18.01
Calculus
Jason
Starr
Fall
2005
Lecture
2.
September
9,
2005
Homework.
Problem
Set
1
Part
I:
(f)–(h);
Part
II:
Problems
3.
Practice
Problems.
Course
Reader:
1C-2,
1C-3,
1C-4,
1D-3,
1D-5.
1.
Tangent
lines
to
graphs.
For
y
=
f
(
x
),
the
equation
of
the
secant
line
through
(
x
0
,
f
(
x
0
))
and
(
x
0
+
Δ
x,
f
(
x
0
+
Δ
x
))
is,
y
=
f
(
x
0
+
Δ
x
)
−
f
(
x
0
)
(
x
−
x
0
)
+
f
(
x
0
)
.
Δ
x
In
the
limit,
the
equation
of
the
tangent
line
through
(
x
0
,
f
(
x
0
))
is,
y
=
f
(
x
0
)(
x
−
x
0
)
+
y
0
.
Example.
For
the
parabola
y
=
x
2
,
the
derivative
is,
y
(
x
0
)
=
2
x
0
.
The
equation
of
the
tangent
line
is,
y
=
2
x
0
(
x
−
x
0
)
=
2
x
0
x
−
x
2
0
.
For
instance,
the
equation
of
the
tangent
line
through
(2
,
4)
is,
y
=
4
x
−
4.
2
Given
a
point
(
x,
y
),
what
are
all
points
(
x
0
,
x
0
)
on
the
parabola
whose
tangent
line
contains
(
x,
y
)?
To
solve,
consider
x
and
y
as
constants
and
solve
for
x
0
.
For
instance,
if
(
x,
y
) =
(1
,
−
3),
this
gives,
2
(
−
3)
=
2
x
0
(1)
−
x
0
,
or,
2
x
0
−
2
x
0
−
3
=
0
.