MAC 2311
SECTION 10.1 NOTES
The functions x=f(t) and y=g(t) together as a system of equations can describe the motion along a curve as
a function of time t. The system of equations is known as a set of parametric equations, and the variable
t, the independent variable of both functions f and g, is called the parameter. The curve traced out by the
path of motion is called the trajectory or graph of the equations.
y
x
The above figure is the graph of a set of parametric equations. The curve could describe the path of motion
of an object beginning at time t=0.
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Eliminating the parameter
Graphs of parametic equations x=f(t) and y=g(t) can be obtained by the process of eliminating the
parameter, which then results in an equation in x and y alone.
EX 1) Sketch the graph of the parametric equations x=3t+2 and y=-t+4 on the interval 2 ≤ t ≤ 10 by
eliminating the parameter. Indicate the direction of increasing t.
Solution: Write t in terms of x and then substitute the expression into the right side of y.
x = 3t + 2 ⇒
⎡1
1
x − 2 = t y = −t + 4 ⇒ y = − ⎢ x − 2
3
⎣3
(
)
(
)⎤⎥ + 4 ⇒ y = − 31 x + 143
⎦
1
14
The graph of the equations is the line y = − x +
on the t interval [2,10]. When t=2, x=8 and y=2. When
3
3
t=10, x=32 and y=6.
y
(32,6)
(8,2)
x
EX 2) Sketch the graph of the parametric equations x= 3 cos(t) and y = 4 sin(t) on the interval 0 ≤ t ≤ 2π by
eliminating the parameter. Indicate the direction of increasing t.
SOLUTION: Unlike example one, the functions x and y are trigonometric. In such cases, trigonometric
identities are useful in eliminating the parameter. We will expression both dependent variables in terms of
cos(t) and sin(t) and then use cos2 t + sin2 t = 1 .
x = 3cos t ⇒ cos t =
x
3
y = 4sint ⇒ sint =
y
4
cos2 t + sin2 t = 1⇒
x 2 y2
+
=1
9 16
x 2 y2
+
= 1 on 0 ≤ t ≤ 2π is a full closed ellipse with x-intercepts ±3 and y9 16
intercepts ±4 . The point (3,0) is where t = 0 and 2π, the point (3,0) is where t=π, and the points (0, ±4 )
indicate t=π/2 and 3π/2.
The graph of the equation
4
y
3
2
1
x
–6
–4
–2
2
4
6
–1
–2
–3
–4
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3) Derivatives of parametric equations
The derivative of the parametric equations x=f(t) and y=g(t) is defined as
dy dy / dt
=
dx dx / dt
EX 3) For the set of parametric equations x = t + cos t and y = 1+ sint , find dy/dx at the point t = π / 6
without eliminating the parameter.
SOLUTION:
π
3
cos
dy d
dx d
dy
6 = 2 = 3
= ⎡1+ sint ⎤⎦ = cos t and
= ⎡ t + cos t ⎤⎦ = 1− sint
=
π
1
dt dt ⎣
dt dt ⎣
dx t=π/6
1− sin
1−
6
2
The positive value of dy/dx at the point indicates that the curve is increasing at the point.