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].

For each of the following parametrizations [math]P(t) = (x(t),y(t))[/math], find the derived vector [math]\vec dP(t)[/math] for an arbitrary value of [math]t[/math] in the domain. Draw the vectors [math]\vec dP(0)[/math], [math]\vec dP(1)[/math], and [math]\vec dP(2)[/math] in the [math]xy[/math]-plane.

• [math]\dilemma{x(t) = t^2-1,} {y(t) = t^3, \quad -1\leq t \leq3.}[/math]
• [math]\dilemma{x(t) = \frac12(e^t+e^{-t}),} {y(t) = \frac12(e^t-e^{-t}), \quad -\infty \lt t \lt \infty.}[/math]
• [math]\dilemma{x(t) = t^2,} {y(t) = \frac23(3t+1)^{\frac32}, \quad -\frac13 \leq t \leq 5.}[/math]
• [math]\dilemma{x(t) = t^2+t+1,} {y(t) = \frac{t^3}3 + t^2 - 1, \quad -\infty \lt t \lt \infty.}[/math]

The cycloid shown in Figure is defined by a parametrization [math]P(\theta) = (x,y)[/math] in which

Compute the derived vector [math]\vec dP(\theta)[/math]. Sketch the curve, and draw the tangent vectors [math]\vec dP(0)[/math], [math]\vec dP\left(\frac\pi2\right)[/math], [math]\vec dP(\pi)[/math], and [math]\vec dP(2\pi)[/math].

Prove that the curve defined parametrically by the equations

where not both [math]d_1[/math] and [math]d_2[/math] are zero, is a straight line. (Note: Check the definition of a straight line given in section \secref{1.5}.)

Converse of Problem Exercise: Prove that, if [math]L[/math] is a straight line in [math]\R^2[/math], then it can be defined by a parametrization [math]P(s) = (x,y)[/math] for which

and not both [math]d_1[/math] and [math]d_2[/math] are zero.

For each of the following parametrizations [math]P(t) = (x(t),y(t))[/math] and values of [math]t_0[/math], compute the derived vector [math]\vec dP(t_0)[/math]. Draw the parametrized curve, the tangent line at [math]t_0[/math], and write an equation in [math]x[/math] and [math]y[/math] of the tangent line.

• [math]P(t) = (t^2+1, t+1)[/math], [math]-\infty \lt t \lt \infty[/math], and [math]t_0 = 2[/math].
• [math]P(t) = (t^2+1, t+1)[/math], [math]-\infty \lt t \lt \infty[/math], and [math]t_0 = 0[/math].
• [math]P(t) = (e^t,t)[/math], [math]-\infty \lt t \lt \infty[/math], and [math]t_0 = \ln 2[/math].
• [math]P(t) = (|t|, t)[/math], [math]-\infty \lt t \lt \infty[/math], and [math]t_0 = 0[/math].

Let [math]P[/math] be the parametrization defined by [math]P(t) = (t^2, \frac12t^2)[/math], for every real number [math]t[/math].

• Write an equation in [math]x[/math] and [math]y[/math] of the tangent line at [math]t=2[/math].
• Describe the vector space of tangent vectors at [math]t=2[/math] and at [math]t=0[/math].

Let [math]f[/math] be a real-valued function which is differentiable at [math]a[/math].

• Write an equation of the line tangent to the graph of [math]f[/math] at [math](a,f(a))[/math].
• Consider the parametrization
  [[math]] P(t) = (t,f(t)) . [[/math]]
  Compute the derived vector [math]\vec dP(a)[/math], and write an equation of the tangent line to the parametrized curve at [math]a[/math].

Let [math]P:[a,b]\goesto\R^2[/math] be a parametrization for which the derivatives [math]x^\prime[/math] and [math]y^\prime[/math] of the coordinate functions are continuous. Prove that the arc length of the curve parametrized by [math]P[/math] is given by