1. If f(x,y) = c, where c is a constant then which of the following is true
Answers
Answer:
Chapter 5 True Or False and Multiple Choice Problems
Answer the following questions.
For each of the following ten statements answer TRUE or FALSE as appropriate:
If
f
is differentiable on
[
−
1
,
1
]
then
f
is continuous at
x
=
0
.
If
f
′
(
x
)
<
0
and
f
"
(
x
)
>
0
for all
x
then
f
is concave down.
The general antiderivative of
f
(
x
)
=
3
x
2
is
F
(
x
)
=
x
3
.
ln
x
exists for any
x
>
1
.
ln
x
=
π
has a unique solution.
e
−
x
is negative for some values of
x
.
ln
e
x
2
=
x
2
for all
x
.
f
(
x
)
=
|
x
|
is differentiable for all
x
.
tan
x
is defined for all
x
.
All critical points of
f
(
x
)
satisfy
f
′
(
x
)
=
0
.
Answer each of the following either TRUE or FALSE.
The function
f
(
x
)
=
{
3
+
sin
(
x
−
2
)
x
−
2
if
x
≠
2
3
if
x
=
2
is continuous at all real numbers
x
.
If
f
′
(
x
)
=
g
′
(
x
)
for
0
<
x
<
1
,
then
f
(
x
)
=
g
(
x
)
for
0
<
x
<
1
.
If
f
is increasing and
f
(
x
)
>
0
on
I
,
then
g
(
x
)
=
1
f
(
x
)
is decreasing on
I
.
There exists a function
f
such that
f
(
1
)
=
−
2
,
f
(
3
)
=
0
,
and
f
′
(
x
)
>
1
for all
x
.
If
f
is differentiable, then
d
d
x
f
(
√
x
)
=
f
′
(
x
)
2
√
x
.
d
d
x
10
x
=
x
10
x
−
1
Let
e
=
exp
(
1
)
as usual. If
y
=
e
2
then
y
′
=
2
e
.
If
f
(
x
)
and
g
(
x
)
are differentiable for all
x
,
then
d
d
x
f
(
g
(
x
)
)
=
f
′
(
g
(
x
)
)
g
′
(
x
)
.
If
g
(
x
)
=
x
5
,
then
lim
x
→
2
g
(
x
)
−
g
(
2
)
x
−
2
=
80
.
An equation of the tangent line to the parabola
y
=
x
2
at
(
−
2
,
4
)
is
y
−
4
=
2
x
(
x
+
2
)
.
d
d
x
tan
2
x
=
d
d
x
sec
2
x
For all real values of
x
we have that
d
d
x
|
x
2
+
x
|
=
|
2
x
+
1
|
.
If
f
is one-to-one then
f
−
1
(
x
)
=
1
f
(
x
)
.
If
x
>
0
,
then
(
ln
x
)
6
=
6
ln
x
.
If
lim
x
→
5
f
(
x
)
=
0
and
lim
x
→
5
g
(
x
)
=
0
,
then
lim
x
→
5
f
(
x
)
g
(
x
)
does not exist.
If the line
x
=
1
is a vertical asymptote of
y
=
f
(
x
)
,
then
f
is not defined at 1.
If
f
′
(
c
)
does not exist and
f
′
(
x
)
changes from positive to negative as
x
increases through
c
,
then
f
(
x
)
has a local minimum at
x
=
c
.
√
a
2
=
a
for all
a
>
0
.
If
f
(
c
)
exists but
f
′
(
c
)
does not exist, then
x
=
c
is a critical point of
f
(
x
)
.