newcommand{\mathds}{\mathbb} [/math]

For each of the following parametrizations and values of [math]t_0[/math], compute [math]P(t_0)[/math] and the derived vector [math]\vec dP(t_0)[/math]. Draw the parametrized curve and each of the tangent vectors [math]\vec dP(t_0)[/math] to the curve.

• [math]P(t) = (x(t),y(t)) = (t-1,t^2), \quad -\infty \lt t \lt \infty[/math]; [math]t_0 = -1[/math], [math]t_0 = 0[/math], and [math]t_0 = 2[/math].
• [math]P(t) = (x(t),y(t)) = (t^2+1,t-1), \quad -\infty \lt t \lt \infty[/math]; [math]t_0 = -1[/math], [math]t_0 = 0[/math], and [math]t_0 = 1[/math].
• [math]P(t) = (t-1,t^3), \quad -\infty \lt t \lt \infty[/math]; [math]t_0 = 0[/math], [math]t_0 = 1[/math], and [math]t_0 = 2[/math].
• [math]P(t) = (x,y) = (e^t,t), \quad -\infty \lt t \lt \infty[/math]; [math]t_0 = 0[/math] and [math]t_0 = \ln2[/math].
• [math]P(t) = (3\cos t,2\sin t), \quad -\infty \lt t \lt \infty[/math]; [math]t_0 = 0[/math], [math]t_0 = \frac\pi4[/math], and [math]t_0 = \frac\pi2[/math].
• [math]P(t) = (x(t),y(t)) = (t-1,t^2), \quad -\infty \lt t \lt \infty[/math]; [math]t_0 = -1[/math], [math]t_0 = 0[/math], and [math]t_0 = 2[/math].
• [math]P(t) = (t^2,t^3), \quad -\infty \lt t \lt \infty[/math]; [math]t_0 = -1[/math], [math]t_0 = 0[/math], and [math]t_0 = 2[/math].
• [math]P(t) = (t-1,2t+4), \quad -2 \leq t \leq 2[/math]; [math]t_0 = -1[/math], [math]t_0 = 0[/math], and [math]t_0 = 1[/math].