Homework 3: Parametrized Curves

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Online Problems

Several parametrized curves are graphed below, and the arrow indicates the direction in which the parameter increases. Which is the graph of the path $\vec {x}(t) = (4-t, 2t+1)$ , for $-1\leq t\leq 1$ ?

Several parametrized curves are graphed below, and the arrow indicates the direction in which the parameter increases. Which is the graph of the path $\vec {x}(t) = (t^2, t^3)$ , for $-1\leq t\leq 1$ ?

Consider the curve below.

Which of the following are parametrizations for the curve? Select all that apply.

$\vec {x}(t) = (\cos t, \sin t)$ , for $0\leq t\leq 2\pi$ $\vec {x}(t) = (\sin t , \cos t)$ , for $0\leq t\leq \pi$ $\vec {x}(t) = (\cos t, -\sin t)$ , for $0\leq t\leq \pi$ $\vec {x}(t) = (-\cos t, \sin t)$ , for $0\leq t\leq \pi$ $\vec {x}(t) = (\sin t , \cos t)$ , for $\pi /2\leq t\leq \pi /2$ $\vec {x}(t) = (\cos t, \sin t)$ , for $\pi \leq t\leq 2\pi$ $\vec {x}(t) = (-\sqrt {1-t^2}, t)$ for $-1\leq t\leq 1$ $\vec {x}(t) = (\sqrt {1-t^2}, t)$ for $0\leq t\leq 1$ $\vec {x}(t) = (t,-\sqrt {1-t^2})$ for $-1\leq t\leq 1$

Consider the curve below.

Which of the following are parametrizations for the curve? Select all that apply.

$\vec {x}(t) = (t,\sin t)$ for $0\leq t\leq 2\pi$ $\vec {x}(t) = (\arcsin t, t)$ for $-1\leq t\leq 1$ $\vec {x}(t) = (2t^2,\sin (2t^2))$ for $-1\leq t\leq 1$ $\vec {x}(t) = \left (t, \cos \left (\frac {\pi /2}(t-1)\right )\right )$ for $0\leq t\leq 2$ $\vec {x}(t) = \left (\frac {\pi }{2}t, \cos \left (\frac {\pi /2}t\right )\right )$ for $0\leq t\leq 2$ $\vec {x}(t) = \left (\frac {\pi }{2}t, \cos \left (\frac {\pi /2}(t-1)\right )\right )$ for $0\leq t\leq 2$ $\vec {x}(t) = \left (\frac {\pi }{2}t, \cos \left (\frac {\pi /2}(t-1)\right )\right )$ for $0\leq t\leq 4$

Consider the path $\vec {x}(t) = (3\cos (t), -2\sin (t))$ , for $t\in \mathbb {R}$ .

Compute the velocity of $\vec {x}$ . $\vec {v}(t) = \answer {(-3\sin (t), -2\cos (t))}$ Compute the speed of $\vec {x}$ . $\|\vec {x}'(t)\| = \answer {\sqrt {13}}$ Compute the acceleration of $\vec {x}$ . $\vec {a}(t) = \answer {(-3\cos (t), 2\cos (t))}$

Two ants are running on the top of a table. Their paths are described by $\vec {x}(t) = (t^2+1,2t-1)$ and $\vec {y}(t) = (\sqrt {t+3}, t),$ with coordinates in inches, for $t\geq 0$ in seconds.

At what time do the ants collide? $t = \answer {1}$

Where do the ants collide? $(x,y) = \answer {(2,1)}$

Consider the curve $\vec {x}(t) = (4t+2, 1-3t)$ for $t\in \mathbb {R}$ .

Compute the velocity. $\vec {v}(t) = \answer {(4,-3)}$

Compute the speed. $\|\vec {x}'(t)\| = \answer {4}$ Compute the acceleration. $\vec {a}(t) = \answer {(0,0)}$

Consider the curve $\vec {x}(t) = (2\cos t, 5\sin t, t^2)$ for $t\in \mathbb {R}$ .

Find the velocity. $\vec {v}(t) = \answer {(-2\sin t, 5\cos t, 2t)}$ Find the velocity when $t = \pi$ . $\vec {v}(\pi ) = \answer {(0,-5,2\pi )}$ Find a parametrization for the tangent line to $\vec {x}$ at the point where $t = \pi$ , so that $L(0) = \vec {x}(\pi )$ . $L(t) = \answer {(-2,5,\pi ^2) + t(0,-5,2\pi )}$

Consider the curve $\vec {x}(t) = (t, t^2, t^3)$ for $t\in \mathbb {R}$ .

Find the velocity. $\vec {v}(t) = \answer {(1, 2t, 3t^2)}$ Find the velocity when $t = 2$ . $\vec {v}(\pi ) = \answer {(1,4,12)}$ Find a parametrization for the tangent line to $\vec {x}$ at the point where $t = 2$ , so that $L(0) = \vec {x}(2)$ . $L(t) = \answer {(2,4,8) + t(1,4,12)}$

Consider the curve $\vec {x}(t) = (t, te^t, e^{t^2})$ for $t\in \mathbb {R}$ .

Find the velocity. $\vec {v}(t) = \answer {(1, e^t+te^t, 2te^{t^2})}$ Find the velocity when $t = 0$ . $\vec {v}(\pi ) = \answer {(1,1,0)}$ Find a parametrization for the tangent line to $\vec {x}$ at the point where $t = 0$ , so that $L(0) = \vec {x}(0)$ . $L(t) = \answer {(0,0,1) + t(1,1,0)}$

Written Problems

(a): Graph the surface $z^2 = x^2 +y^2$ and the curve $\vec {x}(t) = (t\cos (t), t\sin (t), t)$ for $-5\leq t\leq 5$ .
(b): Verify algebraically that the curve lies on the surface.

Prove the following product rule for cross products.

Let $\vec {x}$ and $\vec {y}$ be paths in $\mathbb {R}^3$ , then $(\vec {x}\times \vec {y})'(t) = \vec {x}'(t)\times \vec {y}(t) + \vec {x}\times \vec {y}'(t),$ for $t$ such that $x'(t)$ and $y'(t)$ exist.

Professional Problem

(a): Let $\vec {x}(t)$ be a curve lying on a sphere in $\mathbb {R}^n$ of radius $C$ . Prove that $\vec {x}(t)$ and $\vec {x}'(t)$ are perpendicular.
(b): For the curve $\vec {x}(t) = (\cos (t), \sin ^2(t), \cos (t)\sin (t))$ , verify computationally that $\vec {x}(t)$ lies on the sphere, and that $\vec {x}'(t)$ is perpendicular to $\vec {x}(t)$ .