2.1.E: Curves (Exercises)

Exercise \(\PageIndex{1}\)

Plot the curves parametrized by the following functions over the specified intervals \(I\).

(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) \(h(t)=(t \sin (3 t), t \cos (3 t)), I=[-\pi, \pi]\)

Exercise \(\PageIndex{2}\)

Plot the curves parametrized by the following functions over the specified intervals \(I\).

(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) \(h(t)=(\cos (2 t), \sin (2 t), \sqrt{t}), I=[0,9]\)

Exercise \(\PageIndex{3}\)

Plot the curves parametrized by the following functions over the specified intervals \(I\).

(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) \(h(t)=((1+2 \cos (t)) \cos (t),(1+2 \cos (t)) \sin (t)), I=[0,2 \pi]\)

Exercise \(\PageIndex{4}\)

Suppose \(g: \mathbb{R} \rightarrow \mathbb{R}\) and we define \(f: \mathbb{R} \rightarrow \mathbb{R}^{2}\) by \(f(t)=(t, g(t))\). Describe the curve parametrized by \(f\).

Answer

The curve parametrized by \(f\) is the graph of \(g\).

Exercise \(\PageIndex{5}\)

For each of the following, 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) \(\mathbf{x}_{n}=\left(\frac{2 n-1}{n^{2}+1}, \frac{3 n+4}{n+1}, 4-\frac{6}{n^{2}}, \frac{6 n+1}{2 n^{2}+5}\right)\)

Answer

(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) \(\lim _{n \rightarrow \infty} \mathbf{x}_{n}=(0,3,4,0)\)

Exercise \(\PageIndex{6}\)

Let \(f: \mathbb{R} \rightarrow \mathbb{R}^{3}\) be defined by

\[ f(t)=\left(\frac{\sin (t)}{t}, \cos (t), 3 t^{2}\right). \nonumber \]

Evaluate the following.

(a) \(\lim _{t \rightarrow \pi} f(t)\)

(b) \(\lim _{t \rightarrow 1} f(t)\)

(c) \(\lim _{t \rightarrow 0} f(t)\)

Answer

(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) \(\lim _{t \rightarrow 0} f(t)=(1,1,0)\)

Exercise \(\PageIndex{7}\)

Discuss the continuity of each of the following functions.

(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) \(g(t)=(\cos (4 t), 1-\sqrt{3 t+1}, \sin (5 t), \sec (t))\)

Exercise \(\PageIndex{8}\)

Let \(f: \mathbb{R} \rightarrow \mathbb{R}^{3}\) be defined by \(f(t)=\left(t^{2}, 3 t, 2 t+1\right)\). Find

\[ \lim _{h \rightarrow 0} \frac{f(t+h)-f(t)}{h} .\nonumber \]

Answer

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