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tan x
Calculus
AB
Assignment
Practice
Analyzing
Functions
Defined
by
Definite
Integrals
1
Copyright © 2021 Apex Learning. See Terms of Use for further information. Images of the TI-84 calculator are used with the permission of Texas Instruments Incorporated. Copyright © 2011 Texas Instruments Incorporated.
1.
Practice
finding
the
derivatives
of
functions
that
are
defined
by
definite
integrals.
A.
Let
g
(
u
) = ⎰
u
3 1
dx
.
Then
dg
=
3 2
+
x
2
du
B.
Find
d
⎰
7
(
e
3
t
+ 1
)
dt
.
dx
—
cos
(
3
x
)
C.
Let
F
(
x
) = ⎰
cos
( s
)
ds
.
Find
F
'
(
—
/4
)
.
—
tan x
4
x
Calculus
AB
Assignment
Practice
Analyzing
Functions
Defined
by
Definite
Integrals
2
Copyright © 2021 Apex Learning. See Terms of Use for further information. Images of the TI-84 calculator are used with the permission of Texas Instruments Incorporated. Copyright © 2011 Texas Instruments Incorporated.
2.
The
cost
of
producing
x
units
of
a
certain
commodity
is
given
by
P
(
x
) = 1000 + ⎰
0 MC
(
s
)
ds
,
where
P
is
in
dollars
and
M
(
x
) is
marginal
cost
in
dollars
per
unit.
A.
Suppose
the
marginal
cost
at
a
production
level
of
500 units
is
$10 per
unit,
and
the
cost
of
producing
500 units
is
$12, 000 (that
is,
M
(
500
) = 10 and
P
(
500
) = 12000
).
Use
a
tangent
line
approximation
to
estimate
the
cost
of
producing
only
497 units.
dt
dt
⎰
Calculus
AB
Assignment
Practice
Analyzing
Functions
Defined
by
Definite
Integrals
3
Copyright © 2021 Apex Learning. See Terms of Use for further information. Images of the TI-84 calculator are used with the permission of Texas Instruments Incorporated. Copyright © 2011 Texas Instruments Incorporated.
B.
Suppose
the
production
schedule
is
such
that
the
company
produces
five
units
each
day.
That
is,
the
number
of
units
produced
is
x
= 5
t
,
where
t
is
in
days,
and
t
= 0 corresponds
to
the
beginning
of
production.
Write
an
equation
for
the
cost
of
production
P
as
a
function
of
time
t
.
C.
Use
your
equation
for
P
(
t
) from
part
B
to
find
dP
.
Be
sure
to
indicate
units
and
describe
what
dP
represents,
practically
speaking.
3.
Let
F
(
x
) = x
3
t
2
(
cos
(
t
3
) + 2
)
dt
.
—
2
A.
Using
the
First
Fundamental
Theorem
of
Calculus,
find
F
'
(
x
)
.
(Yes,
this
is
as
easy
as
it
seems.)
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⎰
Calculus
AB
Assignment
Practice
Analyzing
Functions
Defined
by
Definite
Integrals
4
Copyright © 2021 Apex Learning. See Terms of Use for further information. Images of the TI-84 calculator are used with the permission of Texas Instruments Incorporated. Copyright © 2011 Texas Instruments Incorporated.
B.
Starting
with
F
(
x
) = x
3
t
2
(
cos
(
t
3
) + 2
)
dt
,
use
the
substitution
u
(
t
) = t
3
to
rewrite
the
—
2
definite
integral.
You
should
get
a
new
(equivalent)
expression
for
F
(
x
)
,
which
consists
of
this
new
definite
integral.
C.
Using
your
new
expression
for
F
(
x
) from
part
B
and
Leibniz's
Rule,
find
F
'
(
x
)
.
2
1
-2
-1
1
t
2
3
0
-1
-2
-3
Calculus
AB
Assignment
Practice
Analyzing
Functions
Defined
by
Definite
Integrals
5
Copyright © 2021 Apex Learning. See Terms of Use for further information. Images of the TI-84 calculator are used with the permission of Texas Instruments Incorporated. Copyright © 2011 Texas Instruments Incorporated.
=
⎰
4.
The
function
f
(
t
) has
domain
[
—
2, 3
] on
which
it
is
continuous.
A
graph
of
y
= f
(
t
) is
given
below:
y
= f
(
t
)
5
x
+
2
Define
a
new
function
h
(
x
) by
h
(
x
)
2
f
(
t
)
dt
.
5
Calculus
AB
Assignment
Practice
Analyzing
Functions
Defined
by
Definite
Integrals
6
Copyright © 2021 Apex Learning. See Terms of Use for further information. Images of the TI-84 calculator are used with the permission of Texas Instruments Incorporated. Copyright © 2011 Texas Instruments Incorporated.
A.
What
is
the
domain
of
h
(
x
) (that
is,
on
which
x
values
is
the
function
h
(
x
) defined)?
domain of h(x) is
B.
Find
h
'
( —
3
)
.
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Calculus
AB
Assignment
Practice
Analyzing
Functions
Defined
by
Definite
Integrals
7
Copyright © 2021 Apex Learning. See Terms of Use for further information. Images of the TI-84 calculator are used with the permission of Texas Instruments Incorporated. Copyright © 2011 Texas Instruments Incorporated.
C.
At
what
x
value(s)
does
the
function
h
(
x
) attain
a
relative
maximum?
for relative maximum , h'(x)=0 and h'(x) changes from positive to negative
=>f(5x+2)=0
=>5x+2=-2,5x+2=0,5x+2=3
=>x=-4
/5
,x=-2
/5
, x=1
/5
but only at x=-2
/5
, h'(x) changes from positive to negative(since f(t) changes
changes from positive to negative at t=0) .
therefore h(x) attains relative maximum at
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