7.1 - MATH
1081
 Monday,
April
4
 
...

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Unformatted text preview: MATH
1081
 Monday,
April
4
 
 Chapter
7
–
Section
1
 
 ANTIDERIVATIVES
 Homework
#11
(due
4/11):
 








 
 Clicker
Check­in:

Choose
any
letter
to
check
in
now.
 
 Section
7.1
#16,
20,
54,
60
 Section
7.2
#18,
24,
32,
40,
42
 Just
when
you
thought
you
were
getting
all
these
derivatives
 under
control,
we
are
going
to
flip
things
on
their
head…
 
 Suppose
that
we
know
that

 
 f ' ( x ) = 2 x .
 
 Can
we
figure
out
what
the
original
 f ( x ) 
was?
 
 Is
there
only
one
possible
 f ( x ) 
for
which
 f ' ( x ) = 2 x ?
 
 If
 F ' ( x ) = f ( x ) ,
then
we
call
 F ( x )
the
antiderivative
of
 f ( x ) .
 
 That
is,
the
antiderivative
of
 f ( x ) 
is
the
function
 F ( x ) 
that
we
 would
take
the
derivative
of
to
get
 f ( x ) .
 
 
 Further,
there
is
a
theorem
which
states
if
both
 F ( x )
and
 G ( x ) 
 are
antiderivatives
of
 f ( x ) ,
then
 F ( x ) − G ( x ) = C 
for
some
 constant
 C .
 
 That
is,
once
we
have
found
one
an
antiderivative
of
a
function
 f ( x ) ,
we
know
that
up
to
adding
a
constant
to
it,
this
is
the
 only
function
that
will
work
as
an
antiderivative.
 The
notation
that
we
will
use
to
indicate
that
we
are
finding
 the
antiderivative
of
a
function
is
called
an
integral
sign.
 
 It
looks
like
 ∫ 
(kind
of
an
elongated
“S”,
which
we
will
see
a
 connection
to
in
a
later
section).
 
 Similar
to
when
we
found
derivatives,
we
need
to
know
with
 respect
to
which
variable
we
are
working.

So
this
integral
sign
 ALWAYS
comes
paired
with
something
of
the
form
 dx ,
 indicating
that
 x 
is
the
variable
we
are
interested
in.
 
 So,
for
example,
the
antiderivative
of
the
function
 2 x ,
with
 respect
to
the
variable
 x 
is
written
as ∫ 2 x dx .
 
 The
act
of
finding
the
function
to
which
this
is
equal
is
called
 antidifferentiation.

It
is
also
called
integration.
 
 
 We
can
also
refer
to
 ∫ f ( x ) dx 
as
an
indefinite
integral
where
 
 f ( x ) 
is
called
the
integrand
and
 x 
is
the
variable
of
integration.
 Example:
 Evaluate
the
indefinite
integral.
 
 
 
 
 1.

 ∫ 9 x 8 − 4 x + 2 dx 
 ( ) 
 
 
 
 
 
 
 
 
 
 
 ⎛ 1/ 2 1 2 u ⎞ 2.

 ∫ ⎜ u + − e ⎟ du 
 ⎝ ⎠ u 
 
 3.

 ∫ t −2 dt 
 3 t Rules
of
Integration:
 
 Rule 1: The Integral of a Constant k . 
 Rule 2: ∫ k dx = kx + C , The Power Rule where C is any constant 1 n +1 ∫ x dx = n + 1 x + C , ( n ≠ −1) n Rule 3: The Integral of a Constant Multiple of a Function ∫ k ⋅ f ( x ) dx = k ∫ f ( x ) dx , where k is a constant Rule 4: The Sum/Difference Rule ⎣ ⎦ ∫ ⎡ f ( x ) ± g ( x )⎤ dx = ∫ f ( x ) dx ± ∫ g ( x ) dx Rule 5: Integral of the Exponential Function a kx a kx dx = + C, k ≠ 0 ∫ k ( ln a ) Rule 6: Integral of the Function x −1 1 ∫ x dx = ∫ x dx = ln x + C −1 When
we
are
given
the
derivative
of
the
function
and
the
 coordinates
of
a
single
point
of
the
function,
we
can
actually
 determine
the
exact
function
(not
up
to
some
constant).
 
 These
are
often
referred
to
as
initial
value
problems.

To
solve
 them,
we
first
find
the
antiderivative,
then
use
the
added
 information
to
determine
the
value
of
the
constant.
 
 Example:

Suppose
that
the
acceleration
of
a
particle
at
time
 t 

 is
given
by
the
function
 a ( t ) = 18t + 8 .

If
the
velocity
 at
 t = 1
is
 v = 15 
and
the
starting
position
of
the
 particle,
at
time
 t = 0 
is
3,
then
find
the
position
 function
 s ( t ) .
 
 Clicker
Check­out:

Choose
any
letter
to
check
out
now.
 
 
 Next
time
–
Section
7.2
 
 
 ...
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