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  • Calculus: Early Transcendentals
  • Vector Functions

Calculus: Early Transcendentals

James Stewart

Chapter 13

Vector Functions - all with Video Answers

Educators

+ 5 more educators

Section 1

Vector Functions and Space Curves

01:59

Problem 1

Find the domain of the vector function.

$ r(t) = \biggr\langle \ln (t + 1), \frac{t}{\sqrt{9 - t^2}}, 2^t \biggr\rangle $

WZ
Wen Zheng
Numerade Educator
01:11

Problem 2

Find the domain of the vector function.

$ r(t) = \cos t i + \ln t j + \frac{1}{t - 2}\ k $

WZ
Wen Zheng
Numerade Educator
02:17

Problem 3

Find the limit.

$ \lim_{t\to 0} \left(e^{-3t} i + \frac{t^2}{\sin^2 t}\ j + \cos 2t\ k \right) $

Bobby Barnes
Bobby Barnes
University of North Texas
02:22

Problem 4

Find the limit.

$ \lim_{t\to 1} \left(\frac{t^2 - t}{t - 1}\ i + \sqrt{t + 8}\ j + \frac{\sin \pi t}{\ln t}\ k \right) $

WZ
Wen Zheng
Numerade Educator
01:40

Problem 5

Find the limit.

$ \lim_{t\to\infty} \biggr\langle\frac{1 + t^2}{1 - t^2}, \tan^{-1} t , \frac{1 - e^{-2t}}{t} \biggr\rangle $

WZ
Wen Zheng
Numerade Educator
02:20

Problem 6

Find the limit.

$ \lim_{t\to\infty} \biggr\langle te^{-t} , \frac{t^3 + t}{2t^3 - 1}, t \sin \frac{1}{t} \biggr\rangle $

WZ
Wen Zheng
Numerade Educator
00:55

Problem 7

Sketch the curve with the given vector equation. Indicate with an arrow the direction in which $ t $ increases.

$ r(t) = \langle \sin t , t \rangle $

WZ
Wen Zheng
Numerade Educator
01:17

Problem 8

Sketch the curve with the given vector equation. Indicate with an arrow the direction in which $ t $ increases.

$ r(t) = \langle t^2 - 1 , t \rangle $

WZ
Wen Zheng
Numerade Educator
02:23

Problem 9

Sketch the curve with the given vector equation. Indicate with an arrow the direction in which $ t $ increases.

$ r(t) = \langle t , 2 - t , 2t \rangle $

WZ
Wen Zheng
Numerade Educator
01:27

Problem 10

Sketch the curve with the given vector equation. Indicate with an arrow the direction in which $ t $ increases.

$ r(t) = \langle \sin \pi t , t , \cos \pi t \rangle $

Carson Merrill
Carson Merrill
Numerade Educator
02:21

Problem 11

Sketch the curve with the given vector equation. Indicate with an arrow the direction in which $ t $ increases.

$ r(t) = \langle 3 , t , 2 - t^2 \rangle $

WZ
Wen Zheng
Numerade Educator
04:33

Problem 12

Sketch the curve with the given vector equation. Indicate with an arrow the direction in which $ t $ increases.

$ r(t) = 2 \cos t i + 2 \sin t j + k $

KS
Keyan Sheppard
Numerade Educator
01:58

Problem 13

Sketch the curve with the given vector equation. Indicate with an arrow the direction in which $ t $ increases.

$ r(t) = t^2 i + t^4 j + t^6 k $

WZ
Wen Zheng
Numerade Educator
02:37

Problem 14

Sketch the curve with the given vector equation. Indicate with an arrow the direction in which $ t $ increases.

$ r(t) = \cos t i - \cos t j + \sin t k $

WZ
Wen Zheng
Numerade Educator
06:10

Problem 15

Draw the projections of the curve on the three coordinate planes. Use these projections to help sketch the curve.

$ r(t) = \langle t , \sin t , 2 \cos t \rangle $

Linda Hand
Linda Hand
Numerade Educator
02:16

Problem 16

Draw the projections of the curve on the three coordinate planes. Use these projections to help sketch the curve.

$ r(t) = \langle t , t , t^2 \rangle $

WZ
Wen Zheng
Numerade Educator
02:11

Problem 17

Find a vector equation and parametric equations for the line segment that joins $ P $ to $ Q $.

$ P (2, 0, 0), Q (6, 2, -2) $

Manish Kumar ( Iit K )
Manish Kumar ( Iit K )
Numerade Educator
01:34

Problem 18

Find a vector equation and parametric equations for the line segment that joins $ P $ to $ Q $.

$ P (-1, 2, -2), Q (-3, 5, 1) $

Carson Merrill
Carson Merrill
Numerade Educator
01:18

Problem 19

Find a vector equation and parametric equations for the line segment that joins $ P $ to $ Q $.

$ P (0, -1, 1), Q (\frac{1}{2}, \frac{1}{3}, \frac{1}{4}) $

WZ
Wen Zheng
Numerade Educator
01:28

Problem 20

Find a vector equation and parametric equations for the line segment that joins $ P $ to $ Q $.

$ P (a, b, c), Q (u, v, w) $

WZ
Wen Zheng
Numerade Educator
01:09

Problem 21

Match the parametric equations with the graphs (labeled I-VI). Give reasons for your choices.

$ x = t \cos t $ , $ y = t $ , $ z = t \sin t $ , $ t \ge 0 $

WZ
Wen Zheng
Numerade Educator
01:29

Problem 22

Match the parametric equations with the graphs (labeled I-VI). Give reasons for your choices.

$ x = \cos t $ , $ y = \sin t $ , $ z = \frac{1}{(1 + t^2)} $

WZ
Wen Zheng
Numerade Educator
01:06

Problem 23

Match the parametric equations with the graphs (labeled I-VI). Give reasons for your choices.

$ x = t $ , $ y = \frac{1}{(1 + t^2)} $ , $ z = t^2 $

Carson Merrill
Carson Merrill
Numerade Educator
01:13

Problem 24

Match the parametric equations with the graphs (labeled I-VI). Give reasons for your choices.

$ x = \cos t $ , $ y = \sin t $ , $ z = \cos 2t $

WZ
Wen Zheng
Numerade Educator
01:19

Problem 25

Match the parametric equations with the graphs (labeled I-VI). Give reasons for your choices.

$ x = \cos 8t $ , $ y = \sin 8t $ , $ z = e^{0.8t} $ , $ t \ge 0 $

Carson Merrill
Carson Merrill
Numerade Educator
01:14

Problem 26

Match the parametric equations with the graphs (labeled I-VI). Give reasons for your choices.

$ x = \cos^2 t $ , $ y = \sin^2 t $ , $ z = t $

WZ
Wen Zheng
Numerade Educator
02:10

Problem 27

Show that the curve with parametric equations $ x = t \cos t $, $ y = t \sin t $, $ z = t $ lies on the cone $ z^2 = x^2 + y^2 $, and use this fact to help sketch the curve.

Carson Merrill
Carson Merrill
Numerade Educator
03:34

Problem 28

Show that the curve with parametric equations $ x = \sin t $, $ y = \cos t $, $ z = \sin^2 t $ is the curve of intersection of the surfaces $ z = x^2 $ and $ x^2 + y^2 = 1 $. Use this fact to help sketch the curve.

WZ
Wen Zheng
Numerade Educator
05:13

Problem 29

Find three different surfaces that contain the curve
$ r(t) = 2t i + e^t j + e^{2t} k $

Aparna Shakti
Aparna Shakti
Numerade Educator
01:10

Problem 30

Find three different surfaces that contain the curve
$ r(t) = t^2 i + \ln tj + (\frac{1}{t}) k $.

Carson Merrill
Carson Merrill
Numerade Educator
02:48

Problem 31

At what points does the curve $ r(t) = ti + (2t - t^2) k $ intersect the paraboloid $ z = x^2 + y^2 $?

Carson Merrill
Carson Merrill
Numerade Educator
01:42

Problem 32

At what points does the helix $ r(t) = \langle \sin t, \cos t, t \rangle $ intersect the sphere $ x^2 + y^2 + z^2 = 5 $?

WZ
Wen Zheng
Numerade Educator
01:03

Problem 33

Use a computer to graph the curve with the given vector equation. Make sure you choose a parameter domain and viewpoints that reveal the true nature of the curve.

$ r(t) = \langle \cos t \sin 2t, \sin t \sin 2t, \cos 2t \rangle $

WZ
Wen Zheng
Numerade Educator
01:04

Problem 34

Use a computer to graph the curve with the given vector equation. Make sure you choose a parameter domain and viewpoints that reveal the true nature of the curve.

$ r(t) = \langle te^t, e^{-t}, t \rangle $

WZ
Wen Zheng
Numerade Educator
01:02

Problem 35

Use a computer to graph the curve with the given vector equation. Make sure you choose a parameter domain and viewpoints that reveal the true nature of the curve.

$ r(t) = \langle \sin 3t \cos t, \frac{1}{4} t, \sin 3t \sin t \rangle $

WZ
Wen Zheng
Numerade Educator
01:06

Problem 36

Use a computer to graph the curve with the given vector equation. Make sure you choose a parameter domain and viewpoints that reveal the true nature of the curve.

$ r(t) = \langle \cos (8 \cos t) \sin t, \sin (8 \cos t) \sin t, \cos t \rangle $

WZ
Wen Zheng
Numerade Educator
00:51

Problem 37

Use a computer to graph the curve with the given vector equation. Make sure you choose a parameter domain and viewpoints that reveal the true nature of the curve.

$ r(t) = \langle \cos 2t, \cos 3t, \cos 4t \rangle $

WZ
Wen Zheng
Numerade Educator
01:30

Problem 38

Graph the curve with parametric equations $ x = \sin t $, $ y = \sin 2t $, $ z = \cos 4t $. Explain its shape by graphing its projections onto the three coordinate planes.

WZ
Wen Zheng
Numerade Educator
01:38

Problem 39

Graph the curve with parametric equations
$$ x = (1 + \cos 16t) \cos t $$
$$ y = (1 + \cos 16t) \sin t $$
$$ z = 1 + \cos 16t $$
Explain the appearance of the graph by showing that it lies on a cone.

WZ
Wen Zheng
Numerade Educator
02:19

Problem 40

Graph the curve with parametric equations
$$ x = \sqrt{1 - 0.25 \cos^2 10t} \cos t $$
$$ y = \sqrt{1 - 0.25 \cos^2 10t} \sin t $$
$$ z = 0.5 \cos 10t $$
Explain the appearance of the graph by showing that it lies on a sphere.

WZ
Wen Zheng
Numerade Educator
02:53

Problem 41

Show that the curve with parametric equations $ x = t^2 $, $ y = 1 - 3t $, $ z = 1 + t^3 $ passes through the points $ (1, 4, 0) $ and $ (9, -8, 28) $ but not through the point $ (4, 7, -6) $.

WZ
Wen Zheng
Numerade Educator
01:21

Problem 42

Find a vector function that represents the curve of intersection of the two surfaces.

The cylinder $ x^2 + y^2 = 4 $ and the surface $ z = xy $

WZ
Wen Zheng
Numerade Educator
01:45

Problem 43

Find a vector function that represents the curve of intersection of the two surfaces.

The cone $ z = \sqrt{x^2 + y^2} $ and the plane $ z = 1 + y $

WZ
Wen Zheng
Numerade Educator
02:40

Problem 44

Find a vector function that represents the curve of intersection of the two surfaces.

The paraboloid $ z = 4x^2 + y^2 $ and the parabolic cylinder $ y = x^2 $

Madi Sousa
Madi Sousa
Numerade Educator
01:30

Problem 45

Find a vector function that represents the curve of intersection of the two surfaces.

The hyperboloid $ z = x^2 - y^2 $ and the cylinder $ x^2 + y^2 = 1 $

Carson Merrill
Carson Merrill
Numerade Educator
View

Problem 46

Find a vector function that represents the curve of intersection of the two surfaces.

The semiellipsoid $ x^2 + y^2 + 4z^2 = 4, y \ge 0 $, and the cylinder $ x^2 + z^2 = 1 $

Prabhat Tyagi
Prabhat Tyagi
Numerade Educator
03:12

Problem 47

Try to sketch by hand the curve of intersection of the circular cylinder $ x^2 + y^2 = 4 $ and the parabolic cylinder $ z = x^2 $. Then find parametric equations for this curve and use these equations and a computer to graph the curve.

WZ
Wen Zheng
Numerade Educator
02:02

Problem 48

Try to sketch by hand the curve of intersection of the parabolic cylinder $ y = x^2 $ and the top half of the ellipsoid $ x^2 + 4y^2 + 4z^2 = 16 $. Then find parametric equations for this curve and use these equations and a computer to graph the curve.

Carson Merrill
Carson Merrill
Numerade Educator
02:18

Problem 49

If two objects travel through space along two different curves, it's often important to know whether they will collide. (Will a missile hit its moving target? Will two aircraft collide?) The curves might intersect, but we need to know whether the objects are in the same position at the same time. Suppose the trajectories of two particles are given by the vector functions
$ r_1 (t) = \langle t^2, 7t - 12, t^2 \rangle $ $ r_2 (t) = \langle 4t - 3, t^2, 5t - 6 \rangle $
for $ t \ge 0 $. Do the particles collide?

Linda Hand
Linda Hand
Numerade Educator
03:52

Problem 50

Two particles travel along the space curves
$ r_1 (t) = \langle t, t^2, t^3 \rangle $ $ r_2 (t) = \langle 1 + 2t, 1 + 6t, 1 + 14t \rangle $
Do the particles collide? Do their paths intersect?

WZ
Wen Zheng
Numerade Educator
33:05

Problem 51

(a) Graph the curve with parametric equations
$$ x = \frac{27}{26} \sin 8t - \frac{8}{39} \sin 18t $$
$$ y = -\frac{27}{26} \cos 8t + \frac{8}{39} \cos 18t $$
$$ z = \frac{144}{65} \sin 5t $$
(b) Show that the curve lies on the hyperboloid of one sheet
$ 144x^2 + 144y^2 - 25z^2 = 100 $.

Oswaldo Jiménez
Oswaldo Jiménez
Numerade Educator
04:17

Problem 52

The view of the trefoil knot shown in Figure 8 is accurate, but it doesn't reveal the whole story. Use the parametric equations
$$ x = (2 + \cos 1.5t) \cos t $$
$$ y = (2 + \cos 1.5t) \sin t $$
$$ z = \sin 1.5t $$
to sketch the curve by hand as viewed from above, with gaps indicating where the curve passes over itself. Start by showing that the projection of the curve onto the $ xy $-plane has polar coordinates $ r = 2 + \cos 1.5t $ and $ \theta = t $, so $ r $ varies between 1 and 3. Then show that $ z $ has maximum and minimum values when the projection is halfway between $ r = 1 $ and $ r = 3 $.
When you have finished your sketch, use a computer to draw the curve with viewpoint directly above and compare with your sketch. Then use the computer to draw the curve from several other viewpoints. You can get a better impression of the curve if you plot a tube with radius 0.2 around the curve. (Use the tubeplot command in Maple or the tubecurve or Tube command in Mathematica.)

WZ
Wen Zheng
Numerade Educator
04:49

Problem 53

Suppose $ u $ and $ v $ are vector functions that possess limits as $ t \to a $ and let $ c $ be a constant. Prove the following properties of limits.
(a) $ \displaystyle \lim_{t \to a} [u(t) + v(t)] = \displaystyle \lim_{t \to a} u(t) + \displaystyle \lim_{t \to a} v(t) $
(b) $ \displaystyle \lim_{t \to a} cu(t) = c \displaystyle \lim_{t \to a} u(t) $
(c) $ \displaystyle \lim_{t \to a} [u(t) \cdot v(t)] = \displaystyle \lim_{t \to a} u(t) \cdot \displaystyle \lim_{t \to a} v(t) $
(d) $ \displaystyle \lim_{t \to a} [u(t) \times v(t)] = \displaystyle \lim_{t \to a} u(t) \times \displaystyle \lim_{t \to a} v(t) $

WZ
Wen Zheng
Numerade Educator
06:47

Problem 54

Show that $ \displaystyle \lim_{t \to a} r(t) = b $ if and only if for every $ \varepsilon > 0 $ there is a number $ \delta > 0 $ such that
if $ 0 < \mid t - a \mid < \delta $ then $ \mid r(t) - b \mid < \varepsilon $

WZ
Wen Zheng
Numerade Educator

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