10.2 Tangents and Areas
If
we
have
parametr
ic
equat
ions
x=f(t)
, y=g(t)
and we substitute
y=F(
x
),
then we have
g(t)=F(
f(t))
.
Now, if
g,F,
and
f
are differentiable functions, the Chain Rule can be used to find
g’(t).
'( )
'(
( ))
)
)
Slope of the tangent to
(
) at ( ,
(
)) is
)
0
gt
g
tF
f
tf
x
f
t F
x
ft
yF
x
x
F
x
Fx
dy
dy
dx
dt
dx
dx
dt
dt
=⋅
=
⋅
⇒
=
=
=≠
Example: Find an equation of the tangent line at t=1 for the curve
.
32
1,
3
xt
y
t
t
=−
= − +
1
t
Example: Find an equation of the tangent to the curve at (3,4)
(a)
without eliminating the
parameter and (b)

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