Download Chapter 12 Vector-Valued Functions

Chapter 12 Vector-Valued Functions
Chapter Summary
Section Topics
12.1
Vector-Valued Functions—Analyze and sketch a space curve given by a vector-valued
function. Extend the concepts of limits and continuity to vector-valued functions.
12.2
Differentiation and Integration of Vector-Valued Functions—Differentiate a
vector-valued function. Integrate a vector-valued function.
12.3
Velocity and Acceleration—Describe the velocity and acceleration associated with
a vector-valued function. Use a vector-valued function to analyze projectile motion.
12.4
Tangent Vectors and Normal Vectors—Find a unit tangent vector at a point on a
space curve. Find the tangential and normal components of acceleration.
12.5
Arc Length and Curvature—Find the arc length of a space curve. Use the arc length
parameter to describe a plane curve or space curve. Find the curvature of a curve at a
point on the curve. Use a vector-valued function to find frictional force.
Chapter Comments
In discussing vector-valued functions with your students, be sure to distinguish them from
real-valued functions. A vector-valued function is a vector, whereas a real-valued function
is a real number. Remind your students that the parameterization of a curve is not unique.
As discussed on page 836 at the end of Example 3, the choice of a parameter determines the
orientation of the curve.
Sections 12.1 and 12.2 of this chapter discuss the domain, limit, continuity, differentiation and
integration of vector-valued functions. Go over all of this carefully because it is the basis for
the applications discussed in Sections 12.3, 12.4 and 12.5. Be sure to point out that when
integrating a vector-valued function, the constant of integration is a vector, not a real number.
Go over all of the ideas presented in Sections 12.3, 12.4 and 12.5. Some, such as arc length,
will be familiar to your students. Choose your assignments carefully in these sections as the
problems are lengthy. Be sure to assign problems with a mix of functions, transcendental as
well as algebraic.
If you are pressed for time, it is not necessary to cover every formula for curvature. However,
do not omit this idea entirely.
Section 12.1 Vector-Valued Functions
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Exercises 1– 8, 27– 42 (1– 8, 23–38 in Calculus 8/e)
Be sure to go over finding the domain of a vector-valued function. Some students find this difficult.
We revised Example 4 to include a discussion on the domain of r (t ).
Exercises 69 –74 (69–74 in Calculus 8/e)
We revised Exercises 69, 70, and 72, and reordered the exercises to improve the grading. We also
rewrote the direction line to find the limit if it exists.
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97
Capstone
Page 841, Exercise 84 You can use this exercise to review the following concepts.
•
Vector-valued functions
•
Analyzing a space curve given by a vector-valued function
Go over the definition of a vector-valued function on page 834. Then write the parametric
equations and the rectangular equations as shown in the solution. Note that the curve represented
by each vector-valued function is an ellipse.
Solution
(a) x = − 3 cos t + 1, y = 5 sin t + 2, z = 4
(x
− 1)
2
+
9
(y
− 2)
2
= 1, z = 4
25
(b) x = 4, y = − 3 cos t + 1, z = 5 sin t + 2
(y
− 1)
2
9
+
(z
− 2)
2
= 1, x = 4
25
(c) x = 3 cos t − 1, y = − 5 sin t − 2, z = 4
(x
+ 1)
2
+
9
(y
+ 2)
2
25
= 1, z = 4
(d) x = − 3 cos 2t + 1, y = 5 sin 2t + 2, z = 4
(x
− 1)
9
2
+
(y
− 2)
25
2
= 1, z = 4
(a) and (d) represent the same graph.
Section 12.2 Differentiation and Integration of
Vector-Valued Functions
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Exercises 1– 8 (1–6 in Calculus 8/e)
Because many students had difficulty with these exercises, we revised Example 1 to demonstrate the
skills necessary to solve these exercises. To give students more practice, we added Exercises 6 and 8.
Also, we rewrote Exercise 7 (old Exercise 6) in component form to pair it with new Exercise 8.
Exercises 11–22 (11– 18 in Calculus 8/e)
To give students more practice, we added Exercises 11–14.
Exercise 32 (28 in Calculus 8/e)
We revised this exercise so that the vectors in the figure could be more easily discerned.
Exercises 45 and 46 (41 and 42 in Calculus 8/e)
We rewrote the direction line to clearly state that students should find the indicated derivatives in
two different ways.
Exercises 77–84 (73–80 in Calculus 8/e)
We revised Theorem 12.2 and these exercises by replacing f with w. Previously, some students
confused the f used in the properties of the derivative and these exercises with the f used in the
i-component of r (t ) = f (t ) i + g (t ) j + h (t )k.
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Capstone
Page 849, Exercise 88 You can use this exercise to review the following concepts.
•
Sketching a curve represented by a vector-valued function
•
Differentiating a vector-valued function
•
Theorem 12.1
Review the definition of the derivative of a vector-valued function (see page 842) and Theorem
12.1 (see page 843). After going over the solution, you can extend this exercise by asking students
to sketch the vector r ′(1). Students had trouble making similar sketches in Exercises 1–8.
Solution
(a) r (t ) = t i + ( 4 − t 2 ) j
y
5
r(1)
3
2
r(1.25)
1
r (1.25) − r (1)
x
−3
(b)
−1
−1
3
r (1) = i + 3j
r (1.25) = 1.25i + 2.4375 j
r (1.25) − r (1) = 0.25i − 0.5625 j
r ′(t ) = i − 2t j
r ′(1) = i − 2 j
r (1.25) − r (1)
1.25 − 1
=
0.25i − 0.5625 j
= i − 2.25 j
0.25
This vector approximates r ′(1).
Section 12.3 Velocity and Acceleration
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Exercises 1– 10 (1–8 in Calculus 8/e)
To give students more practice, we added Exercises 4 and 6.
Exercises 11–20 (9–16 in Calculus 8/e)
To give students more practice, we added Exercises 18 and 20.
Exercises 23–28 (19–22 in Calculus 8/e)
To give students more practice, we added Exercises 26 and 28.
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99
Capstone
Page 858, Exercise 60 You can use this exercise to review the following concepts.
•
Describing the velocity associated with a vector-valued function
•
Finding the speed of a particle moving on an elliptical path
•
Describing the acceleration associated with a vector-valued function
Review the definitions of velocity and acceleration on page 851. Then go over the solution below.
Solution
r (t ) = a cos ω t i + b sin ω t j
(a) r ′(t ) = v (t ) = − aω sin ω t i + bω cos ω t j
Speed = v (t ) =
a 2ω 2 sin 2 ω t + b 2ω 2 cos 2 ω t
(b) a (t ) + v ′(t ) = − aω 2 cos ω t i + bω 2 sin ω t j
= ω 2 ( − a cos ω t i − b sin ω t j)
= − ω 2 r (t )
Section 12.4 Tangent Vectors and Normal Vectors
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Exercises 49–54 (49–52 in Calculus 8/e)
To give students more practice, we added Exercises 51 and 52.
Exercises 55–62 (53–56 in Calculus 8/e)
To give students more practice, we added Exercises 57, 58, 60, and 62.
Exercises 63–66 (57 and 58 in Calculus 8/e)
To give students more practice, we added Exercises 64 and 66.
Capstone
Page 867, Exercise 70 You can use this exercise to review the following concepts.
•
Describing the velocity and acceleration associated with a vector-valued function
•
Finding a unit tangent vector
•
Finding a principal unit normal vector (if it exists)
If necessary, review velocity and acceleration in Section 12.3. Go over the definitions of the unit
tangent vector (see page 859) and the principal unit normal vector (see page 860). Then go over
the solution on the next page.
100
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Solution
r (t ) = 3t i + 4t j
v (t ) = r ′(t ) = 3i + 4 j, v (t ) =
9 + 16 = 5
a (t ) = v ′ (t ) = 0
T (t ) =
v (t )
v (t )
=
3
4
i + j
5
5
T′(t ) = 0 ⇒ N (t ) does not exist.
The path is a line. The speed is constant (5).
Section 12.5 Arc Length and Curvature
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Exercises 41–44 (New)
To give students practice finding the curvature of a curve at a point, we added Exercises 41–44.
Exercises 45–54 (41– 46 in Calculus 8/e)
To give students more practice, we added Exercises 49, 50, 53, and 54.
Exercises 63 –70 (55–60 in Calculus 8/e)
To give students more practice, we added Exercises 69 and 70.
Capstone
Page 879, Exercise 78 You can use this exercise to review the following concepts.
•
Finding the arc length of a space curve
•
Theorem 12.6
•
Finding the curvature of a plane curve at a point on the curve
•
Theorem 12.9
•
Analyzing the curvature of a curve
Go over Theorem 12.6 and part (a). If you wish to skip the integration steps, show your students
how to set up the integral and then use a graphing utility or CAS to find the result. Then go over
the definition of curvature (see page 872) and Theorem 12.9 (see page 874). Note in part (b) that
we use the rectangular equations that correspond to the given vector-valued function.
Solution
r (t ) = t i + t 2 j
(a)
dx
dy
= 1,
= 2t
dt
dt
s =
2
∫0
1 + 4t 2 dt =
1 2
2 ∫0
1 + 4t 2 ( 2) dt (u = 2t )
2
1 1⎡
⋅ 2t 1 + 4t 2 + ln 2t + 1 + 4t 2 ⎤ (Theorem 8.2)
⎦⎥ 0
2 2 ⎣⎢
1
= ⎡4 17 + ln 4 + 17 ⎤ ≈ 4.647
⎦
4⎣
=
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101
(b) Let y = x 2 , y ′ = 2 x, y ′′ = 2
At t = 0, x = 0, y = 0, y ′ = 0, y ′′ = 2
K =
2
[1 + 0]3 2
= 2
At t = 1, x = 1, y = 1, y ′ = 2, y ′′ = 2
K =
2
2 32
⎡1 + ( 2) ⎤
⎣
⎦
=
2
32
5
≈ 0.179
At t = 2, x = 2, y = 4, y ′ = 4, y ′′ = 2
K =
2
[1 + 16]
32
=
2
≈ 0.0285
173 2
(c) As t changes from 0 to 2, the curvature decreases.
102
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Chapter 12 Project
Throwing a Shot-Put
The path of a shot-put thrown at an angle θ is r (t ) = (v0 cos θ )t i + ⎡⎣h + (v0 sin θ )t −
1 gt 2 ⎤ j
2
⎦
where v0 is the initial speed, h is the initial height, t is the time in seconds and g ≈ 32 feet per
second per second is the acceleration due to gravity. (This formula neglects air resistance.)
Exercises
In Exercises 1–5, a shot-put is thrown from an initial height of 6 feet with an initial velocity
of 40 feet per second and an initial angle of 35°.
1. Find the vector-valued function r (t ) that gives the position of the shot-put.
2. At what time is the shot-put at its maximum height?
3. What is the maximum height of the shot-put?
4. How long is the shot-put in the air?
5. What is the horizontal distance traveled by the shot-put?
In Exercises 6–10, a shot-put is thrown from an initial height of 6 feet with an initial
velocity of 40 feet per second and an initial angle of 45°.
6. Find the vector-valued function r (t ) that gives the position of the shot-put.
7. At what time is the shot-put at its maximum height?
8. What is the maximum height of the shot-put?
9. How long is the shot-put in the air?
10. What is the horizontal distance traveled by the shot-put?
11. Write an equation for t, the time when a shot-put thrown from an initial height h with an
initial velocity of v0 and an initial angle of θ hits the ground.
12. Write an equation for the horizontal distance traveled by a shot-put thrown from an initial
height h with an initial velocity of v0 and an initial angle of θ .
In Exercises 13 and 14, a shot-put is thrown from an initial height of 6 feet with an initial
velocity of 40 feet per second.
13. Write an equation for the time t when the shot-put hits the ground in terms of θ , the angle
at which it is thrown.
14. Write an equation for the horizontal distance traveled by the shot-put in terms of θ .
15. Use a graphing utility to graph the equation you wrote in Exercise 14. Where does the
maximum occur? What does this mean in the context of the problem?
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103