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Math 53: Multivariable Calculus
Solutions for Worksheet 8/31/11: Tangent Lines, Area, and Arclength (Oh my!)
Exercise 0.1.
(Stewart 10.2.25) Consider the curve
x
= cos
t
,
y
=
cos
t
sin
t
, 0
≤
t <
2
π
. Find all tangent lines to the point (0
,
0) (hap
pening for diﬀerent values of
t
).
Solution.
First we need to ﬁnd which values of
t
give us the point (0
,
0).
This means we wish to simultaneously solve the equations cos
t
= 0 and
sin
t
cos
t
= 0 for 0
≤
t <
2
π
. The only values in this range for which
cos
t
= 0 are
t
=
π/
2 and
t
= 3
π/
2, so these are the only possible
values of
t
for our point. Since sin
t
cos
t
is also 0 for these values of
t
,
we have that
t
=
π/
2 and
t
= 3
π/
2 both give us the point (0
,
0).
We now wish to ﬁnd the slope of the tangent line at time
t
; as we
saw in section and in lecture, this is just
g
0
(
t
)
f
0
(
t
)
(assuming that expression
makes sense). Since
f
(
t
) = cos
t
and
g
(
t
) = sin
t
cos
t
, we have
f
0
(
t
) =

sin
t
and
g
0
(
t
) = cos
2
t

sin
2
t
(by the product rule). Thus we have
g
0
(
t
)
f
0
(
t
)
=
cos
2
t

sin
2
t

sin
t
.
At time
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 Spring '07
 Hutchings
 Calculus, Multivariable Calculus

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