2. 11 Vector-Valued Functions 11 11.1 VECTOR-VALUED FUNCTIONS 11.2
THE CALCULUS OF VECTOR-VALUED FUNCTIONS
11.3
MOTION IN SPACE
11.4
CURVATURE
11.5
TANGENT AND NORMAL VECTORS
11.6
PARAMETRIC SURFACES 11
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3. VECTOR-VALUED FUNCTIONS
11.1
VECTOR-VALUED FUNCTIONS
Preliminaries
For the circuitous path of
the airplane, it is
convenient to describe
the airplane’s location at
any given time by the end
point of a vector whose
initial point is located at
the origin (a position vector).
Notice that a function that gives us a vector in V3 for each time t would do the job nicely. This is the concept of a vector-valued function.
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4. r(t) = f (t)i + g(t)j + h(t)k, (1.1)
11.1
VECTOR-VALUED FUNCTIONS
1.1
A vector-valued function r(t) is a mapping from its domain D ⊂ to its range R ⊂ V3, so that for each t in D, r(t) = v for exactly one vector v ∈ R.
We can always write a vector-valued function as
r(t) = f (t)i + g(t)j + h(t)k, (1.1)
for some scalar functions f, g and h (called the component functions of r).
We can likewise define a vector-valued function r(t) in V2 by r(t) = f (t)i + g(t)j, for some scalar functions f and g.
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5. VECTOR-VALUED FUNCTIONS
11.1
VECTOR-VALUED FUNCTIONS
Preliminaries
For each t, we regard r(t) as a position vector.
The endpoint of r(t) then can be viewed as tracing out a curve.
Observe that for r(t) as defined in (1.1), this curve is the same as that described by the parametric equations
x = f (t), y = g(t) and z = h(t).
In three dimensions, such a curve is referred to as a space curve.
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Slide 5

6. VECTOR-VALUED FUNCTIONS
11.1
VECTOR-VALUED FUNCTIONS
1.1
Sketching the Curve Defined by a Vector-Valued Function
Sketch a graph of the curve traced out by the endpoint of the two-dimensional vector-valued function
r(t) = (t + 1)i + (t2 − 2)j.
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7. VECTOR-VALUED FUNCTIONS
11.1
VECTOR-VALUED FUNCTIONS
1.1
Sketching the Curve Defined by a Vector-Valued Function
Substituting some values for t, we have r(0) = i − 2j = 1,−2,
r(2) = 3i + 2j = 3, 2 and
r(−2) = −1, 2.
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8. VECTOR-VALUED FUNCTIONS
11.1
VECTOR-VALUED FUNCTIONS
1.1
Sketching the Curve Defined by a Vector-Valued Function
The endpoints of all position vectors r(t) lie on the curve C, described parametrically by
Eliminate the parameter:
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9. VECTOR-VALUED FUNCTIONS
11.1
VECTOR-VALUED FUNCTIONS
1.1
Sketching the Curve Defined by a Vector-Valued Function
The small arrows marked on the graph indicate the orientation, that is, the direction of increasing values
of t. If the curve describes the path of an object, then the orientation indicates the direction in which the object traverses the path.
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10. VECTOR-VALUED FUNCTIONS
11.1
VECTOR-VALUED FUNCTIONS
1.2
A Vector-Valued Function Defining an Ellipse
Sketch a graph of the curve traced out by the endpoint of the vector-valued function
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Slide 10

11. VECTOR-VALUED FUNCTIONS
11.1
VECTOR-VALUED FUNCTIONS
1.2
A Vector-Valued Function Defining an Ellipse
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Slide 11

12. VECTOR-VALUED FUNCTIONS
11.1
VECTOR-VALUED FUNCTIONS
1.3
A Vector-Valued Function Defining an Elliptical Helix
Plot the curve traced out by the vector-valued function
r(t) = sin ti − 3 cos tj + 2tk, t ≥ 0.
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Slide 12

13. VECTOR-VALUED FUNCTIONS
11.1
VECTOR-VALUED FUNCTIONS
1.3
A Vector-Valued Function Defining an Elliptical Helix
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14. VECTOR-VALUED FUNCTIONS
11.1
VECTOR-VALUED FUNCTIONS
1.4
A Vector-Valued Function Defining a Line
Plot the curve traced out by the vector-valued function
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15. VECTOR-VALUED FUNCTIONS
11.1
VECTOR-VALUED FUNCTIONS
1.4
A Vector-Valued Function Defining a Line
Recognize these equations as parametric equations for the straight line parallel to the vector 2,−3,−4 and passing through the point (3, 5, 2).
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16. VECTOR-VALUED FUNCTIONS
11.1
VECTOR-VALUED FUNCTIONS
1.5
Matching a Vector-Valued Function to Its Graph
Match each of the vector-valued functions
f1(t) = cos t, ln t, sin t,
f2(t) = t cos t, t sin t, t,
f3(t) = 3 sin 2t, t, t and
f4(t) = 5 sin3 t, 5 cos3 t, t
with the corresponding computer-generated graph (on the following slide).
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17. VECTOR-VALUED FUNCTIONS
11.1
VECTOR-VALUED FUNCTIONS
1.5
Matching a Vector-Valued Function to Its Graph
f1(t) = cos t, ln t, sin t,
f2(t) = t cos t, t sin t, t,
f3(t) = 3 sin 2t, t, t and
f4(t) = 5 sin3 t, 5 cos3 t, t
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18. VECTOR-VALUED FUNCTIONS
11.1
VECTOR-VALUED FUNCTIONS
1.5
Matching a Vector-Valued Function to Its Graph f1 f4
f1(t) = cos t, ln t, sin t,
f2(t) = t cos t, t sin t, t,
f3(t) = 3 sin 2t, t, t and
f4(t) = 5 sin3 t, 5 cos3 t, t f2 f3
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19. VECTOR-VALUED FUNCTIONS
11.1
VECTOR-VALUED FUNCTIONS
Arc Length in
A natural question to ask about a curve is, “How long is it?”
Note that the plane curve traced out exactly once by the endpoint of the vector-valued function r(t) = f (t), g(t), for t ∈ [a, b] is the same as the curve defined parametrically by x = f (t), y = g(t).
Recall from section 10.3 that if f, f' , g and g' are all continuous for t ∈ [a, b], the arc length is given by
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20. VECTOR-VALUED FUNCTIONS
11.1
VECTOR-VALUED FUNCTIONS
Arc Length in
Consider a space curve traced out by the endpoint of the vector-valued function r(t) = f (t), g(t), h(t), where f, f' , g, g', h and h' are all continuous for t ∈ [a, b] and where the curve is traversed exactly once as t increases from a to b.
The arc length of the space curve is given by:
(See the text for the derivation of this result.)
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21. VECTOR-VALUED FUNCTIONS
11.1
VECTOR-VALUED FUNCTIONS
1.6
Computing Arc Length in
Find the arc length of the curve traced out by the endpoint of the vector-valued function r(t) = 2t, ln t, t2, for
1 ≤ t ≤ e.
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22. VECTOR-VALUED FUNCTIONS
11.1
VECTOR-VALUED FUNCTIONS
1.6
Computing Arc Length in
First, notice that for x(t) = 2t, y(t) = ln t and z(t) = t2, we have x'(t) = 2, y'(t) = 1/t and z'(t) = 2t, and the curve is traversed exactly once for 1 ≤ t ≤ e.
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Slide 22

23. VECTOR-VALUED FUNCTIONS
11.1
VECTOR-VALUED FUNCTIONS
1.6
Computing Arc Length in
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24. VECTOR-VALUED FUNCTIONS
11.1
VECTOR-VALUED FUNCTIONS
1.7
Approximating Arc Length in
Find the arc length of the curve traced out by the endpoint of the vector-valued function r(t) = e2t , sin t, t,
for 0 ≤ t ≤ 2.
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25. VECTOR-VALUED FUNCTIONS
11.1
VECTOR-VALUED FUNCTIONS
1.7
Approximating Arc Length in
First, note that for x(t) = e2t , y(t) = sin t and z(t) = t, we have x'(t) = 2e2t , y'(t) = cos t and z'(t) = 1, and that the curve is traversed exactly once for 0 ≤ t ≤ 2
Approximate the integral using Simpson’s Rule or the numerical integration routine built into your calculator:
s ≈ 53.8.
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26. VECTOR-VALUED FUNCTIONS
11.1
VECTOR-VALUED FUNCTIONS
1.8
Finding Parametric Equations for an Intersection of Surfaces
Find the arc length of the portion of the curve determined by the intersection of the cone
and the plane y + z = 2 in the first octant.
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Slide 26

27. VECTOR-VALUED FUNCTIONS
11.1
VECTOR-VALUED FUNCTIONS
1.8
Finding Parametric Equations for an Intersection of Surfaces
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Slide 27

28. VECTOR-VALUED FUNCTIONS
11.1
VECTOR-VALUED FUNCTIONS
1.8
Finding Parametric Equations for an Intersection of Surfaces
The portion of the parabola in the first octant must have x ≥ 0 (so
t ≥ 0), y ≥ 0 (so t2 ≤ 4) and z ≥ 0 (always true).
This occurs if 0 ≤ t ≤ 2.
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Slide 28

29. VECTOR-VALUED FUNCTIONS
11.1
VECTOR-VALUED FUNCTIONS
1.8
Finding Parametric Equations for an Intersection of Surfaces
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Slide 29