Saltar al contenido principal

# 2.1.E: Curvas (Ejercicios)

$$\newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} }$$

$$\newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}}$$

$$\newcommand{\id}{\mathrm{id}}$$ $$\newcommand{\Span}{\mathrm{span}}$$

( \newcommand{\kernel}{\mathrm{null}\,}\) $$\newcommand{\range}{\mathrm{range}\,}$$

$$\newcommand{\RealPart}{\mathrm{Re}}$$ $$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$

$$\newcommand{\Argument}{\mathrm{Arg}}$$ $$\newcommand{\norm}[1]{\| #1 \|}$$

$$\newcommand{\inner}[2]{\langle #1, #2 \rangle}$$

$$\newcommand{\Span}{\mathrm{span}}$$

$$\newcommand{\id}{\mathrm{id}}$$

$$\newcommand{\Span}{\mathrm{span}}$$

$$\newcommand{\kernel}{\mathrm{null}\,}$$

$$\newcommand{\range}{\mathrm{range}\,}$$

$$\newcommand{\RealPart}{\mathrm{Re}}$$

$$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$

$$\newcommand{\Argument}{\mathrm{Arg}}$$

$$\newcommand{\norm}[1]{\| #1 \|}$$

$$\newcommand{\inner}[2]{\langle #1, #2 \rangle}$$

$$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\AA}{\unicode[.8,0]{x212B}}$$

$$\newcommand{\vectorA}[1]{\vec{#1}} % arrow$$

$$\newcommand{\vectorAt}[1]{\vec{\text{#1}}} % arrow$$

$$\newcommand{\vectorB}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} }$$

$$\newcommand{\vectorC}[1]{\textbf{#1}}$$

$$\newcommand{\vectorD}[1]{\overrightarrow{#1}}$$

$$\newcommand{\vectorDt}[1]{\overrightarrow{\text{#1}}}$$

$$\newcommand{\vectE}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash{\mathbf {#1}}}}$$

$$\newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} }$$

$$\newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}}$$

Ejercicio$$\PageIndex{1}$$

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$$\PageIndex{7}$$

(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

$$\lim _{h \rightarrow 0} \frac{f(t+h)-f(t)}{h}=(2 t, 3,2)$$

This page titled 2.1.E: Curvas (Ejercicios) is shared under a CC BY-NC-SA 1.0 license and was authored, remixed, and/or curated by Dan Sloughter via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.