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Math Problems
Calculus
Find derivatives of logarithmic functions
f
(
x
)
=
−
x
2
+
7
x
−
10
6
x
+
11
f(x)=\frac{-x^{2}+7 x-10}{6 x+11}
f
(
x
)
=
6
x
+
11
−
x
2
+
7
x
−
10
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The functions
f
f
f
and
g
g
g
are given by
\newline
f
(
x
)
=
1
2
log
10
(
x
−
3
)
g
(
x
)
=
3
ln
(
x
+
2
)
\begin{array}{l} f(x)=\frac{1}{2} \log _{10}(x-3) \\ g(x)=3 \ln (x+2) \end{array}
f
(
x
)
=
2
1
lo
g
10
(
x
−
3
)
g
(
x
)
=
3
ln
(
x
+
2
)
\newline
(A) Solve
f
(
x
)
=
1
f(x)=1
f
(
x
)
=
1
for values of
x
x
x
in the domain of
f
f
f
.
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Find the area under the graph of
f
f
f
over the interval
[
−
2
,
4
]
[-2,4]
[
−
2
,
4
]
.
\newline
f
(
x
)
=
{
x
2
+
6
x
≤
2
5
x
x
>
2
f(x)=\left\{\begin{array}{ll} x^{2}+6 & x \leq 2 \\ 5 x & x>2 \end{array}\right.
f
(
x
)
=
{
x
2
+
6
5
x
x
≤
2
x
>
2
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What value of
y
y
y
is a solution to this equation?
\newline
3
y
=
3
3y = 3
3
y
=
3
\newline
Choices:
\newline
(A)
y
=
1
y = 1
y
=
1
\newline
(B)
y
=
3
y = 3
y
=
3
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If
y
(
x
)
=
(
x
x
x
)
,
x
>
0
y(x)=\left(x^{x^{x}}\right), x>0
y
(
x
)
=
(
x
x
x
)
,
x
>
0
then
d
2
x
d
y
2
+
20
\frac{d^{2} x}{d y^{2}}+20
d
y
2
d
2
x
+
20
at
x
=
1
x=1
x
=
1
is equal to:
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3
3
3
.
g
(
x
)
=
ln
(
cos
(
x
3
)
)
g(x)=\ln \left(\cos \left(x^{3}\right)\right)
g
(
x
)
=
ln
(
cos
(
x
3
)
)
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Differentiate
f
(
x
)
=
cos
(
ln
6
x
)
f(x)=\cos(\ln 6x)
f
(
x
)
=
cos
(
ln
6
x
)
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Find
k
′
(
x
)
k'(x)
k
′
(
x
)
.
\newline
k
(
x
)
=
e
x
(
−
x
3
5
)
k(x)=e^{x}(-x^{\frac{3}{5}})
k
(
x
)
=
e
x
(
−
x
5
3
)
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Find all critical points of the function
g
(
θ
)
=
sin
2
(
6
θ
)
.
g(\theta)=\sin^{2}(6\theta).
g
(
θ
)
=
sin
2
(
6
θ
)
.
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Find all critical points of the function
\newline
f
(
x
)
=
cos
−
1
(
x
)
+
3
x
f(x)=\cos^{-1}(x)+3x
f
(
x
)
=
cos
−
1
(
x
)
+
3
x
for
−
1
<
x
<
1.
-1 < x < 1.
−
1
<
x
<
1.
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What is the particular solution to the differential equation
d
y
d
x
=
(
1
−
y
)
2
x
+
1
\frac{dy}{dx}=\frac{(1-y)^{2}}{x+1}
d
x
d
y
=
x
+
1
(
1
−
y
)
2
with the initial condition
y
(
0
)
=
5
y(0)=5
y
(
0
)
=
5
?
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What is the particular solution to the differential equation
d
y
d
x
\frac{dy}{dx}
d
x
d
y
=
4
x
2
e
2
y
\frac{4}{x^{2}e^{2y}}
x
2
e
2
y
4
with the initial condition
y
(
1
)
=
0
y(1)=0
y
(
1
)
=
0
?
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Find the inverse of the function
\newline
f
(
x
)
=
3
x
2
−
27
,
x
≥
0
f(x)=3x^{2}-27,\,x \geq 0
f
(
x
)
=
3
x
2
−
27
,
x
≥
0
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Find the derivative of
\newline
f
(
x
)
=
−
4
x
(
3
x
4
)
.
f(x)=-4x(3^{x^{4}}).
f
(
x
)
=
−
4
x
(
3
x
4
)
.
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Integrate
1
1
+
x
2
\frac{1}{1+x^2}
1
+
x
2
1
for a limit
[
0
,
1
]
[0,1]
[
0
,
1
]
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What is the particular solution to the differential equation
\newline
d
y
d
x
=
(
x
−
3
)
y
\frac{dy}{dx}=(x-3)y
d
x
d
y
=
(
x
−
3
)
y
with the initial condition
\newline
y
(
0
)
=
2
y(0)=2
y
(
0
)
=
2
?
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What is the particular solution to the differential equation
d
y
d
x
=
3
+
y
1
−
2
x
\frac{dy}{dx}=\frac{3+y}{1-2x}
d
x
d
y
=
1
−
2
x
3
+
y
with the initial condition
y
(
0
)
=
0
y(0)=0
y
(
0
)
=
0
?
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Find the interval where
k
(
x
)
=
x
2
e
x
k(x)=x^{2} e^{x}
k
(
x
)
=
x
2
e
x
is concave down.
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Find the derivative of
g
(
x
)
=
3
cos
−
1
(
x
)
g(x)=3 \cos ^{-1}(x)
g
(
x
)
=
3
cos
−
1
(
x
)
at the point
x
=
1
4
x=\frac{1}{4}
x
=
4
1
.
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f
′
(
x
)
=
−
4
x
2
and
f
(
2
)
=
4
f
(
1
)
=
\begin{array}{l}f^{\prime}(x)=-\frac{4}{x^{2}} \text { and } f(2)=4 \\ f(1)=\end{array}
f
′
(
x
)
=
−
x
2
4
and
f
(
2
)
=
4
f
(
1
)
=
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f
(
x
)
=
3
x
+
2
m
(
x
)
=
3
x
h
(
x
)
=
3
x
2
−
5
x
+
4
f
(
m
(
h
(
1
)
)
)
=
?
\begin{array}{l}f(x)=3 x+2 \\ m(x)=\frac{3}{x} \\ h(x)=3 x^{2}-5 x+4 \\ f(m(h(1)))=?\end{array}
f
(
x
)
=
3
x
+
2
m
(
x
)
=
x
3
h
(
x
)
=
3
x
2
−
5
x
+
4
f
(
m
(
h
(
1
)))
=
?
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98
98
98
)
−
7
log
(
x
−
10
)
=
0
-7 \log (x-10)=0
−
7
lo
g
(
x
−
10
)
=
0
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h
(
x
)
=
4
x
h(x)=\sqrt{4x}
h
(
x
)
=
4
x
\newline
The function
h
h
h
is defined. What is the value of
h
(
9
)
h(9)
h
(
9
)
?
\newline
Choose
1
1
1
answer:
\newline
(A)
3
\text{(A)}\ 3
(A)
3
\newline
(B)
6
\text{(B)}\ 6
(B)
6
\newline
(C)
12
\text{(C)}\ 12
(C)
12
\newline
(D)
36
\text{(D)}\ 36
(D)
36
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Calculate the derivative of
\newline
f
(
x
)
=
sinh
(
x
7
)
.
f(x)=\sinh(x^{7}).
f
(
x
)
=
sinh
(
x
7
)
.
\newline
(Use symbolic notation and fractions where needed.)
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3
log
(
x
)
+
3
log
(
3
)
3\log(x)+3\log(3)
3
lo
g
(
x
)
+
3
lo
g
(
3
)
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lim
x
→
0
(
ln
(
cos
3
x
)
ln
(
cos
2
x
)
)
\lim_{x \to 0}\left(\frac{\ln(\cos 3x)}{\ln(\cos 2x)}\right)
x
→
0
lim
(
ln
(
cos
2
x
)
ln
(
cos
3
x
)
)
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Which value for the constant
c
c
c
makes
z
=
−
5
4
z=-\frac{5}{4}
z
=
−
4
5
an extraneous solution in the following equation?
\newline
\begin{align*} \sqrt{4z+9} &= cz+8 \ c &= \square \end{align*}
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Determine the following limit in simplest form. If the limit is infinite, state that the limit does not exist (DNE).
\newline
lim
x
→
∞
13
x
3
+
47
x
7
+
x
10
3
7
x
+
3
x
3
\lim _{x \rightarrow \infty} \frac{\sqrt[3]{13 x^{3}+47 x^{7}+x^{10}}}{7 x+3 x^{3}}
x
→
∞
lim
7
x
+
3
x
3
3
13
x
3
+
47
x
7
+
x
10
\newline
Answer:
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Determine the following limit in simplest form. If the limit is infinite, state that the limit does not exist (DNE).
\newline
lim
x
→
∞
30
x
8
−
64
x
12
3
9
x
4
+
7
\lim _{x \rightarrow \infty} \frac{\sqrt[3]{30 x^{8}-64 x^{12}}}{9 x^{4}+7}
x
→
∞
lim
9
x
4
+
7
3
30
x
8
−
64
x
12
\newline
Answer:
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Determine the following limit in simplest form. If the limit is infinite, state that the limit does not exist (DNE).
\newline
lim
x
→
∞
27
x
12
−
23
x
8
3
6
x
3
+
9
\lim _{x \rightarrow \infty} \frac{\sqrt[3]{27 x^{12}-23 x^{8}}}{6 x^{3}+9}
x
→
∞
lim
6
x
3
+
9
3
27
x
12
−
23
x
8
\newline
Answer:
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f
(
x
)
=
log
5
(
2
x
−
4
x
+
3
−
5
)
f(x)=\log_{5}\left(\frac{2x-4}{x+3}-5\right)
f
(
x
)
=
lo
g
5
(
x
+
3
2
x
−
4
−
5
)
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Find the domain of the following logarithmic function
\newline
f
(
x
)
=
log
5
(
2
x
−
4
x
+
3
−
5
)
f(x)=\log_{5}\left(\frac{2x-4}{x+3}-5\right)
f
(
x
)
=
lo
g
5
(
x
+
3
2
x
−
4
−
5
)
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Determine whether the function
f
(
x
)
f(x)
f
(
x
)
is continuous at
x
=
−
3
x=-3
x
=
−
3
.
\newline
f
(
x
)
=
{
18
−
4
x
2
,
x
>
−
3
−
9
+
3
x
,
x
≤
−
3
f(x)=\left\{\begin{array}{ll} 18-4 x^{2}, & x>-3 \\ -9+3 x, & x \leq-3 \end{array}\right.
f
(
x
)
=
{
18
−
4
x
2
,
−
9
+
3
x
,
x
>
−
3
x
≤
−
3
\newline
f
(
x
)
f(x)
f
(
x
)
is discontinuous at
x
=
−
3
x=-3
x
=
−
3
\newline
f
(
x
)
f(x)
f
(
x
)
is continuous at
x
=
−
3
x=-3
x
=
−
3
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Determine whether the function
f
(
x
)
f(x)
f
(
x
)
is continuous at
x
=
3
x=3
x
=
3
.
\newline
f
(
x
)
=
{
10
−
3
x
2
,
x
>
3
−
9
−
3
x
,
x
≤
3
f(x)=\left\{\begin{array}{ll} 10-3 x^{2}, & x>3 \\ -9-3 x, & x \leq 3 \end{array}\right.
f
(
x
)
=
{
10
−
3
x
2
,
−
9
−
3
x
,
x
>
3
x
≤
3
\newline
f
(
x
)
f(x)
f
(
x
)
is discontinuous at
x
=
3
x=3
x
=
3
\newline
f
(
x
)
f(x)
f
(
x
)
is continuous at
x
=
3
x=3
x
=
3
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Determine whether the function
f
(
x
)
f(x)
f
(
x
)
is continuous at
x
=
3
x=3
x
=
3
.
\newline
f
(
x
)
=
{
7
−
2
x
2
,
x
≤
3
−
8
−
x
,
x
>
3
f(x)=\left\{\begin{array}{ll} 7-2 x^{2}, & x \leq 3 \\ -8-x, & x>3 \end{array}\right.
f
(
x
)
=
{
7
−
2
x
2
,
−
8
−
x
,
x
≤
3
x
>
3
\newline
f
(
x
)
f(x)
f
(
x
)
is continuous at
x
=
3
x=3
x
=
3
\newline
f
(
x
)
f(x)
f
(
x
)
is discontinuous at
x
=
3
x=3
x
=
3
Get tutor help
Determine whether the function
f
(
x
)
f(x)
f
(
x
)
is continuous at
x
=
−
3
x=-3
x
=
−
3
.
\newline
f
(
x
)
=
{
15
−
x
2
,
x
<
3
15
−
2
x
,
x
≥
3
f(x)=\left\{\begin{array}{ll} 15-x^{2}, & x<3 \\ 15-2 x, & x \geq 3 \end{array}\right.
f
(
x
)
=
{
15
−
x
2
,
15
−
2
x
,
x
<
3
x
≥
3
\newline
f
(
x
)
f(x)
f
(
x
)
is continuous at
x
=
−
3
x=-3
x
=
−
3
\newline
f
(
x
)
f(x)
f
(
x
)
is discontinuous at
x
=
−
3
x=-3
x
=
−
3
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Determine whether the function
f
(
x
)
f(x)
f
(
x
)
is continuous at
x
=
2
x=2
x
=
2
.
\newline
f
(
x
)
=
{
1
−
3
x
2
,
x
>
2
−
5
−
3
x
,
x
≤
2
f(x)=\left\{\begin{array}{ll} 1-3 x^{2}, & x>2 \\ -5-3 x, & x \leq 2 \end{array}\right.
f
(
x
)
=
{
1
−
3
x
2
,
−
5
−
3
x
,
x
>
2
x
≤
2
\newline
f
(
x
)
f(x)
f
(
x
)
is discontinuous at
x
=
2
x=2
x
=
2
\newline
f
(
x
)
f(x)
f
(
x
)
is continuous at
x
=
2
x=2
x
=
2
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Determine whether the function
f
(
x
)
f(x)
f
(
x
)
is continuous at
x
=
−
2
x=-2
x
=
−
2
.
\newline
f
(
x
)
=
{
9
−
5
x
2
,
x
≤
−
2
−
5
+
3
x
,
x
>
−
2
f(x)=\left\{\begin{array}{ll} 9-5 x^{2}, & x \leq-2 \\ -5+3 x, & x>-2 \end{array}\right.
f
(
x
)
=
{
9
−
5
x
2
,
−
5
+
3
x
,
x
≤
−
2
x
>
−
2
\newline
f
(
x
)
f(x)
f
(
x
)
is discontinuous at
x
=
−
2
x=-2
x
=
−
2
\newline
f
(
x
)
f(x)
f
(
x
)
is continuous at
x
=
−
2
x=-2
x
=
−
2
Get tutor help
Determine whether the function
f
(
x
)
f(x)
f
(
x
)
is continuous at
x
=
3
x=3
x
=
3
.
\newline
f
(
x
)
=
{
7
+
x
2
,
x
≥
−
3
12
−
2
x
,
x
<
−
3
f(x)=\left\{\begin{array}{ll} 7+x^{2}, & x \geq-3 \\ 12-2 x, & x<-3 \end{array}\right.
f
(
x
)
=
{
7
+
x
2
,
12
−
2
x
,
x
≥
−
3
x
<
−
3
\newline
f
(
x
)
f(x)
f
(
x
)
is discontinuous at
x
=
3
x=3
x
=
3
\newline
f
(
x
)
f(x)
f
(
x
)
is continuous at
x
=
3
x=3
x
=
3
Get tutor help
Determine whether the function
f
(
x
)
f(x)
f
(
x
)
is continuous at
x
=
3
x=3
x
=
3
.
\newline
f
(
x
)
=
{
13
−
x
2
,
x
>
3
10
−
2
x
,
x
<
3
f(x)=\left\{\begin{array}{ll} 13-x^{2}, & x>3 \\ 10-2 x, & x<3 \end{array}\right.
f
(
x
)
=
{
13
−
x
2
,
10
−
2
x
,
x
>
3
x
<
3
\newline
f
(
x
)
f(x)
f
(
x
)
is continuous at
x
=
3
x=3
x
=
3
\newline
f
(
x
)
f(x)
f
(
x
)
is discontinuous at
x
=
3
x=3
x
=
3
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Determine whether the function
f
(
x
)
f(x)
f
(
x
)
is continuous at
x
=
2
x=2
x
=
2
.
\newline
f
(
x
)
=
{
6
+
3
x
2
,
x
≥
2
20
−
2
x
,
x
<
2
f(x)=\left\{\begin{array}{ll} 6+3 x^{2}, & x \geq 2 \\ 20-2 x, & x<2 \end{array}\right.
f
(
x
)
=
{
6
+
3
x
2
,
20
−
2
x
,
x
≥
2
x
<
2
\newline
f
(
x
)
f(x)
f
(
x
)
is continuous at
x
=
2
x=2
x
=
2
\newline
f
(
x
)
f(x)
f
(
x
)
is discontinuous at
x
=
2
x=2
x
=
2
Get tutor help
Find
(
f
∘
g
)
(
0
)
(f \circ g)(0)
(
f
∘
g
)
(
0
)
.
f
(
x
)
=
6
x
f(x) = 6x
f
(
x
)
=
6
x
,
g
(
x
)
=
x
2
+
4
x
g(x) = x^{2} + 4x
g
(
x
)
=
x
2
+
4
x
Get tutor help
Find the average value
f
ave
f_{\text{ave}}
f
ave
of the function
f
f
f
on the given interval.
\newline
f
(
x
)
=
3
x
2
+
4
x
,
[
−
1
,
2
]
f(x)=3x^{2}+4x, \quad [-1,2]
f
(
x
)
=
3
x
2
+
4
x
,
[
−
1
,
2
]
Get tutor help
y
=
ln
tan
(
π
4
+
x
2
)
y=\ln\tan\left(\frac{\pi}{4}+\frac{x}{2}\right)
y
=
ln
tan
(
4
π
+
2
x
)
Get tutor help
Find the derivative of the following function.
\newline
y
=
log
3
(
−
8
x
6
−
6
x
5
)
y=\log _{3}\left(-8 x^{6}-6 x^{5}\right)
y
=
lo
g
3
(
−
8
x
6
−
6
x
5
)
\newline
Answer:
y
′
=
y^{\prime}=
y
′
=
Get tutor help
Find the derivative of the following function.
\newline
y
=
log
5
(
x
3
−
2
x
2
)
y=\log _{5}\left(x^{3}-2 x^{2}\right)
y
=
lo
g
5
(
x
3
−
2
x
2
)
\newline
Answer:
y
′
=
y^{\prime}=
y
′
=
Get tutor help
Find the derivative of the following function.
\newline
y
=
log
3
(
8
x
3
+
x
2
)
y=\log _{3}\left(8 x^{3}+x^{2}\right)
y
=
lo
g
3
(
8
x
3
+
x
2
)
\newline
Answer:
y
′
=
y^{\prime}=
y
′
=
Get tutor help
Find the derivative of the following function.
\newline
y
=
log
3
(
3
x
2
+
8
x
)
y=\log _{3}\left(3 x^{2}+8 x\right)
y
=
lo
g
3
(
3
x
2
+
8
x
)
\newline
Answer:
y
′
=
y^{\prime}=
y
′
=
Get tutor help
Find the derivative of the following function.
\newline
y
=
log
2
(
−
9
x
3
)
y=\log _{2}\left(-9 x^{3}\right)
y
=
lo
g
2
(
−
9
x
3
)
\newline
Answer:
y
′
=
y^{\prime}=
y
′
=
Get tutor help
Let
y
=
ln
(
sin
(
x
)
)
y=\ln (\sin (x))
y
=
ln
(
sin
(
x
))
.
\newline
Find
d
y
d
x
\frac{d y}{d x}
d
x
d
y
.
\newline
Choose
1
1
1
answer:
\newline
(A)
cos
(
x
)
sin
(
x
)
\frac{\cos (x)}{\sin (x)}
s
i
n
(
x
)
c
o
s
(
x
)
\newline
(B)
1
sin
(
x
)
\frac{1}{\sin (x)}
s
i
n
(
x
)
1
\newline
(C)
1
cos
(
x
)
\frac{1}{\cos (x)}
c
o
s
(
x
)
1
\newline
(D)
ln
(
cos
(
x
)
)
\ln (\cos (x))
ln
(
cos
(
x
))
Get tutor help
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