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Transcript of CHAPTER 11 Vector-Valued Functions Slide 2 © The McGraw-Hill Companies, Inc. Permission required...
CHAPTER
11Vector-Valued Functions
11
Slide 2© The McGraw-Hill Companies, Inc. Permission required for reproduction or display.
11.1 VECTOR-VALUED FUNCTIONS11.2 THE CALCULUS OF VECTOR-VALUED FUNCTIONS11.3 MOTION IN SPACE11.4 CURVATURE11.5 TANGENT AND NORMAL VECTORS11.6 PARAMETRIC SURFACES
11.1 VECTOR-VALUED FUNCTIONS
Preliminaries
Slide 3© The McGraw-Hill Companies, Inc. Permission required for reproduction or display.
For the circuitous path ofthe airplane, it is convenient to describe the airplane’s location at any given time by the endpoint 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.
DEFINITION
11.1 VECTOR-VALUED FUNCTIONS
1.1
Slide 4© The McGraw-Hill Companies, Inc. Permission required for reproduction or display.
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 asr(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.
11.1 VECTOR-VALUED FUNCTIONS
Preliminaries
Slide 5© The McGraw-Hill Companies, Inc. Permission required for reproduction or display.
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.
EXAMPLE
11.1 VECTOR-VALUED FUNCTIONS
1.1 Sketching the Curve Defined by a Vector-Valued Function
Slide 6© The McGraw-Hill Companies, Inc. Permission required for reproduction or display.
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.
EXAMPLE
Solution
11.1 VECTOR-VALUED FUNCTIONS
1.1 Sketching the Curve Defined by a Vector-Valued Function
Slide 7© The McGraw-Hill Companies, Inc. Permission required for reproduction or display.
Substituting some values for t, we have r(0) = i − 2j = 1,−2,r(2) = 3i + 2j = 3, 2 and r(−2) = −1, 2.
EXAMPLE
Solution
11.1 VECTOR-VALUED FUNCTIONS
1.1 Sketching the Curve Defined by a Vector-Valued Function
Slide 8© The McGraw-Hill Companies, Inc. Permission required for reproduction or display.
The endpoints of all position vectors r(t) lie on the curve C, described parametrically by
Eliminate the parameter:
EXAMPLE
Solution
11.1 VECTOR-VALUED FUNCTIONS
1.1 Sketching the Curve Defined by a Vector-Valued Function
Slide 9© The McGraw-Hill Companies, Inc. Permission required for reproduction or display.
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.
EXAMPLE
11.1 VECTOR-VALUED FUNCTIONS
1.2 A Vector-Valued Function Defining an Ellipse
Slide 10© The McGraw-Hill Companies, Inc. Permission required for reproduction or display.
Sketch a graph of the curve traced out by the endpoint of the vector-valued function
EXAMPLE
Solution
11.1 VECTOR-VALUED FUNCTIONS
1.2 A Vector-Valued Function Defining an Ellipse
Slide 11© The McGraw-Hill Companies, Inc. Permission required for reproduction or display.
EXAMPLE
11.1 VECTOR-VALUED FUNCTIONS
1.3 A Vector-Valued Function Defining an Elliptical Helix
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Plot the curve traced out by the vector-valued function r(t) = sin ti − 3 cos tj + 2tk, t ≥ 0.
EXAMPLE
Solution
11.1 VECTOR-VALUED FUNCTIONS
1.3 A Vector-Valued Function Defining an Elliptical Helix
Slide 13© The McGraw-Hill Companies, Inc. Permission required for reproduction or display.
EXAMPLE
11.1 VECTOR-VALUED FUNCTIONS
1.4 A Vector-Valued Function Defining a Line
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Plot the curve traced out by the vector-valued function
EXAMPLE
Solution
11.1 VECTOR-VALUED FUNCTIONS
1.4 A Vector-Valued Function Defining a Line
Slide 15© The McGraw-Hill Companies, Inc. Permission required for reproduction or display.
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).
EXAMPLE
11.1 VECTOR-VALUED FUNCTIONS
1.5 Matching a Vector-Valued Function to Its Graph
Slide 16© The McGraw-Hill Companies, Inc. Permission required for reproduction or display.
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).
EXAMPLE
11.1 VECTOR-VALUED FUNCTIONS
1.5 Matching a Vector-Valued Function to Its Graph
Slide 17© The McGraw-Hill Companies, Inc. Permission required for reproduction or display.
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
EXAMPLE
Solution
11.1 VECTOR-VALUED FUNCTIONS
1.5 Matching a Vector-Valued Function to Its Graph
Slide 18© The McGraw-Hill Companies, Inc. Permission required for reproduction or display.
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
f1
f3
f4
11.1 VECTOR-VALUED FUNCTIONS
Arc Length in
Slide 19© The McGraw-Hill Companies, Inc. Permission required for reproduction or display.
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
11.1 VECTOR-VALUED FUNCTIONS
Arc Length in
Slide 20© The McGraw-Hill Companies, Inc. Permission required for reproduction or display.
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.)
EXAMPLE
11.1 VECTOR-VALUED FUNCTIONS
1.6 Computing Arc Length in
Slide 21© The McGraw-Hill Companies, Inc. Permission required for reproduction or display.
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.
EXAMPLE
Solution
11.1 VECTOR-VALUED FUNCTIONS
1.6 Computing Arc Length in
Slide 22© The McGraw-Hill Companies, Inc. Permission required for reproduction or display.
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.
EXAMPLE
Solution
11.1 VECTOR-VALUED FUNCTIONS
1.6 Computing Arc Length in
Slide 23© The McGraw-Hill Companies, Inc. Permission required for reproduction or display.
EXAMPLE
11.1 VECTOR-VALUED FUNCTIONS
1.7 Approximating Arc Length in
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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.
EXAMPLE
Solution
11.1 VECTOR-VALUED FUNCTIONS
1.7 Approximating Arc Length in
Slide 25© The McGraw-Hill Companies, Inc. Permission required for reproduction or display.
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.
EXAMPLE
11.1 VECTOR-VALUED FUNCTIONS
1.8 Finding Parametric Equations for an Intersection of Surfaces
Slide 26© The McGraw-Hill Companies, Inc. Permission required for reproduction or display.
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.
EXAMPLE
Solution
11.1 VECTOR-VALUED FUNCTIONS
1.8 Finding Parametric Equations for an Intersection of Surfaces
Slide 27© The McGraw-Hill Companies, Inc. Permission required for reproduction or display.
EXAMPLE
Solution
11.1 VECTOR-VALUED FUNCTIONS
1.8 Finding Parametric Equations for an Intersection of Surfaces
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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.
EXAMPLE
Solution
11.1 VECTOR-VALUED FUNCTIONS
1.8 Finding Parametric Equations for an Intersection of Surfaces
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