Mixing
Application of derivatives on trig functions
$begingroup$
The forward and backward movement of an air molecule in still air during the transmission of a middle C note is given by
$$x = Asin (2272t)$$ where $t$ is the time in seconds and $A$ is the maximum displacement of the molecule. $A$ is related to the loudness of the sound.
Q: How many times does the molecule have a velocity of zero in 1 second?
So I have answered this question in two ways but they give me different answers and I would like to know why one works and the other one doesn't.
Method 1:
- Derive the function to get $x'=2272Acos(2272t)$
- Calculated the frequency and got a frequency of approx 361
- since cosine has two zeros per period: $361cdot 2 = 723$ times molecule has zero velocity in 1 second.
Method 2:
- Derived function as method 1.
- since cosine has zeros at every $npi/2$, equated $cos(2272t)=npi/2$
- subbed $t=1$ and solved for $n$ and got $1446$ which is double method 1's answer.
The correct answer is method 1 but could anyone clarify why method 2 does not give me the same answer?
derivatives trigonometry
edited Jan 29 at 12:16
Robert Z
101k1071144
asked Jan 29 at 12:06
user415903user415903
295
$endgroup$
add a comment |
$begingroup$
The forward and backward movement of an air molecule in still air during the transmission of a middle C note is given by
$$x = Asin (2272t)$$ where $t$ is the time in seconds and $A$ is the maximum displacement of the molecule. $A$ is related to the loudness of the sound.
Q: How many times does the molecule have a velocity of zero in 1 second?
So I have answered this question in two ways but they give me different answers and I would like to know why one works and the other one doesn't.
Method 1:
- Derive the function to get $x'=2272Acos(2272t)$
- Calculated the frequency and got a frequency of approx 361
- since cosine has two zeros per period: $361cdot 2 = 723$ times molecule has zero velocity in 1 second.
Method 2:
- Derived function as method 1.
- since cosine has zeros at every $npi/2$, equated $cos(2272t)=npi/2$
- subbed $t=1$ and solved for $n$ and got $1446$ which is double method 1's answer.
The correct answer is method 1 but could anyone clarify why method 2 does not give me the same answer?
derivatives trigonometry
edited Jan 29 at 12:16
Robert Z
101k1071144
asked Jan 29 at 12:06
user415903user415903
295
$endgroup$
add a comment |
$begingroup$
The forward and backward movement of an air molecule in still air during the transmission of a middle C note is given by
$$x = Asin (2272t)$$ where $t$ is the time in seconds and $A$ is the maximum displacement of the molecule. $A$ is related to the loudness of the sound.
Q: How many times does the molecule have a velocity of zero in 1 second?
So I have answered this question in two ways but they give me different answers and I would like to know why one works and the other one doesn't.
Method 1:
- Derive the function to get $x'=2272Acos(2272t)$
- Calculated the frequency and got a frequency of approx 361
- since cosine has two zeros per period: $361cdot 2 = 723$ times molecule has zero velocity in 1 second.
Method 2:
- Derived function as method 1.
- since cosine has zeros at every $npi/2$, equated $cos(2272t)=npi/2$
- subbed $t=1$ and solved for $n$ and got $1446$ which is double method 1's answer.
The correct answer is method 1 but could anyone clarify why method 2 does not give me the same answer?
derivatives trigonometry
edited Jan 29 at 12:16
Robert Z
101k1071144
asked Jan 29 at 12:06
user415903user415903
295
$endgroup$
The forward and backward movement of an air molecule in still air during the transmission of a middle C note is given by
$$x = Asin (2272t)$$ where $t$ is the time in seconds and $A$ is the maximum displacement of the molecule. $A$ is related to the loudness of the sound.
Q: How many times does the molecule have a velocity of zero in 1 second?
So I have answered this question in two ways but they give me different answers and I would like to know why one works and the other one doesn't.
Method 1:
- Derive the function to get $x'=2272Acos(2272t)$
- Calculated the frequency and got a frequency of approx 361
- since cosine has two zeros per period: $361cdot 2 = 723$ times molecule has zero velocity in 1 second.
Method 2:
- Derived function as method 1.
- since cosine has zeros at every $npi/2$, equated $cos(2272t)=npi/2$
- subbed $t=1$ and solved for $n$ and got $1446$ which is double method 1's answer.
The correct answer is method 1 but could anyone clarify why method 2 does not give me the same answer?
derivatives trigonometry
derivatives trigonometry
edited Jan 29 at 12:16
Robert Z
101k1071144
asked Jan 29 at 12:06
user415903user415903
295
edited Jan 29 at 12:16
Robert Z
101k1071144
asked Jan 29 at 12:06
user415903user415903
295
edited Jan 29 at 12:16
Robert Z
101k1071144
edited Jan 29 at 12:16
Robert Z
101k1071144
edited Jan 29 at 12:16
Robert Z
101k1071144
101k1071144
asked Jan 29 at 12:06
user415903user415903
295
asked Jan 29 at 12:06
user415903user415903
295
asked Jan 29 at 12:06
user415903user415903
295
295
add a comment |
add a comment |
1 Answer
1
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$begingroup$
The function $cos(x)$ has a zero at $x=frac{pi}{2}+npi$ for any integer $n$. There are two zeros per period $T=2pi$. In saying that the cosine has a zeros at every $npi/2$, you doubled the number of such zeros.
answered Jan 29 at 12:11
Robert ZRobert Z
101k1071144
$endgroup$
$begingroup$
ah, such a silly error. Thank you for the clarification!
$endgroup$
– user415903
Jan 29 at 12:21
add a comment |
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1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
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active
oldest
votes
$begingroup$
The function $cos(x)$ has a zero at $x=frac{pi}{2}+npi$ for any integer $n$. There are two zeros per period $T=2pi$. In saying that the cosine has a zeros at every $npi/2$, you doubled the number of such zeros.
answered Jan 29 at 12:11
Robert ZRobert Z
101k1071144
$endgroup$
$begingroup$
ah, such a silly error. Thank you for the clarification!
$endgroup$
– user415903
Jan 29 at 12:21
add a comment |
$begingroup$
The function $cos(x)$ has a zero at $x=frac{pi}{2}+npi$ for any integer $n$. There are two zeros per period $T=2pi$. In saying that the cosine has a zeros at every $npi/2$, you doubled the number of such zeros.
answered Jan 29 at 12:11
Robert ZRobert Z
101k1071144
$endgroup$
$begingroup$
ah, such a silly error. Thank you for the clarification!
$endgroup$
– user415903
Jan 29 at 12:21
add a comment |
$begingroup$
The function $cos(x)$ has a zero at $x=frac{pi}{2}+npi$ for any integer $n$. There are two zeros per period $T=2pi$. In saying that the cosine has a zeros at every $npi/2$, you doubled the number of such zeros.
answered Jan 29 at 12:11
Robert ZRobert Z
101k1071144
$endgroup$
The function $cos(x)$ has a zero at $x=frac{pi}{2}+npi$ for any integer $n$. There are two zeros per period $T=2pi$. In saying that the cosine has a zeros at every $npi/2$, you doubled the number of such zeros.
answered Jan 29 at 12:11
Robert ZRobert Z
101k1071144
edited Jan 29 at 12:18
answered Jan 29 at 12:11
Robert ZRobert Z
101k1071144
answered Jan 29 at 12:11
Robert ZRobert Z
101k1071144
answered Jan 29 at 12:11
Robert ZRobert Z
101k1071144
101k1071144
$begingroup$
ah, such a silly error. Thank you for the clarification!
$endgroup$
– user415903
Jan 29 at 12:21
add a comment |
$begingroup$
ah, such a silly error. Thank you for the clarification!
$endgroup$
– user415903
Jan 29 at 12:21
$begingroup$
ah, such a silly error. Thank you for the clarification!
$endgroup$
– user415903
Jan 29 at 12:21
$begingroup$
ah, such a silly error. Thank you for the clarification!
$endgroup$
– user415903
Jan 29 at 12:21
add a comment |
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