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Math Problems
Calculus
Intermediate Value Theorem
Let
f
f
f
be a continuous function on the closed interval
[
1
,
5
]
[1,5]
[
1
,
5
]
, where
f
(
1
)
=
1
f(1)=1
f
(
1
)
=
1
and
f
(
5
)
=
−
3
f(5)=-3
f
(
5
)
=
−
3
.
\newline
Which of the following is guaranteed by the Intermediate Value Theorem?
\newline
Choose
1
1
1
answer:
\newline
(A)
f
(
c
)
=
2
f(c)=2
f
(
c
)
=
2
for at least one
c
c
c
between
1
1
1
and
5
5
5
\newline
(B)
f
(
c
)
=
−
2
f(c)=-2
f
(
c
)
=
−
2
for at least one
c
c
c
between
−
3
-3
−
3
and
1
1
1
\newline
(C)
f
(
c
)
=
−
2
f(c)=-2
f
(
c
)
=
−
2
for at least one
c
c
c
between
1
1
1
and
5
5
5
\newline
(D)
f
(
c
)
=
2
f(c)=2
f
(
c
)
=
2
for at least one
c
c
c
between
−
3
-3
−
3
and
1
1
1
Get tutor help
Let
g
g
g
be a continuous function on the closed interval
[
−
1
,
3
]
[-1,3]
[
−
1
,
3
]
, where
g
(
−
1
)
=
−
2
g(-1)=-2
g
(
−
1
)
=
−
2
and
g
(
3
)
=
−
5
g(3)=-5
g
(
3
)
=
−
5
.
\newline
Which of the following is guaranteed by the Intermediate Value Theorem?
\newline
Choose
1
1
1
answer:
\newline
(A)
g
(
c
)
=
−
3
g(c)=-3
g
(
c
)
=
−
3
for at least one
c
c
c
between
−
5
-5
−
5
and
−
2
-2
−
2
\newline
(B)
g
(
c
)
=
0
g(c)=0
g
(
c
)
=
0
for at least one
c
c
c
between
−
5
-5
−
5
and
−
2
-2
−
2
\newline
(C)
g
(
c
)
=
0
g(c)=0
g
(
c
)
=
0
for at least one
c
c
c
between
−
1
-1
−
1
and
3
3
3
\newline
(D)
g
(
c
)
=
−
3
g(c)=-3
g
(
c
)
=
−
3
for at least one
c
c
c
between
−
1
-1
−
1
and
3
3
3
Get tutor help
Let
\newline
f
(
x
)
=
{
ln
(
−
x
)
+
3
for
x
<
−
3
ln
(
−
x
+
3
)
for
−
3
≤
x
<
3
f(x)=\left\{\begin{array}{ll} \ln (-x)+3 & \text { for } x<-3 \\ \ln (-x+3) & \text { for }-3 \leq x<3 \end{array}\right.
f
(
x
)
=
{
ln
(
−
x
)
+
3
ln
(
−
x
+
3
)
for
x
<
−
3
for
−
3
≤
x
<
3
\newline
Is
f
f
f
continuous at
x
=
−
3
x=-3
x
=
−
3
?
\newline
Choose
1
1
1
answer:
\newline
(A) Yes
\newline
(B)
N
o
\mathrm{No}
No
Get tutor help
Let
\newline
f
(
x
)
=
{
ln
(
x
)
for
0
<
x
≤
2
x
2
ln
(
2
)
for
x
>
2
f(x)=\left\{\begin{array}{ll}\ln (x) & \text { for } 0<x \leq 2 \\ x^{2} \ln (2) & \text { for } x>2\end{array}\right.
f
(
x
)
=
{
ln
(
x
)
x
2
ln
(
2
)
for
0
<
x
≤
2
for
x
>
2
\newline
Is
f
f
f
continuous at
x
=
2
x=2
x
=
2
?
\newline
Choose
1
1
1
answer:
\newline
(A) Yes
\newline
(B)
N
o
\mathrm{No}
No
Get tutor help
Let
\newline
g
(
x
)
=
{
1
cos
(
x
)
for
−
π
2
<
x
<
0
cos
(
x
+
π
)
for
x
≥
0
g(x)=\left\{\begin{array}{ll} \frac{1}{\cos (x)} & \text { for }-\frac{\pi}{2}<x<0 \\ \cos (x+\pi) & \text { for } x \geq 0 \end{array}\right.
g
(
x
)
=
{
c
o
s
(
x
)
1
cos
(
x
+
π
)
for
−
2
π
<
x
<
0
for
x
≥
0
\newline
Is
g
g
g
continuous at
x
=
0
x=0
x
=
0
?
\newline
Choose
1
1
1
answer:
\newline
(A) Yes
\newline
(B) No
Get tutor help
Let
\newline
h
(
x
)
=
{
2
x
−
1
for
x
<
1
2
1
−
x
for
x
≥
1
h(x)=\left\{\begin{array}{ll} 2^{x-1} & \text { for } x<1 \\ 2^{1-x} & \text { for } x \geq 1 \end{array}\right.
h
(
x
)
=
{
2
x
−
1
2
1
−
x
for
x
<
1
for
x
≥
1
\newline
Is
h
h
h
continuous at
x
=
1
x=1
x
=
1
?
\newline
Choose
1
1
1
answer:
\newline
(A) Yes
\newline
(B) No
Get tutor help
Let
\newline
g
(
x
)
=
{
(
x
−
2
)
2
for
x
≤
2
2
−
x
2
for
x
>
2
g(x)=\left\{\begin{array}{ll} (x-2)^{2} & \text { for } x \leq 2 \\ 2-x^{2} & \text { for } x>2 \end{array}\right.
g
(
x
)
=
{
(
x
−
2
)
2
2
−
x
2
for
x
≤
2
for
x
>
2
\newline
Is
g
g
g
continuous at
x
=
2
x=2
x
=
2
?
\newline
Choose
1
1
1
answer:
\newline
(A) Yes
\newline
(B)
N
o
\mathrm{No}
No
Get tutor help
Let
\newline
g
(
x
)
=
{
e
x
for
x
≤
−
1
−
e
−
x
for
x
>
−
1
g(x)=\left\{\begin{array}{ll} e^{x} & \text { for } x \leq-1 \\ -e^{-x} & \text { for } x>-1 \end{array}\right.
g
(
x
)
=
{
e
x
−
e
−
x
for
x
≤
−
1
for
x
>
−
1
\newline
Is
g
g
g
continuous at
x
=
−
1
x=-1
x
=
−
1
?
\newline
Choose
1
1
1
answer:
\newline
(A) Yes
\newline
(B) No
Get tutor help
Let
\newline
h
(
x
)
=
{
e
2
x
for
x
<
0
e
5
x
for
x
≥
0
h(x)=\left\{\begin{array}{ll} e^{2 x} & \text { for } x<0 \\ e^{5 x} & \text { for } x \geq 0 \end{array}\right.
h
(
x
)
=
{
e
2
x
e
5
x
for
x
<
0
for
x
≥
0
\newline
Is
h
h
h
continuous at
x
=
0
x=0
x
=
0
?
\newline
Choose
1
1
1
answer:
\newline
(A) Yes
\newline
(B) No
Get tutor help
Let
\newline
h
(
x
)
=
{
6
x
+
5
for
−
5
<
x
<
−
3
x
2
−
6
for
x
≥
−
3
h(x)=\left\{\begin{array}{cl}\frac{6}{x+5} & \text { for }-5<x<-3 \\ x^{2}-6 & \text { for } x \geq-3\end{array}\right.
h
(
x
)
=
{
x
+
5
6
x
2
−
6
for
−
5
<
x
<
−
3
for
x
≥
−
3
\newline
Is
h
h
h
continuous at
x
=
−
3
x=-3
x
=
−
3
?
\newline
Choose
1
1
1
answer:
\newline
(A) Yes
\newline
(B) No
Get tutor help
Let
\newline
g
(
x
)
=
{
sin
(
x
)
for
x
<
0
x
2
for
x
≥
0
g(x)=\left\{\begin{array}{ll}\sin (x) & \text { for } x<0 \\ x^{2} & \text { for } x \geq 0\end{array}\right.
g
(
x
)
=
{
sin
(
x
)
x
2
for
x
<
0
for
x
≥
0
\newline
Is
g
g
g
continuous at
x
=
0
x=0
x
=
0
?
\newline
Choose
1
1
1
answer:
\newline
(A) Yes
\newline
(B) No
Get tutor help
Let
\newline
h
(
x
)
=
{
−
x
+
1
for
x
<
1
2
x
for
x
≥
1
h(x)=\left\{\begin{array}{ll} \sqrt{-x+1} & \text { for } x<1 \\ \sqrt{2 x} & \text { for } x \geq 1 \end{array}\right.
h
(
x
)
=
{
−
x
+
1
2
x
for
x
<
1
for
x
≥
1
\newline
Is
h
h
h
continuous at
x
=
1
x=1
x
=
1
?
\newline
Choose
1
1
1
answer:
\newline
(A) Yes
\newline
(B) No
Get tutor help
Let
\newline
f
(
x
)
=
{
2
x
2
for
x
≤
−
1
x
+
3
cos
(
x
+
1
)
for
−
1
<
x
<
π
−
2
2
f(x)=\left\{\begin{array}{ll}\frac{2}{x^{2}} & \text { for } x \leq-1 \\ \frac{x+3}{\cos (x+1)} & \text { for }-1<x<\frac{\pi-2}{2}\end{array}\right.
f
(
x
)
=
{
x
2
2
c
o
s
(
x
+
1
)
x
+
3
for
x
≤
−
1
for
−
1
<
x
<
2
π
−
2
\newline
Is
f
f
f
continuous at
x
=
−
1
x=-1
x
=
−
1
?
\newline
Choose
1
1
1
answer:
\newline
(A) Yes
\newline
(B)
N
o
\mathrm{No}
No
Get tutor help
Let
\newline
h
(
x
)
=
{
sin
(
x
)
for
x
<
0
x
+
π
for
x
≥
0
h(x)=\left\{\begin{array}{ll} \sin (x) & \text { for } x<0 \\ \sqrt{x+\pi} & \text { for } x \geq 0 \end{array}\right.
h
(
x
)
=
{
sin
(
x
)
x
+
π
for
x
<
0
for
x
≥
0
\newline
Is
h
h
h
continuous at
x
=
0
x=0
x
=
0
?
\newline
Choose
1
1
1
answer:
\newline
(A) Yes
\newline
(B)
N
o
\mathrm{No}
No
Get tutor help
Let
\newline
g
(
x
)
=
{
5
−
x
for
x
≤
3
2
e
3
−
x
for
x
>
3
g(x)=\left\{\begin{array}{ll}5-x & \text { for } x \leq 3 \\ 2 e^{3-x} & \text { for } x>3\end{array}\right.
g
(
x
)
=
{
5
−
x
2
e
3
−
x
for
x
≤
3
for
x
>
3
\newline
Is
g
g
g
continuous at
x
=
3
x=3
x
=
3
?
\newline
Choose
1
1
1
answer:
\newline
(A) Yes
\newline
(B)
N
o
\mathrm{No}
No
Get tutor help
Let
\newline
g
(
x
)
=
{
1
x
for
x
≤
−
2
−
cos
(
x
+
2
)
2
for
x
>
−
2
g(x)=\left\{\begin{array}{ll} \frac{1}{x} & \text { for } x \leq-2 \\ -\frac{\cos (x+2)}{2} & \text { for } x>-2 \end{array}\right.
g
(
x
)
=
{
x
1
−
2
c
o
s
(
x
+
2
)
for
x
≤
−
2
for
x
>
−
2
\newline
Is
g
g
g
continuous at
x
=
−
2
x=-2
x
=
−
2
?
\newline
Choose
1
1
1
answer:
\newline
(A) Yes
\newline
(B) No
Get tutor help
Let
\newline
h
(
x
)
=
{
e
x
+
2
for
x
≤
−
2
(
x
+
2
)
e
for
x
>
−
2
h(x)=\left\{\begin{array}{ll} e^{x+2} & \text { for } x \leq-2 \\ (x+2)^{e} & \text { for } x>-2 \end{array}\right.
h
(
x
)
=
{
e
x
+
2
(
x
+
2
)
e
for
x
≤
−
2
for
x
>
−
2
\newline
Is
h
h
h
continuous at
x
=
−
2
x=-2
x
=
−
2
?
\newline
Choose
1
1
1
answer:
\newline
(A) Yes
\newline
(B)
N
o
\mathrm{No}
No
Get tutor help
Let
\newline
f
(
x
)
=
{
log
(
3
x
)
for
0
<
x
<
3
(
4
−
x
)
log
(
9
)
for
x
≥
3
f(x)=\left\{\begin{array}{ll}\log (3 x) & \text { for } 0<x<3 \\ (4-x) \log (9) & \text { for } x \geq 3\end{array}\right.
f
(
x
)
=
{
lo
g
(
3
x
)
(
4
−
x
)
lo
g
(
9
)
for
0
<
x
<
3
for
x
≥
3
\newline
Is
f
f
f
continuous at
x
=
3
x=3
x
=
3
?
\newline
Choose
1
1
1
answer:
\newline
(A) Yes
\newline
(B) No
Get tutor help
Which of the following functions are continuous for all real numbers?
\newline
g
(
x
)
=
ln
(
x
)
g(x)=\ln (x)
g
(
x
)
=
ln
(
x
)
\newline
f
(
x
)
=
1
x
f(x)=\frac{1}{x}
f
(
x
)
=
x
1
\newline
Choose
1
1
1
answer:
\newline
A)
g
g
g
only
\newline
(B)
f
f
f
only
\newline
(C) Both
g
g
g
and
f
f
f
\newline
D Neither
g
g
g
nor
f
f
f
Get tutor help
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