Math 1B
§ 10.2 Calculus with Parametric Curves
Tangents
Suppose f and g are differentiable functions and we want to find the tangent line at a point on the curve
where y is also a differentiable function of x. The chain rule gives
!"
!"
!"
!"
= !" ∙ !"
which leads us to …
!"
Derivative for Parametric Equations:
!"
Note: The curve has a horizontal tangent when
The curve has a vertical tangent when
!"
!"
=
!"
!"
!"
!"
!"
!"
𝑖𝑓
!"
!"
≠0
= 0 (provided that
= 0 (provided that
!"
!"
!"
!"
≠ 0)
≠ 0)
Example: Find the equation of the tangent line to the curve at the point corresponding to the given
value of the parameter.
x = t 5 − 4t 3
y = t2
t=2
Example: Find the point on the curve where the tangent line is horizontal or vertical.
x = t 3 − 3t
y = t 3 − 3t 2
Here are the parametric curves from the previous examples:
Second Derivative for Parametric Equations:
!!!
Note: !! ! ≠
!!!
!! !
!
= !"
!"
!"
=
! !"
!" !"
!"
!"
!! !
!!!
!! !
!!!
!"
!!!
Example: Find !" and !! ! . For which values of t is the curve concave upward?
x = t2 + 1
y = et − 1
Areas
!
We know that the area under a curve 𝑦 = 𝐹(𝑥) from a to b is 𝐴 = ! 𝐹 𝑥 𝑑𝑥 . If the curve is traced out
once by the parametric equations 𝑥 = 𝑓(𝑡) and 𝑦 = 𝑔(𝑡), 𝛼 ≤ 𝑡 ≤ 𝛽, then we can calculate the area
under the parametric curve by using the substitution rule as follows:
𝐴=
!
𝑦
!
𝑑𝑥 =
!
𝑔
!
𝑡 𝑓′(𝑡) 𝑑𝑡
[or
!
𝑔
!
𝑡 𝑓′(𝑡) 𝑑𝑡]
Example: Find the area under the parametric curve given by the parametric equations
x = 6 (θ − sin θ )
y = 6 (1− cosθ )
0 ≤ θ ≤ 2π
Arc Length
If a curve C is described by the parametric equations 𝑥 = 𝑓(𝑡), 𝑦 = 𝑔(𝑡), and 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
!
𝐿=
!
𝑑𝑥
𝑑𝑡
Example: Find the length of the parametric curve.
x = et + e−t
y = 5 − 2t
0≤t ≤3
!
𝑑𝑦
+
𝑑𝑡
!
𝑑𝑡
Surface Area
If the curve given by the parametric equations 𝑥 = 𝑓(𝑡), 𝑦 = 𝑔(𝑡), 𝛼 ≤ 𝑡 ≤ 𝛽, is rotated about the xaxis, where 𝑓′, 𝑔′ are continuous and 𝑔 𝑡 ≥ 0, then the area of the resulting surface is given by
!
𝑆=
2𝜋𝑦
!
𝑑𝑥
𝑑𝑡
!
𝑑𝑦
+
𝑑𝑡
!
𝑑𝑡
Note: The general formulas 𝑆 = ∫ 2𝜋𝑦 𝑑𝑠 and 𝑆 = ∫ 2𝜋𝑥 𝑑𝑠 are still valid, but for parametric curves
we use
𝑑𝑠 =
𝑑𝑥
𝑑𝑡
!
𝑑𝑦
+
𝑑𝑡
!
Example: Find the area of the surface generated by rotating the given parametric curve about the xaxis.
0 ≤ θ ≤ π2
y = sin 3 θ
x = cos3 θ