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Math Problems
Precalculus
Write equations of sine and cosine functions using properties
Let
u
=
−
5
i
+
2
j
u=-5i+2j
u
=
−
5
i
+
2
j
and
v
=
−
5
i
−
5
j
v=-5i-5j
v
=
−
5
i
−
5
j
. Decompose
u
u
u
into two vectors
u
1
u_{1}
u
1
and
u
2
u_{2}
u
2
, where
u
1
u_{1}
u
1
is parallel to
v
v
v
and
u
2
u_{2}
u
2
is orthogonal to
v
v
v
. Write your answers in the form
a
i
+
b
j
ai+bj
ai
+
bj
.
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Find the average value of the functions on the given interval.
\newline
Average value of
f
(
x
)
=
x
f(x)=x
f
(
x
)
=
x
on
[
5
,
13
]
[5,13]
[
5
,
13
]
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Let
f
f
f
be a differentiable function such that
f
(
2
)
=
1
f(2)=1
f
(
2
)
=
1
and
f
′
(
x
)
=
sin
(
x
2
−
5
)
f^{\prime}(x)=\sin \left(x^{2}-5\right)
f
′
(
x
)
=
sin
(
x
2
−
5
)
.
\newline
What is the value of
f
(
6
)
f(6)
f
(
6
)
? Use a graphing calculator and round your answer to three decimal places.
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A study of urban density shows that the population density
C
\mathrm{C}
C
(in persons
/
k
m
2
/ \mathrm{km}^{2}
/
km
2
) is related to the distance a (in
k
m
\mathrm{km}
km
) from the city centre by
log
e
C
=
log
e
z
−
v
a
+
t
a
4
\log _{e} \mathrm{C}=\log _{e} \mathrm{z}-\mathrm{va}+\mathrm{ta}^{4}
lo
g
e
C
=
lo
g
e
z
−
va
+
ta
4
, where
z
,
v
\mathrm{z}, \mathrm{v}
z
,
v
, and
t
\mathrm{t}
t
are positive constants. Solve for
C
\mathrm{C}
C
as a function of a.
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The formula for the perimeter of a rectangle is
P
=
2
l
+
2
w
P=2 l+2 w
P
=
2
l
+
2
w
. Solve the formula for
l
l
l
in terms of
P
P
P
and
w
w
w
.
\newline
Answer:
l
=
l=
l
=
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A formula for converting degrees Celsius
(
C
)
(C)
(
C
)
to degrees Fahrenheit
(
F
)
(F)
(
F
)
is given by the formula
F
=
9
5
C
+
32
F=\frac{9}{5} C+32
F
=
5
9
C
+
32
. Solve the formula for
C
C
C
in terms of
F
F
F
.
\newline
Answer:
C
=
C=
C
=
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A formula to find the period,
T
T
T
, of a cyclic phenomena based on the frequency,
f
f
f
, is
T
=
1
f
T=\frac{1}{f}
T
=
f
1
. Solve the formula for
f
f
f
in terms of
T
T
T
.
\newline
Answer:
f
=
f=
f
=
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The formula for the perimeter of a rectangle is
P
=
2
l
+
2
w
P=2 l+2 w
P
=
2
l
+
2
w
. Solve the formula for
w
w
w
in terms of
P
P
P
and
l
l
l
.
\newline
Answer:
w
=
w=
w
=
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