Parametric Equations and Calculus
If a smooth curve C is given by the equations
x f t
and y g t
,
then the slope of C at the point
x, y
is given by
dy
dx
dy
dt
dx
dt
where
dx
dt
0
,
and the second derivative is given by
d
2
y
dx
2
d
dx
dy
dx
d
dt
dy
dx
dx
dt
.
Ex. 1 (Noncalculator)
Given the parametric equations
x 2 t and y 3t
2
2t
, find
dy
dx
and
d
2
y
dx
2
.
__________________________________________________________________________________
Ex. 2 (Noncalculator)
Given the parametric equations
x 4cost and y 3sint
, write an equation of the tangent line to the
curve at the point where
t
3
4
.
__________________________________________________________________________________
Ex 3 (Noncalculator)
Find all points of horizontal and vertical tangency given the parametric equations
x t
2
t, y t
2
3t 5.
__________________________________________________________________________________
Ex. 4 (Noncalculator)
Set up an integral expression for the arc length of the curve given by the parametric
equations
Do not evaluate.
CALCULUS BC
WORKSHEET ON PARAMETRICS AND CALCULUS
Work these on notebook paper. Do not use your calculator.
On problems 1 – 5, find
dy
dx
and
d
2
y
dx
2
.
1.
x t
2
, y t
2
6t 5
2.
x t
2
1, y 2t
3
t
2
3.
x t, y 3t
2
2t
4.
x lnt, y t
2
t
5.
x 3sint 2, y 4cost 1
_____________________________________________________________________________
6. A curve C is defined by the parametric equations
x t
2
t 1, y t
3
t
2
.
(a) Find
dy
dx
in terms of t.
(b) Find an equation of the tangent line to C at the point where t = 2.
7. A curve C is defined by the parametric equations
x 2cost, y 3sint
.
(a) Find
dy
dx
in terms of t.
(b) Find an equation of the tangent line to C at the point where t =
4
.
______________________________________________________________________________
On problems 8 – 10, find:
(a)
dy
dx
in terms of t.
(b) all points of horizontal and vertical tangency
8.
x t 5, y t
2
4t
9.
x t
2
t 1, y t
3
3t
10.
x 3 2cost, y 1 4sint
______________________________________________________________________________
On problems 11 - 12, a curve C is defined by the parametric equations given. For each problem,
write an integral expression that represents the length of the arc of the curve over the given interval.
11.
x t
2
, y t
3
, 0 t 2
12.
x e
2t
1, y 3t 1, 2 t 2
Answers to Worksheet on Parametrics and Calculus
2
2
2
3
3
2 6 3 3
1. 1 ;
2 2 2
dy t d y
t
dx t t dx t t
______________________________________________________________________________
2
2
3
2. 3 1;
2
dy d y
t
dt dx t
________________________________________________________________________________
2
2
11
31
22
22
11
22
6 2 18 2
3. 12 4 ; 36 4
11
22
dy t d y t t
t t t
dx dx
tt


________________________________________________________________________________
2
2
22
2 1 4 1
4. 2 ; 4
11
dy t d y t
t t t t
dx dx
tt

________________________________________________________________________________
2
2
2
3
4
sec
4sin 4 4
3
5. tan ; sec
3cos 3 3cos 9
t
dy t d y
tt
dx t dx t
_______________________________________________________________________________
6. (a)
dy
dx
3t
2
2t
2t 1
(b) When t = 2,
2
3 2 2 2 8
, 5, 4
2 2 1 5
dy
xy
dx

so the tangent line equation is
y 4
8
5
x 5
________________________________________________________________________________
7. (a)
dy
dx
3cost
2sint
3
2
cott
(b) When
t
4
,
dy
dx
3
2
cot
4
3
2
, x 2, y
3 2
2
so the tangent line equation is
y
3 2
2
3
2
x 2
________________________________________________________________________________
8. (a)
dy
dx
2t 4
1
(b) A horizontal tangent occurs when
0 and 0
dy dx
dt dt

so a horizontal tangent
occurs when
2t 4 0
which is at t= 2. When t = 2, x = 7 and y =
4 so a horizontal
tangent occurs at the point
7, 4
. A vertical tangent occurs when
0 and 0
dx dy
dt dt

.
Since
1 0
, there is no point of vertical tangency on this curve.
9. (a)
dy
dx
3t
2
3
2t 1
(b) A horizontal tangent occurs when
0 and 0
dy dx
dt dt

so a horizontal tangent
occurs when
3t
2
3 0
which is at
t 1
. When t = 1 , x = 1 and y =
2, and when
t 1
, x = 3 and y = 2 so a horizontal tangent occurs at the points
1, 2
and 3, 2
A vertical tangent occurs when
0 and 0
dx dy
dt dt

so a vertical tangent occurs when
2t 1 0 so t
1
2
. When
t
1
2
,
x
3
4
and y
11
8
so a vertical tangent occurs at the
point
3
4
,
11
8
.
______________________________________________________________________________
10. (a)
dy
dx
4cost
2sint
(b) A horizontal tangent occurs when
0 and 0
dy dx
dt dt

so a horizontal tangent
occurs when
4cost 0
which is at
t
2
and
3
2
. When t =
2
, x = 3 and y = 3, and when
t
3
2
, x = 3 and y =
5 so a horizontal tangent occurs at the points
3, 3
and 3, 5
.
A vertical tangent occurs when
0 and 0
dx dy
dt dt

so a vertical tangent occurs when
2sint 0 so t 0 and
. When
0,t
x 5 and y 1
and when
t
, x 1 and y 1
so a vertical tangent occurs at the points
5, 1
and 1, 1
.
_______________________________________________________________________________
11.
s
dx
dt
2
dy
dt
2
dt
a
b
2t
2
3t
2
 
2
dt
0
2
or 4t
2
9t
4
dt
0
2
_______________________________________________________________________________
12.
s
dx
dt
2
dy
dt
2
dt
a
b
2e
2t
2
3
2
dt
2
2
or 4e
4t
9 dt
2
2