Math
134
Summer2
2020
Class
Work
Tuesday
07.28
For
today’s
class
work,
try
#1
and
#2.
Our
goal
is
to
learn
a
bit
composition
of
functions,
so
that
we
can
understand
the
CHAIN
RULE
FOR
DERIVATIVES.
1.
Suppose
you
go
into
a
Casino.
There
are
three
gaming
tables
you
like
to
play.
Here
are
the
results
you
will
happen
tonight
when
you
go
to
each
table,
(assuming
that
you
bring
\$x
to
the
table).
Fill
in
the
right
hand
column
of
this
chart:
TABLE
RESULT
RESULT
AS
A
FUNCTION
A
LOSE
HALF
YOUR
MONEY
f(x)
=
B
TRIPLE
YOUR
MONEY
g(x)
=
C
LOSE
\$50
h(x)
=
2.
Now
let’s
remember
FUNCTION
COMPOSITION.
Refer
to
the
chart.
(i)
Write
the
function
g

°f
(x)
=
g(f(x),
the
result
of
going
to
table
A
and
then
to
table
B.
Simplify
as
much
as
possible.
(ii)
Do
the
same
for
B
then
A.
Write
the
result
as
compositions
of
functions.
(iii)
How
A
then
C?
And
what
C
then
A?
(iv)And
suppose
you
go
to
B
then
C?
C
then
B?
3.
While
we’re
at
it:
(i)
If
you
enter
the
Casino
with
\$x,
and
want
to
play
each
table
exactly
once.
(a)
How
many
different
orders
can
you
play
the
three
tables?
(b)
List
them
in
dictionary
order,
beginning
with
ABC.
(ii)
So
let’s
try
writing
all
the
composition
(chains)
of
functions,
beginning
with
the
one
associated
with
the
order
ABC.
4.
Just
for
practice
in
composing
functions,
suppose
m(x)
=
x2
and
n
(x)
=
3x-­2.
(i)
Write
m(m(x))=?
(ii)
m(n(x))=?
(iii)
n(m(x))=?
(iv)
n
(n(x))=?
5.
Next,
we
will
discuss
the
chain
(of
functions)
rule
for
derivatives….
See
class
discussion.
6.
Last,
let’s
find
the
derivatives
for
the
four
compositions
you
found
in
#4.
7.
Now
(combining
the
product
and
chain
rules),
let’s
try
finding
the
derivative
of
f(x)
=
(3×2

x)5
*
(2x-­1).
8.
And
if
g(x)
=
(3×2

x)5
/
(2x-­1),
find
g

(x)
=?
9.
Let’s
suppose
u(x)
=
1-­x;
v(x)
=
(x2-­1)2
(i)
Find
u

(x);
(ii)
Now
use
the
CHAIN
RULE
from
to
find
v

(x);
(iii)
Now
use
the
product
rule
to
find
[
u
*
v
]

=
[
(1-­x)
(x2
-­1)2
]
‘;
So
now
suppose
we
the
graph
of
f(x)
=
(1-­x)
(x2
-­1)2
.
(iv)
Write
the
coordinates
as
an
ordered
pair
for
the
point
whose
x-­‐coordinate
is
2.
(v)
What
is
the
slope
of
the
tangent
line
to
y
=
f(x)
at
that
point?
(vi)And
what
is
the
equation
of
that
tangent
line?
10.
Next,
let
us
look
at
exponential
functions.
Consider
f(x)
=
ex
.
(i)
Fill
out
the
table:
x
-­3
-­2
0
1
2
3
f(x)
(x,
f(x))
as
an
ordered
pair
(ii)
What
is
the
limit
as
x!

−∞
for
our
f(x)?
(iii)
So
which
HORIZONTAL
ASYMPTOTE
can
we
draw,
when
we
try
to
graph
our
function?
(iii)
Now
draw
axes,
and
use
(i),
(ii),
and
(iii)
to
graph
this
function.
11.
Next,
suppose
u(x)
=
2x,
v(x)
=
ex.
(i)
Find
f(x)
=
u(v(x)
).
(ii)
Next
find
g(x)=
v(u(x)
).
12.
Next,
we
will
discuss
in
class
the
differentiation
properties
of
the
exponential
function.
13.
Use
our
discussion
results
to
find
f
’(x)
and
g

(x)
for
the
functions
from
11.
14.
(i)
For
the
functions
f
(x)
and
g
(x)
,
find
the
values
f(0)
and
g(0).
(ii)
Let’s
find
the
tangent
lines
to
the
graphs
of
y
=
f(x)
and
y
=
g(x)
at
the
points
where
x
=
0.