Math 251
Analytic Geometry and Calculus II
Section 12.1: Parametric Equations
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Topics
Sketch the graph of a curve given in parametric form.
Convert between rectangular and parametric form of equations.
Find the slope of a tangent line to a curve given in parametric form.
Determine arc length for curves given in parametric form.
Parametric Equations
Idea
Imagine that a particle moves along the curve C shown in the figure below. It is impossible
to describe C by an equation of the form y = f (x) because C fails the Vertical Line Test.
But the x− and y−coordinates of the particle are functions of time t and so we can write
x = f (t) and y = g(t). Such a pair of equations is often a convenient way of describing a
curve.
Example 1
x = t2 − 2t
Sketch and identify the curve defined by the parametric equations
y =t+1
Math 251
Analytic Geometry and Calculus II
Example 2
x = sin t
Sketch the curve represented by
. Write this equation in rectangular (Cartesian) form.
y = cos t
What type of curve is it? Give an interval for t over which the curve is traversed once.
Example 3

θ


x
=
2
cos


4
Sketch the curve represented by
. Write this equation in rectangular (Cartesian)

θ


y = 5 sin
4
form. What type of curve is it? Give an interval for t over which the curve is traversed once.
Example 4
x = 1 + 5 cos(2θ)
. Write this equation in rectangular (Cartey = 3 − 4 sin(2θ)
sian) form. What type of curve is it? Give an interval for t over which the curve is traversed
once.
Sketch the curve represented by
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Math 251
Analytic Geometry and Calculus II
Example 5
π
π
x = tan2 θ
with − < θ < . Write this equation in rectangular
y = sec θ
2
2
(Cartesian) form and give the domain and range.
Sketch the curve represented by
2
Converting Between Rectangular and Parametric For
Idea
Given a rectangular (Cartesian) equation, we may with to create a corresponding set of
parametric equations. There are an infinite number of possibilities.
Example 6
Write a set of parametric equations for the rectangular equation y =
2
.
x−1
Example 7
Write a set of parametric equations for the vertical line x = 4.
Example 8
Write a set of parametric equation for the line that passes through (1, 5) and (3, 2). (You will have
to do this sort of thing in Calculus III).
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Math 251
Analytic Geometry and Calculus II
General Form of a Line Trough Two Points (x1 , y1 ) and (x2 , y2 )
x = (x2 − x1 )t + x1
y = (y2 − y1 )t + y1
Example 9
Write a set of parametric equation for the line that passes through (−8, 1) and (10, −5).
Example 10
Find a set of parametric equations that satisfies the given condition: y = x2 + 3, and t = 2 at the
point (1, 4).
Example 11
Each of the following sets of parametric equations gives the position of a moving particle at time
t.
x = t3
y=t
x = −t3
y = −t
x=√
t3/2
y= t
x = e−3t
y = e−t
Each set gives the same function when eliminating the parameter. Describe the differences.
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Math 251
3
Analytic Geometry and Calculus II
Calculus and Parametric Equations
Definition
Suppose f and g are differentiable functions and we want to find the tangent line at a point
on the parametric curve defined by the equations x = f (t) and y = g(t). The derivative is
given by
dy
dy
= dt
dx
dx
dt
if
dx
̸= 0
dt
• The first derivative is a function of t.
dy
• The curve has horizontal tangent when
= 0.
dt
dx
• The curve has vertical tangent when
= 0.
dt
The second derivative is given by,
d dy
d dy
d2 y
dt dx
=
=
2
dx
dx
dx dx
dt
if
dx
̸= 0
dt
Example 12
x = t2
Find the slope of the curve at
Given the curve represented by equations
y = 3t − 5
a) time t = 2.
b) the point (9, 4)
c) Find
d2 y
.
dx2
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Math 251
Analytic Geometry and Calculus II
Example 13
dy
d2 y
Find
and 2 for the curve given by
dx
dx
x = sin2 t
y = cos t
Example 14
Find Find the slope, the equation of the tangent line, and describe the concavity, for the curve
x = t2 − t
given by
y = t3
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Math 251
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Analytic Geometry and Calculus II
Arc Length - Parametric
Definition
If a curve C is described by the parametric equations x = f (t) and y = g(t), α ≤ t ≤ β ,
where f ′ and g ′ are continuous on [α, β] and C is traversed exactly once as t increases from
α to β, then the length of C is
s
Z b 2 2
dx
dy
L=
+
dt
dt
dt
a
Example 15
x = t2 + 1
Find the length of the curve given by
for −1 ≤ t ≤ 0.
y = 4t3 + 3
Example 15
Derive the formula
for the circumference of a circle with radius a by finding the arc length for the
x = a cos θ
curve given by
with 0 ≤ θ ≤ 2π.
y = a sin θ
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