4.1 - MATH
1081
 Wednesday,
February
9
 


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Unformatted text preview: MATH
1081
 Wednesday,
February
9
 
 Chapter
4
–
Section
1
 
 TECHNIQUES
FOR
FINDING
 DERIVATIVES
 Homework
#4
(due
2/14):
 
 








 
 Clicker
Check­in:

Choose
any
letter
to
check
in
now.
 
 Section
3.4
#18,
48,
56
 Section
4.1
#34,
56,
62,
72
 Last
time,
we
defined
the
derivative
function
 f ' ( x ) 
as
a
limit.

 In
particular,
when
the
limit
exists,

 
 f ( x + h) − f ( x) 




 .
 f ' ( x ) = lim h→ 0 h 
 
 By
the
way
it
is
defined,
it
is
not
difficult
to
understand
the
 relationship
between
the
derivative
function
and
the
slope
of
 the
tangent
line
to
a
curve
at
a
point
(and
other
interpretations
 of
the
rate
of
change
of
a
function
at
a
point).
 
 The
act
of
finding
a
derivative
is
called
differentiation.


 
 Because
of
the
history
of
the
development
of
calculus,
as
well
 as
the
various
fields
with
applications
requiring
calculus,
the
 notation
signifying
the
derivative
of
a
function
does
vary.
 
 Some
common
notations
that
we
will
see
in
this
class
are
 
 dy d 

 
 f '( x )
 
 
 ⎡ f ( x ) ⎤ 
 
 
 Dx ⎡ f ( x ) ⎤ . ⎣ ⎦ ⎦ dx dx ⎣ For
the
purpose
of
actually
finding
the
derivative
 f ' ( x ) 
of
a
 function,
the
limit
definition
requires
cumbersome
(if
not
 impossible)
algebraic
manipulations.
 
 After
finding
a
number
of
derivatives,
you
may
have
already
 detected
some
patterns.

We
will
now
introduce
a
number
of
 rules
of
differentiation
that
will
simplify
the
process
of
finding
 the
derivative
of
a
function.
 
 
 Note
that
mastering
these
rules
will
be
crucial
for
your
success
 in
this
course.

The
most
effective
method
to
gain
this
mastery
 is
practice,
practice,
practice.
 Rules
of
Differentiation
 
 1. Constant
Rule
 
 
 
 If
 f ( x ) = k ,
where
 k 
is
a
constant,
then
 f ' ( x ) = 0 .
 
 
 We
saw
an
example
of
this
last
time,
where
 f ( x ) = 3.

To
 prove
this
rule
in
general,
just
return
to
the
proof
for
 f ( x ) = 3
 and
replace
every
“3”
with
“ k ”.
 
 
 
 
 If
 f ( x ) = x n 
for
any
real
number
 n ,
then
 f ' ( x ) = nx n −1 .
 
 
 In
some
settings,
this
is
referred
to
as
the
“drop‐down
rule”,
 since
we
drop
the
power
down
in
front
and
subtract
1
from
the
 exponent.
 
 The
proof
for
a
general
real
number
 n 
requires
more
advanced
 analysis.

So,
we
will
prove
only
the
case
that
 n 
is
a
positive
 integer
here.
 
 Rules
of
Differentiation
 
 2. Power
Rule
 Rules
of
Differentiation
 
 3. Constant
Times
a
Function
Rule
 
 Let
 k 
be
a
real
number.

If
 g ' ( x ) 
exists,
then
the
 derivative
of
 f ( x ) = k ⋅ g ( x ) 
is
 f ' ( x ) = k ⋅ g ' ( x ) .
 
 
 In
other
words,
coefficients
to
expressions
just
get
dragged
 along.

So,
if
you
know
that
the
derivative
of
 g ( x ) = x 3 
is
 g ' ( x ) = 3x 2 ,
then
the
derivative
of
 f ( x ) = 7 x 3 
is
 f ' ( x ) = 21x 2 .
 
 The
proof
of
this
rule
is
easy
by
just
factoring
out
the
 k .
 Rules
of
Differentiation
 
 4. Sum
or
Difference
Rule
 
 If
 f ( x ) = u ( x ) ± v ( x ),
and
 u ' ( x )
and
 v ' ( x ) 
both
exist,
then
 f ' ( x ) = u ' ( x ) ± v ' ( x ) .
 
 
 This
rule
simply
says
that
if
you
have
function
defined
as
a
sum
 (or
difference)
of
different
terms,
then
you
can
find
the
 derivative
of
the
function
by
just
finding
the
derivative
of
each
 term
separately.
 
 The
proof
of
this
rule
is
also
easy
with
just
a
little
 rearrangement.
 Example:
Find
the
derivative
of
the
following
functions.
 
 
 
 1.

 f ( x ) = x 5 − x 4 + 9 x 3 − 2 x 2 + 8 x + 1
 
 
 
 
 
 
 
 
 
 
 
 x3 3 2.

 g( x ) = − 2
 2x 1 3 x+ 7 
 3.

 h( x ) = x−4 52 2 x Example:

A
person
standing
on
top
of
a
161
feet
tall
building

 tosses
a
ball
up
into
the
air
so
that
when
it
begins
to



 drop
back
down,
it
will
miss
the
top
of
the
building
 and
fall
all
the
way
to
the
street
below.
 
 Suppose
the
height
(in
feet)
of
the
ball
above
the
 street
after
t
seconds
is
given
by
the
function
 
 
 
 
 
 
 
 
 s(t ) = −16t 2 + 64 t + 161.
 Example:
 
 
 
 s(t ) = −16t 2 + 64 t + 161
 
 1. What
is
the
height
of
the
ball
(above
the
street)
after
4
 seconds?


 
 2. What
is
the
average
velocity
of
the
ball
over
the
interval
 [ 4, 5 ]?


 
 3. When
will
the
ball
hit
the
street?


 
 4. What
is
the
velocity
of
the
ball
at
the
moment
that
it
hits
 the
street?


 
 5. At
what
moment
is
the
velocity
of
the
ball
equal
to
0?

 How
far
above
the
street
is
the
ball
at
that
moment?
 Clicker
Check­out:

Choose
any
letter
to
check
out
now.
 
 
 
 Quiz
in
recitation
tomorrow
(Sections
3.1‐3.3).
 
 Next
time
–
Section
4.2
 ...
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This note was uploaded on 05/01/2011 for the course MATH 1081 taught by Professor Johanson during the Spring '08 term at Colorado.

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