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Math Problems
Calculus
Find instantaneous rates of change
If
x
+
7
x+7
x
+
7
is a factor of the polynomial function
g
g
g
, what is the value of
g
(
7
)
g(7)
g
(
7
)
?
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Find the instantaneous rate of change of
g
(
x
)
=
−
18
g(x) = -18
g
(
x
)
=
−
18
at
x
=
−
12
x = -12
x
=
−
12
.
\newline
Write your answer as an integer or a fraction. Simplify any fractions.
\newline
____
\newline
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Find the instantaneous rate of change of
k
(
x
)
=
16
k(x) = 16
k
(
x
)
=
16
at
x
=
−
7
x = -7
x
=
−
7
.
\newline
Write your answer as an integer or a fraction. Simplify any fractions.
\newline
____
\newline
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Find the instantaneous rate of change of
k
(
x
)
=
19
k(x) = 19
k
(
x
)
=
19
at
x
=
−
7
x = -7
x
=
−
7
.
\newline
Write your answer as an integer or a fraction. Simplify any fractions.
\newline
____
\newline
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Find the instantaneous rate of change of
g
(
x
)
=
−
2
g(x) = -2
g
(
x
)
=
−
2
at
x
=
19
x = 19
x
=
19
.
\newline
Write your answer as an integer or a fraction. Simplify any fractions.
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Find the instantaneous rate of change of
k
(
x
)
=
−
12
k(x) = -12
k
(
x
)
=
−
12
at
x
=
−
3
x = -3
x
=
−
3
.
\newline
Write your answer as an integer or a fraction. Simplify any fractions.
\newline
____
\newline
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Find the instantaneous rate of change of
f
(
x
)
=
x
f(x) = x
f
(
x
)
=
x
at
x
=
−
13
x = -13
x
=
−
13
.
\newline
Write your answer as an integer or a fraction. Simplify any fractions.
\newline
____
\newline
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Find the instantaneous rate of change of
k
(
x
)
=
15
k(x) = 15
k
(
x
)
=
15
at
x
=
−
16
x = -16
x
=
−
16
.
\newline
Write your answer as an integer or a fraction. Simplify any fractions.
\newline
____
\newline
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Find the instantaneous rate of change of
g
(
x
)
=
x
g(x) = x
g
(
x
)
=
x
at
x
=
−
8
x = -8
x
=
−
8
.
\newline
Write your answer as an integer or a fraction. Simplify any fractions.
\newline
____
\newline
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Find the instantaneous rate of change of
g
(
x
)
=
x
g(x) = x
g
(
x
)
=
x
at
x
=
−
15
x = -15
x
=
−
15
.
\newline
Write your answer as an integer or a fraction. Simplify any fractions.
\newline
____
\newline
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Find the instantaneous rate of change of
k
(
x
)
=
−
6
k(x) = -6
k
(
x
)
=
−
6
at
x
=
−
11
x = -11
x
=
−
11
.
\newline
Write your answer as an integer or a fraction. Simplify any fractions.
\newline
____
\newline
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Find the instantaneous rate of change of
f
(
x
)
=
−
18
f(x) = -18
f
(
x
)
=
−
18
at
x
=
−
15
x = -15
x
=
−
15
.
\newline
Write your answer as an integer or a fraction. Simplify any fractions.
\newline
____
\newline
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Find the instantaneous rate of change of
k
(
x
)
=
x
k(x) = x
k
(
x
)
=
x
at
x
=
−
3
x = -3
x
=
−
3
.
\newline
Write your answer as an integer or a fraction. Simplify any fractions.
\newline
____
\newline
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Find the instantaneous rate of change of
k
(
x
)
=
−
6
k(x) = -6
k
(
x
)
=
−
6
at
x
=
−
18
x = -18
x
=
−
18
.
\newline
Write your answer as an integer or a fraction. Simplify any fractions.
\newline
____
\newline
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Find the instantaneous rate of change of
g
(
x
)
=
x
g(x) = x
g
(
x
)
=
x
at
x
=
16
x = 16
x
=
16
.
\newline
Write your answer as an integer or a fraction. Simplify any fractions.
\newline
____
\newline
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Find the instantaneous rate of change of
f
(
x
)
=
−
19
f(x) = -19
f
(
x
)
=
−
19
at
x
=
−
17
x = -17
x
=
−
17
.
\newline
Write your answer as an integer or a fraction. Simplify any fractions.
\newline
____
\newline
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Find the instantaneous rate of change of
g
(
x
)
=
14
g(x) = 14
g
(
x
)
=
14
at
x
=
20
x = 20
x
=
20
.
\newline
Write your answer as an integer or a fraction. Simplify any fractions.
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Find the instantaneous rate of change of
k
(
x
)
=
x
k(x) = x
k
(
x
)
=
x
at
x
=
13
x = 13
x
=
13
.
\newline
Write your answer as an integer or a fraction. Simplify any fractions.
\newline
____
\newline
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Find the instantaneous rate of change of
f
(
x
)
=
x
f(x) = x
f
(
x
)
=
x
at
x
=
6
x = 6
x
=
6
.
\newline
Write your answer as an integer or a fraction. Simplify any fractions.
\newline
____
\newline
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Find the instantaneous rate of change of
k
(
x
)
=
x
k(x) = x
k
(
x
)
=
x
at
x
=
3
x = 3
x
=
3
.
\newline
Write your answer as an integer or a fraction. Simplify any fractions.
\newline
____
\newline
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Find the instantaneous rate of change of
f
(
x
)
=
19
f(x) = 19
f
(
x
)
=
19
at
x
=
11
x = 11
x
=
11
.
\newline
Write your answer as an integer or a fraction. Simplify any fractions.
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One leg of a right triangle is decreasing at a rate of
5
5
5
kilometers per hour and the other leg of the triangle is increasing at a rate of
14
14
14
kilometers per hour.
\newline
At a certain instant, the decreasing leg is
3
3
3
kilometers and the increasing leg is
9
9
9
kilometers.
\newline
What is the rate of change of the area of the right triangle at that instant (in square kilometers per hour)?
\newline
Choose
1
1
1
answer:
\newline
(A)
1
1
1
.
5
5
5
\newline
(B)
111
111
111
\newline
(C)
−
1
-1
−
1
.
5
5
5
\newline
(D)
−
111
-111
−
111
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One leg of a right triangle is decreasing at a rate of
5
5
5
kilometers per hour and the other leg of the triangle is increasing at a rate of
14
14
14
kilometers per hour.
\newline
At a certain instant, the decreasing leg is
3
3
3
kilometers and the increasing leg is
9
9
9
kilometers.
\newline
What is the rate of change of the area of the right triangle at that instant (in square kilometers per hour)?
\newline
Choose
1
1
1
answer:
\newline
(A)
−
1
-1
−
1
.
5
5
5
\newline
(B)
111
111
111
\newline
(C)
1
1
1
.
5
5
5
\newline
(D)
−
111
-111
−
111
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The height of a rectangle is increasing at a rate of
11
11
11
centimeters per hour and the width of the rectangle is decreasing at a rate of
9
9
9
centimeters per hour.
\newline
At a certain instant, the height is
3
3
3
centimeters and the width is
8
8
8
centimeters.
\newline
What is the rate of change of the area of the rectangle at that instant (in square centimeters per hour)?
\newline
Choose
1
1
1
answer:
\newline
(A)
−
61
-61
−
61
\newline
(B)
61
61
61
\newline
(C)
99
99
99
\newline
(D)
−
99
-99
−
99
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How does
f
(
t
)
=
1
0
t
f(t) = 10^t
f
(
t
)
=
1
0
t
change over the interval from
t
=
9
t = 9
t
=
9
to
t
=
10
t = 10
t
=
10
?
\newline
Choices:
\newline
(A)
f
(
t
)
f(t)
f
(
t
)
increases by
1
,
000
%
1,000\%
1
,
000%
\newline
(B)
f
(
t
)
f(t)
f
(
t
)
decreases by a factor of
10
10
10
\newline
(C)
f
(
t
)
f(t)
f
(
t
)
increases by
10
%
10\%
10%
\newline
(D)
f
(
t
)
f(t)
f
(
t
)
increases by
t
=
9
t = 9
t
=
9
0
0
0
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Recall
π
\pi
π
is defined as the ratio of the circumference (say c) of a circle to its diameter (say d). That is
π
=
c
d
\pi=\frac{c}{d}
π
=
d
c
. This seems to contradict the fact that
π
\pi
π
is irrational. How will you resolve this contradiction?
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Show for all
N
∈
N
∖
{
0
}
N \in \mathbb{N}\setminus\{0\}
N
∈
N
∖
{
0
}
and all
a
≠
1
a \neq 1
a
=
1
:
\newline
1
1
−
a
=
a
N
1
−
a
+
∑
n
=
0
N
−
1
a
n
\frac{1}{1-a} = \frac{a^N}{1-a} + \sum_{n=0}^{N-1}a^n
1
−
a
1
=
1
−
a
a
N
+
∑
n
=
0
N
−
1
a
n
.
\newline
Use this result to find a formula for
\newline
∑
n
=
0
∞
a
n
\sum_{n=0}^{\infty}a^n
∑
n
=
0
∞
a
n
.
\newline
What assumption did you have to make to obtain a converging result?
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An investor in Treasury securities expects inflation to be
2.0
%
2.0\%
2.0%
in Year
1
1
1
,
2.6
%
2.6\%
2.6%
in Year
2
2
2
, and
3.75
%
3.75\%
3.75%
each year thereafter. Assume that the real risk-free rate is
1.95
%
1.95\%
1.95%
and that this rate will remain constant. Three-year Treasury securities yield
5.20
%
5.20\%
5.20%
, while
5
5
5
-year Treasury securities yield
6.00
%
6.00\%
6.00%
. What is the difference in the maturity risk premiums (MRPs) on the two securities; that is, what is
MRP
5
−
MRP
3
\text{MRP}_5 - \text{MRP}_3
MRP
5
−
MRP
3
?
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The rate of change
d
P
d
t
\frac{d P}{d t}
d
t
d
P
of the number of people infected by a disease is modeled by the following differential equation:
\newline
d
P
d
t
=
45
125404
P
(
800
−
P
)
\frac{d P}{d t}=\frac{45}{125404} P(800-P)
d
t
d
P
=
125404
45
P
(
800
−
P
)
\newline
At
t
=
0
t=0
t
=
0
, the number of people infected by the disease is
214
214
214
and is increasing at a rate of
45
45
45
people per hour. What is the limiting value for the total number of people infected by the disease as time increases?
\newline
Answer:
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The rate of change
d
P
d
t
\frac{d P}{d t}
d
t
d
P
of the number of bacteria in a tank is modeled by the following differential equation:
\newline
d
P
d
t
=
2
9849
P
(
598
−
P
)
\frac{d P}{d t}=\frac{2}{9849} P(598-P)
d
t
d
P
=
9849
2
P
(
598
−
P
)
\newline
At
t
=
0
t=0
t
=
0
, the number of bacteria in the tank is
196
196
196
and is increasing at a rate of
16
16
16
bacteria per minute. At what value of
P
P
P
does the graph of
P
(
t
)
P(t)
P
(
t
)
have an inflection point?
\newline
Answer:
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The rate of change
d
P
d
t
\frac{d P}{d t}
d
t
d
P
of the number of algae in a tank is modeled by the following differential equation:
\newline
d
P
d
t
=
2317
10614
P
(
1
−
P
662
)
\frac{d P}{d t}=\frac{2317}{10614} P\left(1-\frac{P}{662}\right)
d
t
d
P
=
10614
2317
P
(
1
−
662
P
)
\newline
At
t
=
0
t=0
t
=
0
, the number of algae in the tank is
174
174
174
and is increasing at a rate of
28
28
28
algae per minute. At what value of
P
P
P
is
P
(
t
)
P(t)
P
(
t
)
growing the fastest?
\newline
Answer:
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