Calc Midterm 4
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Rutgers University *
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Course
151
Subject
Mathematics
Date
Jan 9, 2024
Type
Pages
11
Uploaded by BailiffRockWolf28
RUTGERS
UNIV.
MIDTERM
QUIZ
©
2020
Suppose
that
f(x)
is
a
differentiable
function
with
f(7)=16
and
f'(7)=4.
Rutgers
Quiz
(vers.
d883s)
©
2020
(a)
Find
the
linearization
L(x)
of
f
centered
at
7.
Rutgers
Quiz
(vers.
d883s)
Rutgers
Quiz
(vers.
d883s)
©
2020
L(x)=
16
+4(x-7)
(b)
Use
the
standard
linear
approximation
to
find
an
estimate
to
f(6.93).
Do
not
round
your
answer.
Rutgers
Quiz
(vers.
d883s)
Rutgers
Quiz
(vers.
d883s)
Rutgers
Quiz
(vers.
d883s)
Rutgers
Quiz
(vers.
d883s)
f(6.93)~
15.72
.
Rutgers
Quiz
(vers.
d883s)
©
2020
RUTGERS
UNIV.
MIDTERM
QUIZ
©
2020
Question
is
complete.
RUTGERS
UNIV.
OFFICIAL
MATH
151
MIDTERM
©
2020
Find
the
absolute
maximum
and
minimum
values
of
f(x)
=
C-12
+
45x
-
64
on
the
interval
[2,4]
and
say
where
they
occur.
Rutgers
Quiz
(vers.
b245r)
Rutgers
Quiz
(vers.
b245r)
Rutgers
Quiz
(vers.
b245r)
RUTGERS
UNIV.
OFFICIAL
MATH
151
MIDTERM
©
2020
Rutgers
Quiz
(vers.
b245r)
(a)
f
has
an
absolute
maximum
of
-
10'.
The
absolute
maximum
occurs
at
the
x-value(s)
x=
3.
Rutgers
Quiz
(vers.
b245r)
(Use
commas
to
separate
values
as
needed.)
Rutgers
Quiz
(vers.
b245r)
(b)
f
has
an
absolute
minimum
of
—14
.
The
absolute
minimum
occurs
at
the
x-value(s)
x=
2.
Rutgers
Quiz
(vers.
b245r)
(Use
commas
to
separate
values
as
needed.)
RUTGERS
UNIV.
OFFICIAL
MATH
151
MIDTERM
©
2020
Question
is
complete.
Tap
on
the red
indicators
to
see
incorrect
answers.
Rutgers
Quiz
(vers.
a807r)
©
2020
Use
the
graph
of
y
=
f'(x)
to
answer
questions
about
the
CONTINUOUS
FUNCTION
f.
(Note:
the
blobs
on
the
ends
of
the
graph
of
y
=f'(x)
are
arrows.)
Again,
the
graph depicted
above
is
for
y
=f'(x).
This
problem
has
parts
(a)-(f).
Rutgers
Quiz
(vers.
a807r)
©
2020
(a)
Find
the
critical
points
for
f.
The
critical
point(s)
for
f(x)
is/are
at
x=
-
1,5‘.
(Use
commas
to
separate
multiple
values,
if
needed.)
f
has
no
critical
points.
Rutgers
Quiz
(vers.
a807r)
©
2020
(b)
Find
the
interval(s)
where
f
is
increasing.
f(x)
is
increasing
on the
interval(s)
(5,
oo)‘.
(Use
interval
notation
and
separate
multiple
intervals
using
commas,
if
needed.)
fis
never
increasing.
Rutgers
Quiz
(vers.
a807r)
©
2020
1A\
Eind
tha
intansmalle)
whara
fie
danrancina
Question
is
complete.
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on
the red
indicators
to
see
incorrect
answers.
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Rutgers
Quiz
(vers.
a807r)
©
2020
Use
the
graph
of
y
=
f'(x)
to
answer
questions
about
the
CONTINUOUS
FUNCTION
f.
(Note:
the
blobs
on
the
ends
of
the
graph
of
y
=f'(x)
are
arrows.)
Again,
the
graph depicted
above
is
for
y
=f'(x).
This
problem
has
parts
(a)-(f).
(c)
Find
the
interval(s)
where
f
is
decreasing.
f
is
never
decreasing.
f(x)
is
decreasing
on
the
interval(s)
(-
oo,
5)‘
(Use
interval
notation
and
separate
multiple
intervals
using
commas,
if
needed.)
Rutgers
Quiz
(vers.
a807r)
©
2020
(d)
Find
the
interval(s)
where
f
is
concave
up.
f
is
never
concave
up.
f(x)
is
concave
up on the
interval(s)
(-0,
=
1),
(3,
oo)‘
(Use
interval
notation
and
separate
multiple
intervals
using
commas,
if
needed.)
Rutgers
Quiz
(vers.
a807r)
©
2020
(e)
Find
the
interval(s)
where
f
is
concave
down.
Question
is
complete.
Tap
on
the red
indicators
to
see
incorrect
answers.
Rutgers
Quiz
(vers.
a807r)
©
2020
Use
the
graph
of
y
=
f'(x)
to
answer
questions
about
the
CONTINUOUS
FUNCTION
f.
(Note:
the
blobs
on
the
ends
of
the
graph
of
y
=f'(x)
are
arrows.)
Again,
the
graph depicted
above
is
for
y
=f'(x).
This
problem
has
parts
(a)-(f).
(e)
Find
the
interval(s)
where
f
is
concave
down.
*
f(x)
is
concave
down
on
the
interval(s)
(-1,
3)
(Use
interval
notation
and
separate
multiple
intervals
using
commas,
if
needed.)
X
tis
never
concave
down.
Rutgers
Quiz
(vers.
a807r)
©
2020
(f)
Find
the
x-value(s)
of
any
inflection
points
for
f.
f
has
no
inflection
points.
f(x)
has
inflection
point(s)
at
the
point(s)
with
x-value(s)
x=
-
1,3‘
(Use
commas
to
separate
multiple
values,
if
needed.)
Rutgers
Quiz
(vers.
a807r)
©
2020
RUTGERS
UNIV.
MIDTERM
QUIZ
©
2020
Question
is
complete.
Tap
on
the red
indicators
to
see
incorrect
answers.
RUTGERS
UNIV.
OFFICIAL
MIDTERM
QUIZ
©
2020
The
graphs
of
y
=f(x)
and
y
=f'(x)
are
sketched
below.
y=f(x)
Q
Rutgers
Quiz
(vers.
a283t)
©
2020
Use
the
information
given
in
the
graphs
above
to
answer
parts
(a)
and
(b).
.
f(x)
(a)
lim
—————
x—2
6x°
—6x-12
Rutgers
Quiz
(vers.
a283t)
©
2020
YA
_
_
(x)
2
exists
and
is
evaluated
as
lim
2
.
.-
18-
x—2
6x°
-6x-12
does
not
exist.
cannot
be
determined
with the
given
information.
®
tim
(In(9f(x))-
In
(6x?-6x-12))
x—2*
Rutgers
Quiz
(vers.
a283t)
©
2020
-*
Question
is
complete.
Tap
on
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indicators
to
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incorrect
answers.
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RUTGERS
UNIV.
OFFICIAL
MIDTERM
QUIZ
©
2020
The
graphs
of
y
=f(x)
and
y
=f'(x)
are
sketched
below.
y=f(x)
Q
Rutgers
Quiz
(vers.
a283t)
©
2020
Use
the
information
given
in
the
graphs
above
to
answer
parts
(a)
and
(b).
cannot
be
determined
with the
given
information.
®)
tim
(In(9f(x))-
In
(6x%
-6x-12))
x—2*
Rutgers
Quiz
(vers.
a283t)
©
2020
*
9.2
exists
and
is
evaluated
as
lim
(In
(9f(x))
=
In
(6x2
-6x-
12))
=In
26+(-6)
-
x—>2*
X
5.
does
not
exist.
cannot
be
determined
with the
given
information.
RUTGERS
UNIV.
OFFICIAL
MIDTERM
QUIZ
©
2020
Question
is
complete.
Tap
on
the red
indicators
to
see
incorrect
answers.
KU
I
GERS
UNIV.
UFFIUIAL
MATH
151
MIU
|
ERM
©
2U2U
You
decide
to
build
a
200
square
foot
rectangular
vegetable
garden.
In
order
to
protect
the
vegetable
garden,
you
decide
to
surround
your
200
it?
vegetable
garden
with
a
mulch
border.
The
border
will
be
1
ft
wide
on
two
of
the
parallel
sides
and
2
ft
wide
on the
remaining
two
parallel
sides.
A
sketch
of
the
plan
is
drawn
below:
Rutgers
Quiz
(vers.
a552t)
Tft
vegetable
garden
h
2
ft
2
ft
w
mulch
1
ft
Ditrnre
Nuiz
lunre
~ARE2#)
RUTGERS
UNIV.
OFFICIAL
MATH
151
MIDTERM
©
2020
Let
w
and
h
be
the
dimensions
of
the
vegetable
garden
as
drawn
in
the
sketch.
(a)
Write
an
equation
in
w
and
h
that
gives
the
constraint.
The
constraint
is
given
by
the
equation
wh
=200
.
Rutgers
Quiz
(vers.
a552t)
(b)
Let
A
denote
the
area
of
the
mulch
border
for
the
vegetable
garden.
Describe
A as
a
function
of
w
ONLY.
As
a
function
of
w
only,
A(w)
=
+2w+8
.
Rutgers
Quiz
(vers.
a552t)
(c)
Find
the
domain
of
A(w).
Write
the
domain
using
interval
notation.
Use
commas,
as
needed,
to
separate
any
intervals.
The
domain
of
A
is
(0,00)‘.
(d)
Find
the
critical
point(s)
of
A.
Use
commas,
as
needed,
to
separate
values.
The
critical
point(s)
of
A
is/are
20'.
Ritnare
Nuiz
luare
aRR24\
Question
is
complete.
Tap
on
the red
indicators
to
see
incorrect
answers.
KU
I
GERS
UNIV.
UFFIUIAL
VMIATH
1971
MID
|
ERVI
©
2UZU
You
decide
to
build
a
200
square
foot
rectangular
vegetable
garden.
In
order
to
protect
the
vegetable
garden,
you
decide
to
surround
your
200
it?
vegetable
garden
with
a
mulch
border.
The
border
will
be
1
ft
wide
on
two
of
the
parallel
sides
and
2
ft
wide
on the
remaining
two
parallel
sides.
A
sketch
of
the
plan
is
drawn
below:
Rutgers
Quiz
(vers.
a552t)
Tft
vegetable
garden
h
2
ft
2
ft
w
mulch
1
ft
Dutgore
Quiz
{uore,
aBEE2L)
tgore
Quiz
(v
(e)
What
does
the
Second
Derivative
Test
tell
us
about
the
nature
of
the
critical
point?
X
Since
A’’
<0
at
the
critical
point,
the
area
of
the
mulch
is
at
a
local
maximum.
Since
A’
>0
at
the
critical
point,
the
area
of
the
mulch
is
at
a
local
maximum.
*
Since
A’
>0
at
the
critical
point,
the
area
of
the
mulich
is
at
a
local
minimum.
Since
A’’
<0
at
the
critical
point,
the
area
of
the
mulch
is
at
a
local
minimum.
The
Second
Derivative
Test
fails
at
the
critical
point,
and
so,
tells
us
nothing
about
the
nature
of
the
point.
(f)
What
dimensions
for
the
vegetable
garden
will
minimize
the
area
of
the
mulch
border?
Give
EXACT
VALUES.
The
area
of
the
mulch
border
is
an
a
minimum
if
the
vegetable
garden
has
dimensions
w=
20
feetby
h=
10
feet.
Rutgers
Quiz
(vers.
a552t)
DIITREDQ
1IN/
NECICIAL
MATH
184
MINTEDM
A
20920
Question
is
complete.
Tap
on
the red
indicators
to
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answers.
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RUTGERS
UNIV.
OFFICIAL
MATH
151
MIDTERM
QUIZ
©
2020
(a)
The
graph
of
y
=f(x)
on
the
interval
[
-
4,4]
is
sketched
below.
y=f(x)
M
Q
+
Q
=z
2_
\
X
r
T
T
T
—
T
a2
2
4
-2
Rutgers
Quiz
(vers.
c491q)
©
2020
The
Mean
Value
Theorem
says
that
the
graph
of
y
=
f(x)
must
have
at
least
one
x-value
c
in
the
open
interval
(
-
4,4)
where
the
tangent
line
to
the
graph
at
x=c
has
I
3
slope
5
.
Rutgers
Quiz
(vers.
c491q)
©
2020
(b)
TRUE
or
FALSE.
If
g
is
a
continuous
function
on
[4,16],
g’
exists
on
(4,16)
with
g(4)
=4
=g(16),
and
g(10)
=
19,
then
there
are
numbers
c
and
d,
4
<c<d
<
16,
such
that
g’(c)
+
g’(d)
=0.
Rutgers
Quiz
(vers.
c491q)
©
2020
*
True
X
False
Rutgers
Quiz
(vers.
c491q)
©
2020
Question
is
complete.
Tap
on
the red
indicators
to
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incorrect
answers.
@
2
Rutgers
Quiz
(vers.
c491q)
©
2020
The
Mean
Value
Theorem
says
that
the
graph
of
y
=f(x)
must
have
at
least
one
x-value
c
in
the
open
interval
(
-
4,4)
where
the
tangent
line
to
the
graph
at
x=c
has
3
slope
3"
Rutgers
Quiz
(vers.
c491q)
©
2020
(b)
TRUE
or
FALSE.
If
g
is
a
continuous
function
on
[4,16],
g’
exists
on
(4,16)
with
g(4)
=4
=g(16),
and
g(10)
=
19,
then
there
are
numbers
¢
and
d,
4
<c<d
<
16,
such
that
g’(c)
+
g’(d)
=
0.
Rutgers
Quiz
(vers.
c491q)
©
2020
*
True
X
False
Rutgers
Quiz
(vers.
c491q)
©
2020
(c)
TRUE
or
FALSE.
There
exists
a
polynomial
h(x)
such
that
h’’(x)
<0
for
all
x
with
h’(11)
=5
and
h’(13)
=6.
Rutgers
Quiz
(vers.
c491q)
©
2020
X
True
*
False
Rutgers
Quiz
(vers.
c491q)
©
2020
RUTGERS
UNIV.
OFFICIAL
MATH
151
MIDTERM
QUIZ
©
2020
Question
is
complete.
Tap
on
the red
indicators
to
see
incorrect
answers.
@