18.01
Calculus
Jason
Starr
Final
Exam
at
9:00am
sharp
Fall
2005
Tuesday,
December
20,
2005
More
18.01
Final
Practice
Problems
Here
are
some
further
practice
problems
with
solutions
for
the
18.01
Final
Exam.
Many
of
these
problems
are
more
diﬃcult
than
problems
on
the
exam.
Goal
1.
Differentiation.
x
1.1
Find
the
equation
of
every
tangent
line
to
the
curve
y
=
e
containing
the
point
(
−
1
,
0).
This
is
not
a
point
on
the
curve.
1.2
Let
a
and
b
be
positive
real
numbers.
Find
the
equation
of
every
tangent
line
to
the
ellipse
with
implicit
equation,
2
2
x
y
+
=
1
,
a
2
b
2
containing
the
point
(2
a,
2
b
).
This
is
not
a
point
on
the
ellipse.
1.3
Let
a
be
a
real
number
different
from
0.
Use
the
definition
of
the
derivative
as
a
limit
of
difference
quotients
to
find
the
derivative
to
the
following
function,
1
f
(
x
)
=
,
x
at
the
point
x
=
a
.
1.4
Use
the
definition
of
the
derivative
as
a
limit
of
difference
quotients
to
find
the
derivative
of
the
following
function,
f
(
x
)
=
tan(
x
)
,
at
the
point
x
=
0.
You
may
use
without
proof
that
the
following
limits
exist
and
have
the
given
values,
sin
(
x
)
1
−
cos(
x
)
lim
=
1
,
lim
=
0
.
x
0
x
x
0
x
→
→
1.5
For
x
>
0,
let
f
(
x
)
be
the
function,
√
x
f
(
x
)
=
e
.
Thus
the
inverse
function,
y
=
f
−
1
(
x
)
,
1

18.01
Calculus
Jason
Starr
Final
Exam
at
9:00am
sharp
Fall
2005
Tuesday,
December
20,
2005
satisfies
the
equations,
e
√
y
=
x,
and
√
y
=
ln(
x
)
.
Compute
the
derivative,
dy
.
dx
Goal
2.
Sketching
graphs.
2.1
Sketch
the
graph
of
the
function,
1
2
1
f
(
x
)
=
+
.
x
−
1
−
x
x
+
1
2.2
Sketch
the
implicit
function,
y
2
−
xy
−
x
2
=
1
.
2.3
Sketch
the
graph
of
the
function,
x
2
x
2
f
(
x
)
=
+
.
x
+
1
x
−
1
Goal
3.
Applications
of
differentiation.
3.1
A
sculpture
has
the
form
of
a
right
triangle.
The
material
used
for
the
vertical
leg
has
twice
the
cost
of
the
material
used
for
the
horizontal
leg.
The
length
of
the
hypotenuse
is
fixed
(thus
its
cost
is
irrelevant).
What
ratio
of
vertical
leg
to
horizontal
leg
minimizes
the
total
cost
of
the
material?
3.2
A
farmer
has
a
fence
running
diagonally
across
her
property
at
a
45
degree
angle
to
the
north-
south
and
east-west
lines.
She
decides
to
build
a
corral
by
adding
a
length
b
−
a
of
fence
running
north-south,
a
length
b
−
a
of
fence
running
east-west,
and
then
connect
the
two
corners
with
2
length
b
of
fence
running
north-south
and
east-west.
Thus,
the
total
new
length
of
fence
needed
is
4
b
−
2
a
,
and
the
corral
has
the
form
of
a
square
of
length
b
,
with
a
small
isosceles
triangle
of
leg
length
a
removed
from
one
corner
(where
the
square
corral
meets
the
pre-existing
diagonal
fence).
What
ratio
of
a
to
b
gives
maximal
area
of
the
corral
for
a
fixed
length
of
new
fence?
3.3
An
icicle
has
the
shape
of
a
right
circular
cone
whose
ratio
of
length
to
base
radius
is
10.
Assuming
the
icicle
melts
at
a
rate
of
1
cubic
centimeter
per
hour,
how
fast
is
the
length
of
the
icicle
decreasing
when
it
is
10
centimeters
long?