Curves given Parametrically
Assume a pair of functions x = x(t), y = y(t) is differentiable on the interior of an
interval I.
When t ranges over I, the point ( x(t), y(t) ) traces out a path in the xy-plane.
We call such a path a parametrized curve and refer to t as the parameter.
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Salas, Hille, Etgen Calculus: One and Several Variables
Copyright 2007 © John Wiley & Sons, Inc. All rights reserved.
Curves given Parametrically
Example 1 Identify the curve parametrized by the functions
x (t) = t +1
t  - ,  
y (t) = 2t -5
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Salas, Hille, Etgen Calculus: One and Several Variables
Copyright 2007 © John Wiley & Sons, Inc. All rights reserved.
Curves given Parametrically
Example 2 Identify the curve parametrized by the functions
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Salas, Hille, Etgen Calculus: One and Several Variables
Copyright 2007 © John Wiley & Sons, Inc. All rights reserved.
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Salas, Hille, Etgen Calculus: One and Several Variables
Copyright 2007 © John Wiley & Sons, Inc. All rights reserved.
Curves given Parametrically
Example 3 Identify the curve parametrized by the functions
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Salas, Hille, Etgen Calculus: One and Several Variables
Copyright 2007 © John Wiley & Sons, Inc. All rights reserved.
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Salas, Hille, Etgen Calculus: One and Several Variables
Copyright 2007 © John Wiley & Sons, Inc. All rights reserved.
Curves given Parametrically
Straight Lines
Given that (x0, y0) = (x1, y1), the functions
parametrize the line that passes through the points (x0, y0) and (x1, y1).
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Salas, Hille, Etgen Calculus: One and Several Variables
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Curves given Parametrically
Ellipses and Circles
Usually we let t range from 0 to 2π and parametrize the ellipse by setting
If b = a, we have a circle. We can parametrize the circle
x2 + y2 = a2
by setting
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Salas, Hille, Etgen Calculus: One and Several Variables
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Curves given Parametrically
Hyperbolas
Take a, b > 0. The functions x(t) = a cosh t, y(t) = b sinh t satisfy the identity
 x  t 
 y  t 

1
a2
b2
2
2
Since x(t) = a cosh t > 0 for all t, as t ranges over the set of real numbers, the point
(x(t), y(t)) traces out the right branch of the hyperbola
x2 y2
 2 1
2
a
b
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Salas, Hille, Etgen Calculus: One and Several Variables
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Curves given Parametrically
Example 4 The line segment that joins the points (1, 2) and (3, 6) is the graph of
the function
y = 2x,
1 ≤ x ≤ 3.
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Salas, Hille, Etgen Calculus: One and Several Variables
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Curves given Parametrically
Represent a traversal of that same line segment but in the opposite direction.
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Salas, Hille, Etgen Calculus: One and Several Variables
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Curves given Parametrically
Example 5 We return to the ellipse
parameter t to measure time measured in seconds.
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and again use the
Salas, Hille, Etgen Calculus: One and Several Variables
Copyright 2007 © John Wiley & Sons, Inc. All rights reserved.
Curves given Parametrically
Example 6 Two particles start at the same instant, the first along the linear path
And the second along the elliptical path
(a) At what points, if any, do the paths intersect?
(b) At what points, if any, do the particles collide?
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Salas, Hille, Etgen Calculus: One and Several Variables
Copyright 2007 © John Wiley & Sons, Inc. All rights reserved.
Curves given Parametrically
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Salas, Hille, Etgen Calculus: One and Several Variables
Copyright 2007 © John Wiley & Sons, Inc. All rights reserved.