Computation of the derivative of a certain cubic root function using only the definition of a derivative
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First post here, so hello everyone.
Here's the problem:
Find the first derivative of: $$sqrt[3]{sin(2x)}$$
But, you can only use the difference quotient...
(i.e. the limit of $frac{f(x+h)-f(x)}{h}$ as $h rightarrow 0$)
This is a particular problem given by my prof. It's not for credit or even an assignment, she gave it as an interestingly difficult problem with some twist and turns. I've worked on it for several hours now and would like to see how others approach solving it as I'm wrapped around an axel after using double-angle ID and clearing the cubes in the numerator. Any advice is appreciated.
Thanks.
calculus derivatives trigonometry
New contributor
$endgroup$
|
show 2 more comments
$begingroup$
First post here, so hello everyone.
Here's the problem:
Find the first derivative of: $$sqrt[3]{sin(2x)}$$
But, you can only use the difference quotient...
(i.e. the limit of $frac{f(x+h)-f(x)}{h}$ as $h rightarrow 0$)
This is a particular problem given by my prof. It's not for credit or even an assignment, she gave it as an interestingly difficult problem with some twist and turns. I've worked on it for several hours now and would like to see how others approach solving it as I'm wrapped around an axel after using double-angle ID and clearing the cubes in the numerator. Any advice is appreciated.
Thanks.
calculus derivatives trigonometry
New contributor
$endgroup$
2
$begingroup$
Can you give some context on why you want to do that with only the difference quotient? Surely you can replicate the proof of the chain rule, but it's not going to be pretty (although pointless all the more).
$endgroup$
– Klaus
10 hours ago
$begingroup$
Sure, Klaus. This is a particular problem given by my prof. It's not for credit of even an assignment, she gave it as an interestingly difficult problem with some twist and turns. I've worked on it for several hours now and would like to see how others approach solving it as I'm wrapped around an axel after using double-angle ID and clearing the cubes in the numerator. Any advice is appreciated.
$endgroup$
– Sawyer Benson
10 hours ago
$begingroup$
Use sin instead of sin in $rmLaTeX$! It's a built-in function to make it look nice :)
$endgroup$
– Cameron Williams
10 hours ago
$begingroup$
I think you'd have to use the generalized binomial theorem for what it's worth. It's been a while since I did non-integer powers with the limit definition so I might be wrong. The integer power case uses the regular binomial theorem+factoring so I think using the generalized binomial theorem is appropriate.
$endgroup$
– Cameron Williams
10 hours ago
1
$begingroup$
Thanks, Cameron! Unfortunately I cannot use that either. I know how to solve it using the chain rule and even L'Hopital's rule, however, the challenge it to solving it algebraically using nothing more than trig IDs (half angle) and trig limit rules (lim of sinx/x = 1). I know it's a pain, but that's the game! Thanks for all the help!
$endgroup$
– Sawyer Benson
10 hours ago
|
show 2 more comments
$begingroup$
First post here, so hello everyone.
Here's the problem:
Find the first derivative of: $$sqrt[3]{sin(2x)}$$
But, you can only use the difference quotient...
(i.e. the limit of $frac{f(x+h)-f(x)}{h}$ as $h rightarrow 0$)
This is a particular problem given by my prof. It's not for credit or even an assignment, she gave it as an interestingly difficult problem with some twist and turns. I've worked on it for several hours now and would like to see how others approach solving it as I'm wrapped around an axel after using double-angle ID and clearing the cubes in the numerator. Any advice is appreciated.
Thanks.
calculus derivatives trigonometry
New contributor
$endgroup$
First post here, so hello everyone.
Here's the problem:
Find the first derivative of: $$sqrt[3]{sin(2x)}$$
But, you can only use the difference quotient...
(i.e. the limit of $frac{f(x+h)-f(x)}{h}$ as $h rightarrow 0$)
This is a particular problem given by my prof. It's not for credit or even an assignment, she gave it as an interestingly difficult problem with some twist and turns. I've worked on it for several hours now and would like to see how others approach solving it as I'm wrapped around an axel after using double-angle ID and clearing the cubes in the numerator. Any advice is appreciated.
Thanks.
calculus derivatives trigonometry
calculus derivatives trigonometry
New contributor
New contributor
edited 21 mins ago
Jean Marie
29.8k42051
29.8k42051
New contributor
asked 10 hours ago
Sawyer BensonSawyer Benson
494
494
New contributor
New contributor
2
$begingroup$
Can you give some context on why you want to do that with only the difference quotient? Surely you can replicate the proof of the chain rule, but it's not going to be pretty (although pointless all the more).
$endgroup$
– Klaus
10 hours ago
$begingroup$
Sure, Klaus. This is a particular problem given by my prof. It's not for credit of even an assignment, she gave it as an interestingly difficult problem with some twist and turns. I've worked on it for several hours now and would like to see how others approach solving it as I'm wrapped around an axel after using double-angle ID and clearing the cubes in the numerator. Any advice is appreciated.
$endgroup$
– Sawyer Benson
10 hours ago
$begingroup$
Use sin instead of sin in $rmLaTeX$! It's a built-in function to make it look nice :)
$endgroup$
– Cameron Williams
10 hours ago
$begingroup$
I think you'd have to use the generalized binomial theorem for what it's worth. It's been a while since I did non-integer powers with the limit definition so I might be wrong. The integer power case uses the regular binomial theorem+factoring so I think using the generalized binomial theorem is appropriate.
$endgroup$
– Cameron Williams
10 hours ago
1
$begingroup$
Thanks, Cameron! Unfortunately I cannot use that either. I know how to solve it using the chain rule and even L'Hopital's rule, however, the challenge it to solving it algebraically using nothing more than trig IDs (half angle) and trig limit rules (lim of sinx/x = 1). I know it's a pain, but that's the game! Thanks for all the help!
$endgroup$
– Sawyer Benson
10 hours ago
|
show 2 more comments
2
$begingroup$
Can you give some context on why you want to do that with only the difference quotient? Surely you can replicate the proof of the chain rule, but it's not going to be pretty (although pointless all the more).
$endgroup$
– Klaus
10 hours ago
$begingroup$
Sure, Klaus. This is a particular problem given by my prof. It's not for credit of even an assignment, she gave it as an interestingly difficult problem with some twist and turns. I've worked on it for several hours now and would like to see how others approach solving it as I'm wrapped around an axel after using double-angle ID and clearing the cubes in the numerator. Any advice is appreciated.
$endgroup$
– Sawyer Benson
10 hours ago
$begingroup$
Use sin instead of sin in $rmLaTeX$! It's a built-in function to make it look nice :)
$endgroup$
– Cameron Williams
10 hours ago
$begingroup$
I think you'd have to use the generalized binomial theorem for what it's worth. It's been a while since I did non-integer powers with the limit definition so I might be wrong. The integer power case uses the regular binomial theorem+factoring so I think using the generalized binomial theorem is appropriate.
$endgroup$
– Cameron Williams
10 hours ago
1
$begingroup$
Thanks, Cameron! Unfortunately I cannot use that either. I know how to solve it using the chain rule and even L'Hopital's rule, however, the challenge it to solving it algebraically using nothing more than trig IDs (half angle) and trig limit rules (lim of sinx/x = 1). I know it's a pain, but that's the game! Thanks for all the help!
$endgroup$
– Sawyer Benson
10 hours ago
2
2
$begingroup$
Can you give some context on why you want to do that with only the difference quotient? Surely you can replicate the proof of the chain rule, but it's not going to be pretty (although pointless all the more).
$endgroup$
– Klaus
10 hours ago
$begingroup$
Can you give some context on why you want to do that with only the difference quotient? Surely you can replicate the proof of the chain rule, but it's not going to be pretty (although pointless all the more).
$endgroup$
– Klaus
10 hours ago
$begingroup$
Sure, Klaus. This is a particular problem given by my prof. It's not for credit of even an assignment, she gave it as an interestingly difficult problem with some twist and turns. I've worked on it for several hours now and would like to see how others approach solving it as I'm wrapped around an axel after using double-angle ID and clearing the cubes in the numerator. Any advice is appreciated.
$endgroup$
– Sawyer Benson
10 hours ago
$begingroup$
Sure, Klaus. This is a particular problem given by my prof. It's not for credit of even an assignment, she gave it as an interestingly difficult problem with some twist and turns. I've worked on it for several hours now and would like to see how others approach solving it as I'm wrapped around an axel after using double-angle ID and clearing the cubes in the numerator. Any advice is appreciated.
$endgroup$
– Sawyer Benson
10 hours ago
$begingroup$
Use sin instead of sin in $rmLaTeX$! It's a built-in function to make it look nice :)
$endgroup$
– Cameron Williams
10 hours ago
$begingroup$
Use sin instead of sin in $rmLaTeX$! It's a built-in function to make it look nice :)
$endgroup$
– Cameron Williams
10 hours ago
$begingroup$
I think you'd have to use the generalized binomial theorem for what it's worth. It's been a while since I did non-integer powers with the limit definition so I might be wrong. The integer power case uses the regular binomial theorem+factoring so I think using the generalized binomial theorem is appropriate.
$endgroup$
– Cameron Williams
10 hours ago
$begingroup$
I think you'd have to use the generalized binomial theorem for what it's worth. It's been a while since I did non-integer powers with the limit definition so I might be wrong. The integer power case uses the regular binomial theorem+factoring so I think using the generalized binomial theorem is appropriate.
$endgroup$
– Cameron Williams
10 hours ago
1
1
$begingroup$
Thanks, Cameron! Unfortunately I cannot use that either. I know how to solve it using the chain rule and even L'Hopital's rule, however, the challenge it to solving it algebraically using nothing more than trig IDs (half angle) and trig limit rules (lim of sinx/x = 1). I know it's a pain, but that's the game! Thanks for all the help!
$endgroup$
– Sawyer Benson
10 hours ago
$begingroup$
Thanks, Cameron! Unfortunately I cannot use that either. I know how to solve it using the chain rule and even L'Hopital's rule, however, the challenge it to solving it algebraically using nothing more than trig IDs (half angle) and trig limit rules (lim of sinx/x = 1). I know it's a pain, but that's the game! Thanks for all the help!
$endgroup$
– Sawyer Benson
10 hours ago
|
show 2 more comments
4 Answers
4
active
oldest
votes
$begingroup$
Let's use $a^3-b^3=(a-b)(a^2+ab+b^2)$ for $a=sqrt[3]{sin 2(x+h)}$ and $b=sqrt[3]{sin 2x}$. Assume that $bneq 0$. This implies that $a^2+ab+b^2neq 0$ and we can divide:
$$frac{sqrt[3]{sin (2x+2h)} - sqrt[3]{sin 2x}}{h} = frac{sin (2x+2h) - sin 2x}{h} cdot frac{1}{a^2+ab+b^2}$$
Now using the identity $sin2(x+h)-sin 2x = 2sin h cos (2x+h)$, continuity of $sin$ and $cos$ and
$$lim_{~hto 0}frac{sin h}{h}=1$$
we get that the derivative is:
$$frac{2cos 2x}{3sin^{2/3} 2x}$$
$endgroup$
$begingroup$
Shouldn't you have $h$ rather than $2h$ in the denominator, and $sin h$ rather than $sin(2h)$ in the numerator?
$endgroup$
– Calum Gilhooley
9 hours ago
$begingroup$
@CalumGilhooley - you're right, this is how usually this is written. However, if you choose $2h$ convention, you get $cos (2x+h)$ instead of $cos (2x+h/2)$, what looks nicer to me.
$endgroup$
– Paweł Czyż
9 hours ago
$begingroup$
The error is in the claimed identity $sin2(x+h)-sin 2x = 2sin 2h cos (2x+h)$. The $2h$ there should be $h$ (unless I've gone mad, which is always a possibility).
$endgroup$
– Calum Gilhooley
9 hours ago
1
$begingroup$
Oh, I see! It's a terrible mistake of mine, thanks for pointing this out!
$endgroup$
– Paweł Czyż
9 hours ago
$begingroup$
nice approach !(+1)
$endgroup$
– G Cab
9 hours ago
|
show 1 more comment
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I have written down all steps in detail, and I have marked the main ingredients using bold face font. This is also a nice typing exercise in Latex.
Let $pgt0$ and consider more generally the difference quotient
$$d= frac{1}{h} left(sin(2x+2h)^p - sin(2x)^p right)$$
Now for the sine of the sum we can write
$$sin(2x+2h)= sin(2x)cos(2h)+cos(2x)sin(2h) \= cos(2h)sin(2x) (1+cot(2x) tan(2h))$$
Next for $hto 0$ we have approximately
$$cos(2h)= 1- 2h^2 + O(h^4)$$
$$tan(2h) = 2h + O(h^3)$$
so that, keeping only the first order in $h$, we get for the $p$-th power
$$sin(2x+2h)^p = sin(2x)^p(1+cot(2x)2h)^p\simeq sin(2x)^p+2 h p sin(2x)^pcot(2x)\=sin(2x)^p+2 h p sin(2x)^{p-1}cos(2x)$$
where we have use the binomial formula $(1+epsilon)^p simeq 1 + p; epsilon $.
Finally we get for $hto 0$
$$d = frac{1}{h} (sin(2x)^p+2 h p sin(2x)^{p-1}cos(2x) - sin(2x)^p) \= frac{1}{h} 2 h p sin(2x)^{p-1}cos(2x)\= 2 p sin(2x)^{p-1}cos(2x)$$
For $p=frac{1}{3}$ this gives
$$d = frac{2}{3}sin(2x)^{-frac{2}{3}}cos(2x)$$
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Can you also find the integral: $int_0^{frac{pi }{2}} sqrt[3]{sin (2 x)} , dx$ ?
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– Dr. Wolfgang Hintze
9 hours ago
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...( by parts )
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– G Cab
9 hours ago
1
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Your use of the Maclaurin expansion seems to sidestep the requirement to only rely on the difference quotient !
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– Jim
4 hours ago
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@ Jim What would you propose instead? At a certain point you need to use this expansion in one form or other. Notice that all answer here use the same expansion, mostly even shorter, only as a limit.
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– Dr. Wolfgang Hintze
2 hours ago
add a comment |
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Are you allowed to use the trigonometric limit identities of
$$lim limits_{u to 0} frac{sin u}{u} = 1 hspace{.5 in} text{and} hspace{.5 in} lim limits_{u to 0} frac{1-cos u}{u} = 0?$$
If not, the link above gives good hints. You'll need to prove these first. If you are allowed to use them, then let's proceed. The first step is to use some algebra to get rid of those pesky third roots:
begin{align*}
&lim limits_{h to 0} frac{sqrt[3]{sin(2x+2h)} - sqrt[3]{sin(2x)}}{h}\
=& lim limits_{h to 0} frac{sqrt[3]{sin(2x+2h)} - sqrt[3]{sin(2x)}}{h} cdot frac{sin(2x+2h)^{2/3} + sin(2x+2h)^{1/3}sin(2x)^{1/3} + sin(2x)^{2/3}}{sin(2x+2h)^{2/3} + sin(2x+2h)^{1/3}sin(2x)^{1/3} + sin(2x)^{2/3}}\
=&lim limits_{h to 0} frac{sin(2x+2h) - sin(2x)}{h(sin(2x+2h)^{2/3} + sin(2x+2h)^{1/3}sin(2x)^{1/3} + sin(2x)^{2/3})}
end{align*}
That denominator looks a little messy, so maybe we could call the part in parentheses
$$W(x,h) = sin(2x+2h)^{2/3} + sin(2x+2h)^{1/3}sin(2x)^{1/3} + sin(2x)^{2/3}.$$
Notice that $W(x,0) = 3sin(2x)^{2/3}$. That will come in handy later on. Now let's rewrite our limit using our $W$ shorthand and use the angle sum identity to move forward:
begin{align*}
lim limits_{h to 0} frac{sin(2x+2h) - sin(2x)}{h cdot W(x,h)} &= lim limits_{h to 0} frac{sin(2x)cos(2h) - cos(2h)sin(2x) - sin(2x)}{h cdot W(x,h)}\
&= lim limits_{h to 0} frac{cos(2h)-1}{h} cdot frac{sin(2x)}{W(x,h)} + limlimits_{h to 0} frac{-sin(2h)}{h} cdot frac{cos(2x)}{W(x,h)}\
&= lim limits_{h to 0} frac{1-cos(2h)}{2h} cdot frac{-2sin(2x)}{W(x,h)} + limlimits_{h to 0} frac{sin(2h)}{2h} cdot frac{-2cos(2x)}{W(x,h)}
end{align*}
Then, by the identities mentioned above, we have
begin{align*}
lim limits_{h to 0} frac{sqrt[3]{sin(2x+2h)} - sqrt[3]{sin(2x)}}{h} &= 0 cdot frac{-2sin(2x)}{W(x,0)} + 1 cdot frac{-2cos(2x)}{W(x,0)}\
&= frac{-2cos(2x)}{3sin(2x)^{2/3}}
end{align*}
which is the answer you're looking for.
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Ryan, I like your method. Though I don’t understand where you derived the multiple of the first step to clear the numerator? How did you come up with this?
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– Sawyer Benson
18 mins ago
add a comment |
$begingroup$
Hint:
Use symmetry! :
$$sin(2x+2h)=sin((2x+h)+h) quad sin(2x)=sin((2x+h)-h)$$
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This is a good idea !
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– Dr. Wolfgang Hintze
9 hours ago
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@Dr.WolfgangHintze: oh yes, to me, symmetry is the core of maths
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– G Cab
9 hours ago
add a comment |
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4 Answers
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active
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4 Answers
4
active
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$begingroup$
Let's use $a^3-b^3=(a-b)(a^2+ab+b^2)$ for $a=sqrt[3]{sin 2(x+h)}$ and $b=sqrt[3]{sin 2x}$. Assume that $bneq 0$. This implies that $a^2+ab+b^2neq 0$ and we can divide:
$$frac{sqrt[3]{sin (2x+2h)} - sqrt[3]{sin 2x}}{h} = frac{sin (2x+2h) - sin 2x}{h} cdot frac{1}{a^2+ab+b^2}$$
Now using the identity $sin2(x+h)-sin 2x = 2sin h cos (2x+h)$, continuity of $sin$ and $cos$ and
$$lim_{~hto 0}frac{sin h}{h}=1$$
we get that the derivative is:
$$frac{2cos 2x}{3sin^{2/3} 2x}$$
$endgroup$
$begingroup$
Shouldn't you have $h$ rather than $2h$ in the denominator, and $sin h$ rather than $sin(2h)$ in the numerator?
$endgroup$
– Calum Gilhooley
9 hours ago
$begingroup$
@CalumGilhooley - you're right, this is how usually this is written. However, if you choose $2h$ convention, you get $cos (2x+h)$ instead of $cos (2x+h/2)$, what looks nicer to me.
$endgroup$
– Paweł Czyż
9 hours ago
$begingroup$
The error is in the claimed identity $sin2(x+h)-sin 2x = 2sin 2h cos (2x+h)$. The $2h$ there should be $h$ (unless I've gone mad, which is always a possibility).
$endgroup$
– Calum Gilhooley
9 hours ago
1
$begingroup$
Oh, I see! It's a terrible mistake of mine, thanks for pointing this out!
$endgroup$
– Paweł Czyż
9 hours ago
$begingroup$
nice approach !(+1)
$endgroup$
– G Cab
9 hours ago
|
show 1 more comment
$begingroup$
Let's use $a^3-b^3=(a-b)(a^2+ab+b^2)$ for $a=sqrt[3]{sin 2(x+h)}$ and $b=sqrt[3]{sin 2x}$. Assume that $bneq 0$. This implies that $a^2+ab+b^2neq 0$ and we can divide:
$$frac{sqrt[3]{sin (2x+2h)} - sqrt[3]{sin 2x}}{h} = frac{sin (2x+2h) - sin 2x}{h} cdot frac{1}{a^2+ab+b^2}$$
Now using the identity $sin2(x+h)-sin 2x = 2sin h cos (2x+h)$, continuity of $sin$ and $cos$ and
$$lim_{~hto 0}frac{sin h}{h}=1$$
we get that the derivative is:
$$frac{2cos 2x}{3sin^{2/3} 2x}$$
$endgroup$
$begingroup$
Shouldn't you have $h$ rather than $2h$ in the denominator, and $sin h$ rather than $sin(2h)$ in the numerator?
$endgroup$
– Calum Gilhooley
9 hours ago
$begingroup$
@CalumGilhooley - you're right, this is how usually this is written. However, if you choose $2h$ convention, you get $cos (2x+h)$ instead of $cos (2x+h/2)$, what looks nicer to me.
$endgroup$
– Paweł Czyż
9 hours ago
$begingroup$
The error is in the claimed identity $sin2(x+h)-sin 2x = 2sin 2h cos (2x+h)$. The $2h$ there should be $h$ (unless I've gone mad, which is always a possibility).
$endgroup$
– Calum Gilhooley
9 hours ago
1
$begingroup$
Oh, I see! It's a terrible mistake of mine, thanks for pointing this out!
$endgroup$
– Paweł Czyż
9 hours ago
$begingroup$
nice approach !(+1)
$endgroup$
– G Cab
9 hours ago
|
show 1 more comment
$begingroup$
Let's use $a^3-b^3=(a-b)(a^2+ab+b^2)$ for $a=sqrt[3]{sin 2(x+h)}$ and $b=sqrt[3]{sin 2x}$. Assume that $bneq 0$. This implies that $a^2+ab+b^2neq 0$ and we can divide:
$$frac{sqrt[3]{sin (2x+2h)} - sqrt[3]{sin 2x}}{h} = frac{sin (2x+2h) - sin 2x}{h} cdot frac{1}{a^2+ab+b^2}$$
Now using the identity $sin2(x+h)-sin 2x = 2sin h cos (2x+h)$, continuity of $sin$ and $cos$ and
$$lim_{~hto 0}frac{sin h}{h}=1$$
we get that the derivative is:
$$frac{2cos 2x}{3sin^{2/3} 2x}$$
$endgroup$
Let's use $a^3-b^3=(a-b)(a^2+ab+b^2)$ for $a=sqrt[3]{sin 2(x+h)}$ and $b=sqrt[3]{sin 2x}$. Assume that $bneq 0$. This implies that $a^2+ab+b^2neq 0$ and we can divide:
$$frac{sqrt[3]{sin (2x+2h)} - sqrt[3]{sin 2x}}{h} = frac{sin (2x+2h) - sin 2x}{h} cdot frac{1}{a^2+ab+b^2}$$
Now using the identity $sin2(x+h)-sin 2x = 2sin h cos (2x+h)$, continuity of $sin$ and $cos$ and
$$lim_{~hto 0}frac{sin h}{h}=1$$
we get that the derivative is:
$$frac{2cos 2x}{3sin^{2/3} 2x}$$
edited 9 hours ago
answered 9 hours ago
Paweł CzyżPaweł Czyż
76711
76711
$begingroup$
Shouldn't you have $h$ rather than $2h$ in the denominator, and $sin h$ rather than $sin(2h)$ in the numerator?
$endgroup$
– Calum Gilhooley
9 hours ago
$begingroup$
@CalumGilhooley - you're right, this is how usually this is written. However, if you choose $2h$ convention, you get $cos (2x+h)$ instead of $cos (2x+h/2)$, what looks nicer to me.
$endgroup$
– Paweł Czyż
9 hours ago
$begingroup$
The error is in the claimed identity $sin2(x+h)-sin 2x = 2sin 2h cos (2x+h)$. The $2h$ there should be $h$ (unless I've gone mad, which is always a possibility).
$endgroup$
– Calum Gilhooley
9 hours ago
1
$begingroup$
Oh, I see! It's a terrible mistake of mine, thanks for pointing this out!
$endgroup$
– Paweł Czyż
9 hours ago
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nice approach !(+1)
$endgroup$
– G Cab
9 hours ago
|
show 1 more comment
$begingroup$
Shouldn't you have $h$ rather than $2h$ in the denominator, and $sin h$ rather than $sin(2h)$ in the numerator?
$endgroup$
– Calum Gilhooley
9 hours ago
$begingroup$
@CalumGilhooley - you're right, this is how usually this is written. However, if you choose $2h$ convention, you get $cos (2x+h)$ instead of $cos (2x+h/2)$, what looks nicer to me.
$endgroup$
– Paweł Czyż
9 hours ago
$begingroup$
The error is in the claimed identity $sin2(x+h)-sin 2x = 2sin 2h cos (2x+h)$. The $2h$ there should be $h$ (unless I've gone mad, which is always a possibility).
$endgroup$
– Calum Gilhooley
9 hours ago
1
$begingroup$
Oh, I see! It's a terrible mistake of mine, thanks for pointing this out!
$endgroup$
– Paweł Czyż
9 hours ago
$begingroup$
nice approach !(+1)
$endgroup$
– G Cab
9 hours ago
$begingroup$
Shouldn't you have $h$ rather than $2h$ in the denominator, and $sin h$ rather than $sin(2h)$ in the numerator?
$endgroup$
– Calum Gilhooley
9 hours ago
$begingroup$
Shouldn't you have $h$ rather than $2h$ in the denominator, and $sin h$ rather than $sin(2h)$ in the numerator?
$endgroup$
– Calum Gilhooley
9 hours ago
$begingroup$
@CalumGilhooley - you're right, this is how usually this is written. However, if you choose $2h$ convention, you get $cos (2x+h)$ instead of $cos (2x+h/2)$, what looks nicer to me.
$endgroup$
– Paweł Czyż
9 hours ago
$begingroup$
@CalumGilhooley - you're right, this is how usually this is written. However, if you choose $2h$ convention, you get $cos (2x+h)$ instead of $cos (2x+h/2)$, what looks nicer to me.
$endgroup$
– Paweł Czyż
9 hours ago
$begingroup$
The error is in the claimed identity $sin2(x+h)-sin 2x = 2sin 2h cos (2x+h)$. The $2h$ there should be $h$ (unless I've gone mad, which is always a possibility).
$endgroup$
– Calum Gilhooley
9 hours ago
$begingroup$
The error is in the claimed identity $sin2(x+h)-sin 2x = 2sin 2h cos (2x+h)$. The $2h$ there should be $h$ (unless I've gone mad, which is always a possibility).
$endgroup$
– Calum Gilhooley
9 hours ago
1
1
$begingroup$
Oh, I see! It's a terrible mistake of mine, thanks for pointing this out!
$endgroup$
– Paweł Czyż
9 hours ago
$begingroup$
Oh, I see! It's a terrible mistake of mine, thanks for pointing this out!
$endgroup$
– Paweł Czyż
9 hours ago
$begingroup$
nice approach !(+1)
$endgroup$
– G Cab
9 hours ago
$begingroup$
nice approach !(+1)
$endgroup$
– G Cab
9 hours ago
|
show 1 more comment
$begingroup$
I have written down all steps in detail, and I have marked the main ingredients using bold face font. This is also a nice typing exercise in Latex.
Let $pgt0$ and consider more generally the difference quotient
$$d= frac{1}{h} left(sin(2x+2h)^p - sin(2x)^p right)$$
Now for the sine of the sum we can write
$$sin(2x+2h)= sin(2x)cos(2h)+cos(2x)sin(2h) \= cos(2h)sin(2x) (1+cot(2x) tan(2h))$$
Next for $hto 0$ we have approximately
$$cos(2h)= 1- 2h^2 + O(h^4)$$
$$tan(2h) = 2h + O(h^3)$$
so that, keeping only the first order in $h$, we get for the $p$-th power
$$sin(2x+2h)^p = sin(2x)^p(1+cot(2x)2h)^p\simeq sin(2x)^p+2 h p sin(2x)^pcot(2x)\=sin(2x)^p+2 h p sin(2x)^{p-1}cos(2x)$$
where we have use the binomial formula $(1+epsilon)^p simeq 1 + p; epsilon $.
Finally we get for $hto 0$
$$d = frac{1}{h} (sin(2x)^p+2 h p sin(2x)^{p-1}cos(2x) - sin(2x)^p) \= frac{1}{h} 2 h p sin(2x)^{p-1}cos(2x)\= 2 p sin(2x)^{p-1}cos(2x)$$
For $p=frac{1}{3}$ this gives
$$d = frac{2}{3}sin(2x)^{-frac{2}{3}}cos(2x)$$
$endgroup$
$begingroup$
Can you also find the integral: $int_0^{frac{pi }{2}} sqrt[3]{sin (2 x)} , dx$ ?
$endgroup$
– Dr. Wolfgang Hintze
9 hours ago
$begingroup$
...( by parts )
$endgroup$
– G Cab
9 hours ago
1
$begingroup$
Your use of the Maclaurin expansion seems to sidestep the requirement to only rely on the difference quotient !
$endgroup$
– Jim
4 hours ago
$begingroup$
@ Jim What would you propose instead? At a certain point you need to use this expansion in one form or other. Notice that all answer here use the same expansion, mostly even shorter, only as a limit.
$endgroup$
– Dr. Wolfgang Hintze
2 hours ago
add a comment |
$begingroup$
I have written down all steps in detail, and I have marked the main ingredients using bold face font. This is also a nice typing exercise in Latex.
Let $pgt0$ and consider more generally the difference quotient
$$d= frac{1}{h} left(sin(2x+2h)^p - sin(2x)^p right)$$
Now for the sine of the sum we can write
$$sin(2x+2h)= sin(2x)cos(2h)+cos(2x)sin(2h) \= cos(2h)sin(2x) (1+cot(2x) tan(2h))$$
Next for $hto 0$ we have approximately
$$cos(2h)= 1- 2h^2 + O(h^4)$$
$$tan(2h) = 2h + O(h^3)$$
so that, keeping only the first order in $h$, we get for the $p$-th power
$$sin(2x+2h)^p = sin(2x)^p(1+cot(2x)2h)^p\simeq sin(2x)^p+2 h p sin(2x)^pcot(2x)\=sin(2x)^p+2 h p sin(2x)^{p-1}cos(2x)$$
where we have use the binomial formula $(1+epsilon)^p simeq 1 + p; epsilon $.
Finally we get for $hto 0$
$$d = frac{1}{h} (sin(2x)^p+2 h p sin(2x)^{p-1}cos(2x) - sin(2x)^p) \= frac{1}{h} 2 h p sin(2x)^{p-1}cos(2x)\= 2 p sin(2x)^{p-1}cos(2x)$$
For $p=frac{1}{3}$ this gives
$$d = frac{2}{3}sin(2x)^{-frac{2}{3}}cos(2x)$$
$endgroup$
$begingroup$
Can you also find the integral: $int_0^{frac{pi }{2}} sqrt[3]{sin (2 x)} , dx$ ?
$endgroup$
– Dr. Wolfgang Hintze
9 hours ago
$begingroup$
...( by parts )
$endgroup$
– G Cab
9 hours ago
1
$begingroup$
Your use of the Maclaurin expansion seems to sidestep the requirement to only rely on the difference quotient !
$endgroup$
– Jim
4 hours ago
$begingroup$
@ Jim What would you propose instead? At a certain point you need to use this expansion in one form or other. Notice that all answer here use the same expansion, mostly even shorter, only as a limit.
$endgroup$
– Dr. Wolfgang Hintze
2 hours ago
add a comment |
$begingroup$
I have written down all steps in detail, and I have marked the main ingredients using bold face font. This is also a nice typing exercise in Latex.
Let $pgt0$ and consider more generally the difference quotient
$$d= frac{1}{h} left(sin(2x+2h)^p - sin(2x)^p right)$$
Now for the sine of the sum we can write
$$sin(2x+2h)= sin(2x)cos(2h)+cos(2x)sin(2h) \= cos(2h)sin(2x) (1+cot(2x) tan(2h))$$
Next for $hto 0$ we have approximately
$$cos(2h)= 1- 2h^2 + O(h^4)$$
$$tan(2h) = 2h + O(h^3)$$
so that, keeping only the first order in $h$, we get for the $p$-th power
$$sin(2x+2h)^p = sin(2x)^p(1+cot(2x)2h)^p\simeq sin(2x)^p+2 h p sin(2x)^pcot(2x)\=sin(2x)^p+2 h p sin(2x)^{p-1}cos(2x)$$
where we have use the binomial formula $(1+epsilon)^p simeq 1 + p; epsilon $.
Finally we get for $hto 0$
$$d = frac{1}{h} (sin(2x)^p+2 h p sin(2x)^{p-1}cos(2x) - sin(2x)^p) \= frac{1}{h} 2 h p sin(2x)^{p-1}cos(2x)\= 2 p sin(2x)^{p-1}cos(2x)$$
For $p=frac{1}{3}$ this gives
$$d = frac{2}{3}sin(2x)^{-frac{2}{3}}cos(2x)$$
$endgroup$
I have written down all steps in detail, and I have marked the main ingredients using bold face font. This is also a nice typing exercise in Latex.
Let $pgt0$ and consider more generally the difference quotient
$$d= frac{1}{h} left(sin(2x+2h)^p - sin(2x)^p right)$$
Now for the sine of the sum we can write
$$sin(2x+2h)= sin(2x)cos(2h)+cos(2x)sin(2h) \= cos(2h)sin(2x) (1+cot(2x) tan(2h))$$
Next for $hto 0$ we have approximately
$$cos(2h)= 1- 2h^2 + O(h^4)$$
$$tan(2h) = 2h + O(h^3)$$
so that, keeping only the first order in $h$, we get for the $p$-th power
$$sin(2x+2h)^p = sin(2x)^p(1+cot(2x)2h)^p\simeq sin(2x)^p+2 h p sin(2x)^pcot(2x)\=sin(2x)^p+2 h p sin(2x)^{p-1}cos(2x)$$
where we have use the binomial formula $(1+epsilon)^p simeq 1 + p; epsilon $.
Finally we get for $hto 0$
$$d = frac{1}{h} (sin(2x)^p+2 h p sin(2x)^{p-1}cos(2x) - sin(2x)^p) \= frac{1}{h} 2 h p sin(2x)^{p-1}cos(2x)\= 2 p sin(2x)^{p-1}cos(2x)$$
For $p=frac{1}{3}$ this gives
$$d = frac{2}{3}sin(2x)^{-frac{2}{3}}cos(2x)$$
edited 9 hours ago
answered 9 hours ago
Dr. Wolfgang HintzeDr. Wolfgang Hintze
3,505618
3,505618
$begingroup$
Can you also find the integral: $int_0^{frac{pi }{2}} sqrt[3]{sin (2 x)} , dx$ ?
$endgroup$
– Dr. Wolfgang Hintze
9 hours ago
$begingroup$
...( by parts )
$endgroup$
– G Cab
9 hours ago
1
$begingroup$
Your use of the Maclaurin expansion seems to sidestep the requirement to only rely on the difference quotient !
$endgroup$
– Jim
4 hours ago
$begingroup$
@ Jim What would you propose instead? At a certain point you need to use this expansion in one form or other. Notice that all answer here use the same expansion, mostly even shorter, only as a limit.
$endgroup$
– Dr. Wolfgang Hintze
2 hours ago
add a comment |
$begingroup$
Can you also find the integral: $int_0^{frac{pi }{2}} sqrt[3]{sin (2 x)} , dx$ ?
$endgroup$
– Dr. Wolfgang Hintze
9 hours ago
$begingroup$
...( by parts )
$endgroup$
– G Cab
9 hours ago
1
$begingroup$
Your use of the Maclaurin expansion seems to sidestep the requirement to only rely on the difference quotient !
$endgroup$
– Jim
4 hours ago
$begingroup$
@ Jim What would you propose instead? At a certain point you need to use this expansion in one form or other. Notice that all answer here use the same expansion, mostly even shorter, only as a limit.
$endgroup$
– Dr. Wolfgang Hintze
2 hours ago
$begingroup$
Can you also find the integral: $int_0^{frac{pi }{2}} sqrt[3]{sin (2 x)} , dx$ ?
$endgroup$
– Dr. Wolfgang Hintze
9 hours ago
$begingroup$
Can you also find the integral: $int_0^{frac{pi }{2}} sqrt[3]{sin (2 x)} , dx$ ?
$endgroup$
– Dr. Wolfgang Hintze
9 hours ago
$begingroup$
...( by parts )
$endgroup$
– G Cab
9 hours ago
$begingroup$
...( by parts )
$endgroup$
– G Cab
9 hours ago
1
1
$begingroup$
Your use of the Maclaurin expansion seems to sidestep the requirement to only rely on the difference quotient !
$endgroup$
– Jim
4 hours ago
$begingroup$
Your use of the Maclaurin expansion seems to sidestep the requirement to only rely on the difference quotient !
$endgroup$
– Jim
4 hours ago
$begingroup$
@ Jim What would you propose instead? At a certain point you need to use this expansion in one form or other. Notice that all answer here use the same expansion, mostly even shorter, only as a limit.
$endgroup$
– Dr. Wolfgang Hintze
2 hours ago
$begingroup$
@ Jim What would you propose instead? At a certain point you need to use this expansion in one form or other. Notice that all answer here use the same expansion, mostly even shorter, only as a limit.
$endgroup$
– Dr. Wolfgang Hintze
2 hours ago
add a comment |
$begingroup$
Are you allowed to use the trigonometric limit identities of
$$lim limits_{u to 0} frac{sin u}{u} = 1 hspace{.5 in} text{and} hspace{.5 in} lim limits_{u to 0} frac{1-cos u}{u} = 0?$$
If not, the link above gives good hints. You'll need to prove these first. If you are allowed to use them, then let's proceed. The first step is to use some algebra to get rid of those pesky third roots:
begin{align*}
&lim limits_{h to 0} frac{sqrt[3]{sin(2x+2h)} - sqrt[3]{sin(2x)}}{h}\
=& lim limits_{h to 0} frac{sqrt[3]{sin(2x+2h)} - sqrt[3]{sin(2x)}}{h} cdot frac{sin(2x+2h)^{2/3} + sin(2x+2h)^{1/3}sin(2x)^{1/3} + sin(2x)^{2/3}}{sin(2x+2h)^{2/3} + sin(2x+2h)^{1/3}sin(2x)^{1/3} + sin(2x)^{2/3}}\
=&lim limits_{h to 0} frac{sin(2x+2h) - sin(2x)}{h(sin(2x+2h)^{2/3} + sin(2x+2h)^{1/3}sin(2x)^{1/3} + sin(2x)^{2/3})}
end{align*}
That denominator looks a little messy, so maybe we could call the part in parentheses
$$W(x,h) = sin(2x+2h)^{2/3} + sin(2x+2h)^{1/3}sin(2x)^{1/3} + sin(2x)^{2/3}.$$
Notice that $W(x,0) = 3sin(2x)^{2/3}$. That will come in handy later on. Now let's rewrite our limit using our $W$ shorthand and use the angle sum identity to move forward:
begin{align*}
lim limits_{h to 0} frac{sin(2x+2h) - sin(2x)}{h cdot W(x,h)} &= lim limits_{h to 0} frac{sin(2x)cos(2h) - cos(2h)sin(2x) - sin(2x)}{h cdot W(x,h)}\
&= lim limits_{h to 0} frac{cos(2h)-1}{h} cdot frac{sin(2x)}{W(x,h)} + limlimits_{h to 0} frac{-sin(2h)}{h} cdot frac{cos(2x)}{W(x,h)}\
&= lim limits_{h to 0} frac{1-cos(2h)}{2h} cdot frac{-2sin(2x)}{W(x,h)} + limlimits_{h to 0} frac{sin(2h)}{2h} cdot frac{-2cos(2x)}{W(x,h)}
end{align*}
Then, by the identities mentioned above, we have
begin{align*}
lim limits_{h to 0} frac{sqrt[3]{sin(2x+2h)} - sqrt[3]{sin(2x)}}{h} &= 0 cdot frac{-2sin(2x)}{W(x,0)} + 1 cdot frac{-2cos(2x)}{W(x,0)}\
&= frac{-2cos(2x)}{3sin(2x)^{2/3}}
end{align*}
which is the answer you're looking for.
$endgroup$
$begingroup$
Ryan, I like your method. Though I don’t understand where you derived the multiple of the first step to clear the numerator? How did you come up with this?
$endgroup$
– Sawyer Benson
18 mins ago
add a comment |
$begingroup$
Are you allowed to use the trigonometric limit identities of
$$lim limits_{u to 0} frac{sin u}{u} = 1 hspace{.5 in} text{and} hspace{.5 in} lim limits_{u to 0} frac{1-cos u}{u} = 0?$$
If not, the link above gives good hints. You'll need to prove these first. If you are allowed to use them, then let's proceed. The first step is to use some algebra to get rid of those pesky third roots:
begin{align*}
&lim limits_{h to 0} frac{sqrt[3]{sin(2x+2h)} - sqrt[3]{sin(2x)}}{h}\
=& lim limits_{h to 0} frac{sqrt[3]{sin(2x+2h)} - sqrt[3]{sin(2x)}}{h} cdot frac{sin(2x+2h)^{2/3} + sin(2x+2h)^{1/3}sin(2x)^{1/3} + sin(2x)^{2/3}}{sin(2x+2h)^{2/3} + sin(2x+2h)^{1/3}sin(2x)^{1/3} + sin(2x)^{2/3}}\
=&lim limits_{h to 0} frac{sin(2x+2h) - sin(2x)}{h(sin(2x+2h)^{2/3} + sin(2x+2h)^{1/3}sin(2x)^{1/3} + sin(2x)^{2/3})}
end{align*}
That denominator looks a little messy, so maybe we could call the part in parentheses
$$W(x,h) = sin(2x+2h)^{2/3} + sin(2x+2h)^{1/3}sin(2x)^{1/3} + sin(2x)^{2/3}.$$
Notice that $W(x,0) = 3sin(2x)^{2/3}$. That will come in handy later on. Now let's rewrite our limit using our $W$ shorthand and use the angle sum identity to move forward:
begin{align*}
lim limits_{h to 0} frac{sin(2x+2h) - sin(2x)}{h cdot W(x,h)} &= lim limits_{h to 0} frac{sin(2x)cos(2h) - cos(2h)sin(2x) - sin(2x)}{h cdot W(x,h)}\
&= lim limits_{h to 0} frac{cos(2h)-1}{h} cdot frac{sin(2x)}{W(x,h)} + limlimits_{h to 0} frac{-sin(2h)}{h} cdot frac{cos(2x)}{W(x,h)}\
&= lim limits_{h to 0} frac{1-cos(2h)}{2h} cdot frac{-2sin(2x)}{W(x,h)} + limlimits_{h to 0} frac{sin(2h)}{2h} cdot frac{-2cos(2x)}{W(x,h)}
end{align*}
Then, by the identities mentioned above, we have
begin{align*}
lim limits_{h to 0} frac{sqrt[3]{sin(2x+2h)} - sqrt[3]{sin(2x)}}{h} &= 0 cdot frac{-2sin(2x)}{W(x,0)} + 1 cdot frac{-2cos(2x)}{W(x,0)}\
&= frac{-2cos(2x)}{3sin(2x)^{2/3}}
end{align*}
which is the answer you're looking for.
$endgroup$
$begingroup$
Ryan, I like your method. Though I don’t understand where you derived the multiple of the first step to clear the numerator? How did you come up with this?
$endgroup$
– Sawyer Benson
18 mins ago
add a comment |
$begingroup$
Are you allowed to use the trigonometric limit identities of
$$lim limits_{u to 0} frac{sin u}{u} = 1 hspace{.5 in} text{and} hspace{.5 in} lim limits_{u to 0} frac{1-cos u}{u} = 0?$$
If not, the link above gives good hints. You'll need to prove these first. If you are allowed to use them, then let's proceed. The first step is to use some algebra to get rid of those pesky third roots:
begin{align*}
&lim limits_{h to 0} frac{sqrt[3]{sin(2x+2h)} - sqrt[3]{sin(2x)}}{h}\
=& lim limits_{h to 0} frac{sqrt[3]{sin(2x+2h)} - sqrt[3]{sin(2x)}}{h} cdot frac{sin(2x+2h)^{2/3} + sin(2x+2h)^{1/3}sin(2x)^{1/3} + sin(2x)^{2/3}}{sin(2x+2h)^{2/3} + sin(2x+2h)^{1/3}sin(2x)^{1/3} + sin(2x)^{2/3}}\
=&lim limits_{h to 0} frac{sin(2x+2h) - sin(2x)}{h(sin(2x+2h)^{2/3} + sin(2x+2h)^{1/3}sin(2x)^{1/3} + sin(2x)^{2/3})}
end{align*}
That denominator looks a little messy, so maybe we could call the part in parentheses
$$W(x,h) = sin(2x+2h)^{2/3} + sin(2x+2h)^{1/3}sin(2x)^{1/3} + sin(2x)^{2/3}.$$
Notice that $W(x,0) = 3sin(2x)^{2/3}$. That will come in handy later on. Now let's rewrite our limit using our $W$ shorthand and use the angle sum identity to move forward:
begin{align*}
lim limits_{h to 0} frac{sin(2x+2h) - sin(2x)}{h cdot W(x,h)} &= lim limits_{h to 0} frac{sin(2x)cos(2h) - cos(2h)sin(2x) - sin(2x)}{h cdot W(x,h)}\
&= lim limits_{h to 0} frac{cos(2h)-1}{h} cdot frac{sin(2x)}{W(x,h)} + limlimits_{h to 0} frac{-sin(2h)}{h} cdot frac{cos(2x)}{W(x,h)}\
&= lim limits_{h to 0} frac{1-cos(2h)}{2h} cdot frac{-2sin(2x)}{W(x,h)} + limlimits_{h to 0} frac{sin(2h)}{2h} cdot frac{-2cos(2x)}{W(x,h)}
end{align*}
Then, by the identities mentioned above, we have
begin{align*}
lim limits_{h to 0} frac{sqrt[3]{sin(2x+2h)} - sqrt[3]{sin(2x)}}{h} &= 0 cdot frac{-2sin(2x)}{W(x,0)} + 1 cdot frac{-2cos(2x)}{W(x,0)}\
&= frac{-2cos(2x)}{3sin(2x)^{2/3}}
end{align*}
which is the answer you're looking for.
$endgroup$
Are you allowed to use the trigonometric limit identities of
$$lim limits_{u to 0} frac{sin u}{u} = 1 hspace{.5 in} text{and} hspace{.5 in} lim limits_{u to 0} frac{1-cos u}{u} = 0?$$
If not, the link above gives good hints. You'll need to prove these first. If you are allowed to use them, then let's proceed. The first step is to use some algebra to get rid of those pesky third roots:
begin{align*}
&lim limits_{h to 0} frac{sqrt[3]{sin(2x+2h)} - sqrt[3]{sin(2x)}}{h}\
=& lim limits_{h to 0} frac{sqrt[3]{sin(2x+2h)} - sqrt[3]{sin(2x)}}{h} cdot frac{sin(2x+2h)^{2/3} + sin(2x+2h)^{1/3}sin(2x)^{1/3} + sin(2x)^{2/3}}{sin(2x+2h)^{2/3} + sin(2x+2h)^{1/3}sin(2x)^{1/3} + sin(2x)^{2/3}}\
=&lim limits_{h to 0} frac{sin(2x+2h) - sin(2x)}{h(sin(2x+2h)^{2/3} + sin(2x+2h)^{1/3}sin(2x)^{1/3} + sin(2x)^{2/3})}
end{align*}
That denominator looks a little messy, so maybe we could call the part in parentheses
$$W(x,h) = sin(2x+2h)^{2/3} + sin(2x+2h)^{1/3}sin(2x)^{1/3} + sin(2x)^{2/3}.$$
Notice that $W(x,0) = 3sin(2x)^{2/3}$. That will come in handy later on. Now let's rewrite our limit using our $W$ shorthand and use the angle sum identity to move forward:
begin{align*}
lim limits_{h to 0} frac{sin(2x+2h) - sin(2x)}{h cdot W(x,h)} &= lim limits_{h to 0} frac{sin(2x)cos(2h) - cos(2h)sin(2x) - sin(2x)}{h cdot W(x,h)}\
&= lim limits_{h to 0} frac{cos(2h)-1}{h} cdot frac{sin(2x)}{W(x,h)} + limlimits_{h to 0} frac{-sin(2h)}{h} cdot frac{cos(2x)}{W(x,h)}\
&= lim limits_{h to 0} frac{1-cos(2h)}{2h} cdot frac{-2sin(2x)}{W(x,h)} + limlimits_{h to 0} frac{sin(2h)}{2h} cdot frac{-2cos(2x)}{W(x,h)}
end{align*}
Then, by the identities mentioned above, we have
begin{align*}
lim limits_{h to 0} frac{sqrt[3]{sin(2x+2h)} - sqrt[3]{sin(2x)}}{h} &= 0 cdot frac{-2sin(2x)}{W(x,0)} + 1 cdot frac{-2cos(2x)}{W(x,0)}\
&= frac{-2cos(2x)}{3sin(2x)^{2/3}}
end{align*}
which is the answer you're looking for.
answered 9 hours ago
Ryan T JohnsonRyan T Johnson
1114
1114
$begingroup$
Ryan, I like your method. Though I don’t understand where you derived the multiple of the first step to clear the numerator? How did you come up with this?
$endgroup$
– Sawyer Benson
18 mins ago
add a comment |
$begingroup$
Ryan, I like your method. Though I don’t understand where you derived the multiple of the first step to clear the numerator? How did you come up with this?
$endgroup$
– Sawyer Benson
18 mins ago
$begingroup$
Ryan, I like your method. Though I don’t understand where you derived the multiple of the first step to clear the numerator? How did you come up with this?
$endgroup$
– Sawyer Benson
18 mins ago
$begingroup$
Ryan, I like your method. Though I don’t understand where you derived the multiple of the first step to clear the numerator? How did you come up with this?
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– Sawyer Benson
18 mins ago
add a comment |
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Hint:
Use symmetry! :
$$sin(2x+2h)=sin((2x+h)+h) quad sin(2x)=sin((2x+h)-h)$$
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This is a good idea !
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– Dr. Wolfgang Hintze
9 hours ago
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@Dr.WolfgangHintze: oh yes, to me, symmetry is the core of maths
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– G Cab
9 hours ago
add a comment |
$begingroup$
Hint:
Use symmetry! :
$$sin(2x+2h)=sin((2x+h)+h) quad sin(2x)=sin((2x+h)-h)$$
$endgroup$
$begingroup$
This is a good idea !
$endgroup$
– Dr. Wolfgang Hintze
9 hours ago
$begingroup$
@Dr.WolfgangHintze: oh yes, to me, symmetry is the core of maths
$endgroup$
– G Cab
9 hours ago
add a comment |
$begingroup$
Hint:
Use symmetry! :
$$sin(2x+2h)=sin((2x+h)+h) quad sin(2x)=sin((2x+h)-h)$$
$endgroup$
Hint:
Use symmetry! :
$$sin(2x+2h)=sin((2x+h)+h) quad sin(2x)=sin((2x+h)-h)$$
answered 9 hours ago
G CabG Cab
19.5k31238
19.5k31238
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This is a good idea !
$endgroup$
– Dr. Wolfgang Hintze
9 hours ago
$begingroup$
@Dr.WolfgangHintze: oh yes, to me, symmetry is the core of maths
$endgroup$
– G Cab
9 hours ago
add a comment |
$begingroup$
This is a good idea !
$endgroup$
– Dr. Wolfgang Hintze
9 hours ago
$begingroup$
@Dr.WolfgangHintze: oh yes, to me, symmetry is the core of maths
$endgroup$
– G Cab
9 hours ago
$begingroup$
This is a good idea !
$endgroup$
– Dr. Wolfgang Hintze
9 hours ago
$begingroup$
This is a good idea !
$endgroup$
– Dr. Wolfgang Hintze
9 hours ago
$begingroup$
@Dr.WolfgangHintze: oh yes, to me, symmetry is the core of maths
$endgroup$
– G Cab
9 hours ago
$begingroup$
@Dr.WolfgangHintze: oh yes, to me, symmetry is the core of maths
$endgroup$
– G Cab
9 hours ago
add a comment |
Sawyer Benson is a new contributor. Be nice, and check out our Code of Conduct.
Sawyer Benson is a new contributor. Be nice, and check out our Code of Conduct.
Sawyer Benson is a new contributor. Be nice, and check out our Code of Conduct.
Sawyer Benson is a new contributor. Be nice, and check out our Code of Conduct.
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2
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Can you give some context on why you want to do that with only the difference quotient? Surely you can replicate the proof of the chain rule, but it's not going to be pretty (although pointless all the more).
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– Klaus
10 hours ago
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Sure, Klaus. This is a particular problem given by my prof. It's not for credit of even an assignment, she gave it as an interestingly difficult problem with some twist and turns. I've worked on it for several hours now and would like to see how others approach solving it as I'm wrapped around an axel after using double-angle ID and clearing the cubes in the numerator. Any advice is appreciated.
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– Sawyer Benson
10 hours ago
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Use sin instead of sin in $rmLaTeX$! It's a built-in function to make it look nice :)
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– Cameron Williams
10 hours ago
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I think you'd have to use the generalized binomial theorem for what it's worth. It's been a while since I did non-integer powers with the limit definition so I might be wrong. The integer power case uses the regular binomial theorem+factoring so I think using the generalized binomial theorem is appropriate.
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– Cameron Williams
10 hours ago
1
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Thanks, Cameron! Unfortunately I cannot use that either. I know how to solve it using the chain rule and even L'Hopital's rule, however, the challenge it to solving it algebraically using nothing more than trig IDs (half angle) and trig limit rules (lim of sinx/x = 1). I know it's a pain, but that's the game! Thanks for all the help!
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– Sawyer Benson
10 hours ago