2.1.E: Curvas (Ejercicios)
- Page ID
- 111727
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Trazar las curvas parametrizadas por las siguientes funciones a lo largo de los intervalos especificados\(I\).
(a)\(f(t)=(3 t+1,2 t-1), I=[-5,5]\)
b)\(g(t)=\left(t, t^{2}\right), I=[-3,3]\)
c)\(f(t)=(3 \cos (t), 3 \sin (t)), I=[0,2 \pi]\)
d)\(h(t)=(3 \cos (t), 3 \sin (t)), I=[0, \pi]\)
(e)\(f(t)=(4 \cos (2 t), 2 \sin (2 t), I=[0, \pi]\)
f)\(g(t)=(-4 \cos (t), 2 \sin (t)), I=[0, \pi]\)
g)\(h(t)=(t \sin (3 t), t \cos (3 t)), I=[-\pi, \pi]\)
Ejercicio\(\PageIndex{2}\)
Trazar las curvas parametrizadas por las siguientes funciones a lo largo de los intervalos especificados\(I\).
(a)\(f(t)=(t+1,2 t-1,3 t), I=[-4,4] \)
b)\(g(t)=(\cos (t), t, \sin (t)), I=[0,4 \pi]\)
c)\(f(t)=(t \cos (2 t), t \sin (2 t), t), I=[-10,10] \)
d)\(h(t)=(\cos (2 t), \sin (2 t), \sqrt{t}), I=[0,9]\)
Ejercicio\(\PageIndex{3}\)
Trazar las curvas parametrizadas por las siguientes funciones a lo largo de los intervalos especificados\(I\).
(a)\(f(t)=(\cos (4 \pi t), \sin (5 \pi t)), I=[-0.5,0.5] \)
b)\(f(t)=(\cos (6 \pi t), \sin (7 \pi t)), I=[-0.5,0.5]\)
c)\(h(t)=\left(\cos ^{3}(t), \sin ^{3}(t)\right), I=[0,2 \pi]\)
d)\(g(t)=(\cos (2 \pi t), \sin (2 \pi t), \sin (4 \pi t)), I=[0,1]\)
(e)\(f(t)=(\sin (4 t) \cos (t), \sin (4 t) \sin (t)), I=[0,2 \pi]\)
f)\(h(t)=((1+2 \cos (t)) \cos (t),(1+2 \cos (t)) \sin (t)), I=[0,2 \pi]\)
Ejercicio\(\PageIndex{4}\)
Supongamos\(g: \mathbb{R} \rightarrow \mathbb{R}\) y definimos\(f: \mathbb{R} \rightarrow \mathbb{R}^{2}\) por\(f(t)=(t, g(t))\). Describir la curva parametrizada por\(f\).
- Contestar
-
La curva parametrizada por\(f\) es la gráfica de\(g\).
Ejercicio\(\PageIndex{5}\)
Para cada una de las siguientes, compute\(\lim _{n \rightarrow \infty} \mathbf{x}_{n}\)
(a)\(\mathbf{x}_{n}=\left(\frac{n+1}{2 n+3}, 3-\frac{1}{n}\right)\)
b)\(\mathbf{x}_{n}=\left(\sin \left(\frac{n-1}{n}\right), \cos \left(\frac{n-1}{n}\right), \frac{n-1}{n}\right)\)
c)\(\mathbf{x}_{n}=\left(\frac{2 n-1}{n^{2}+1}, \frac{3 n+4}{n+1}, 4-\frac{6}{n^{2}}, \frac{6 n+1}{2 n^{2}+5}\right)\)
- Contestar
-
(a)\(\lim _{n \rightarrow \infty} \mathbf{x}_{n}=\left(\frac{1}{2}, 3\right)\)
b)\(\lim _{n \rightarrow \infty} \mathbf{x}_{n}=(\sin (1), \cos (1), 1)\)
c)\(\lim _{n \rightarrow \infty} \mathbf{x}_{n}=(0,3,4,0)\)
Ejercicio\(\PageIndex{6}\)
Dejar\(f: \mathbb{R} \rightarrow \mathbb{R}^{3}\) ser definido por
\[ f(t)=\left(\frac{\sin (t)}{t}, \cos (t), 3 t^{2}\right). \nonumber \]
Evalúe lo siguiente.
(a)\(\lim _{t \rightarrow \pi} f(t)\)
b)\(\lim _{t \rightarrow 1} f(t)\)
c)\(\lim _{t \rightarrow 0} f(t)\)
- Contestar
-
(a)\(\lim _{t \rightarrow \pi} f(t)=\left(0,-1,3 \pi^{2}\right)\)
b)\(\lim _{t \rightarrow 1} f(t)=(\sin (1), \cos (1), 3)\)
c)\(\lim _{t \rightarrow 0} f(t)=(1,1,0)\)
Ejercicio\(\PageIndex{7}\)
Discutir la continuidad de cada una de las siguientes funciones.
(a)\(f(t)=\left(t^{2}+1, \cos (2 t), \sin (3 t)\right. ) \)
b)\(g(t)=(\sqrt{t+1}, \tan (t))\)
c)\(f(t)=\left(\frac{1}{t^{2}-1}, \sqrt{1-t^{2}}, \frac{1}{t}\right)\)
d)\(g(t)=(\cos (4 t), 1-\sqrt{3 t+1}, \sin (5 t), \sec (t))\)
Ejercicio\(\PageIndex{8}\)
Dejar\(f: \mathbb{R} \rightarrow \mathbb{R}^{3}\) ser definido por\(f(t)=\left(t^{2}, 3 t, 2 t+1\right)\). Encuentra
\[ \lim _{h \rightarrow 0} \frac{f(t+h)-f(t)}{h} .\nonumber \]
- Contestar
-
\(\lim _{h \rightarrow 0} \frac{f(t+h)-f(t)}{h}=(2 t, 3,2)\)