Chapter 13 – Vector Functions 13.1 Vector Functions and Space Curves 1 Objectives: Use vector...
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Transcript of Chapter 13 – Vector Functions 13.1 Vector Functions and Space Curves 1 Objectives: Use vector...
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Chapter 13 – Vector Functions13.1 Vector Functions and Space Curves
13.1 Vector Functions and Space Curves
Objectives: Use vector -valued
functions to describe the motion of objects through space
Draw vector functions and their corresponding space curves
© jdannels - http://josephdannels.com/
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13.1 Vector Functions and Space Curves
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Vector FunctionsThe functions that we have been
using so far have been real-valued functions.We study functions whose values are vectors
because such functions are needed to describe curves and surfaces in space.
We will also use vector-valued functions to describe the motion of objects through space. ◦ In particular, we will use them to derive
Kepler’s laws of planetary motion.
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13.1 Vector Functions and Space Curves
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Definition – Vector functionA vector-valued function, or vector
function, is simply a function whose:
◦ Domain is a set of real numbers.
◦ Range is a set of vectors.
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13.1 Vector Functions and Space Curves
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Example 1 – pg. 845 #2Find the domain of the vector
function.
22( ) sin ln(9 )
2
tt t t
t
r i j k
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13.1 Vector Functions and Space Curves
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Definition – Component Functions If f(t), g(t), and h(t) are the components of the
vector r(t), then f, g, and h are real-valued functions called the component functions of r.
We can write: r(t) = ‹f(t), g(t), h(t)› = f(t) i + g(t) j + h(t) k
Note: We usually use the letter t to denote the independent variable because it represents time in most applications of vector functions.
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13.1 Vector Functions and Space Curves
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Definition – Limit of a VectorThe limit of a vector function r is
defined by taking the limits of its component functions as follows.
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13.1 Vector Functions and Space Curves
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Definition - ContinuousA vector function r is continuous at a if:
◦ In view of Definition 1, we see that r is continuous at a if and only if its component functions f, g, and h are continuous at a.
lim ( ) ( )t a
t a
r r
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13.1 Vector Functions and Space Curves
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Example 2Find the limit
0
1 1 1 3lim , ,
1
t
t
e t
t t t
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13.1 Vector Functions and Space Curves
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Space Curve Then, the set C of all points (x, y ,z) in space, where
x = f(t) y = g(t) z = h(t)
and t varies throughout the interval I is called a space curve.
The equations are called parametric equations of C.
t is called a parameter.
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VisualizationVector Functions and Space
CurvesThe Twisted Cubic CurveVisualizing Space Curves
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13.1 Vector Functions and Space Curves
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Example 3Find a vector equation and
parametric equations for the line segment that joins P to Q.
P (-2, 4, 0)
Q (6, -1, 2)
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Example 4 – pg. 846 # 29At what points does the curve
intersect the paraboloid ?
2( ) (2 )t t t t r i k2 2z x y
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Example 5 – pg 847 # 47If 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 the two particles are given by the vector functions
for t0. Do the particles collide?
2 2 21 2( ) ,7 12, ( ) 4 3, ,5 6t t t t t t t t r r
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More Examples
The video examples below are from section 13.1 in your textbook. Please watch them on your own time for extra instruction. Each video is about 2 minutes in length. ◦Example 3◦Example 6
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Demonstrations
Feel free to explore these demonstrations below.
Four Space CurvesEquation of a Line in Vector Form
2D