Fresnel function and finding its maxima (local maxima)
$begingroup$
I have this one problem in my Calculus textbook which consists of finding all of the local maxima of the Fresnel function : $S(x)= int_0^xsin(pi x^2/2) dx$
I solved for the critical points of the derivative $S'(x)=sin(pi x^2/2)$ and found out that there are two local maximums at $x = -2$ and $x = sqrt2$.
My question is : Since I'm working with a sinusoidal function (which means that it's constantly repeating itself on a period) how should I denote my answer?
For example :
Should my answer resemble something of this sort?
$$x = {-2+4n,sqrt2+4n}$$ where $nin Bbb Z$
Moreover, when you graph the derivative of the Fresnel function I see that the period is gradually shrinking so I wonder how this will affect the answer.
P.S. I recently started solving problems that consist of the Fundamental Theorem of Calculus so I know nothing more than that.
calculus definite-integrals
edited Jan 5 at 4:41
Randall
9,71111230
asked Jan 5 at 4:30
DBlykDBlyk
204
$endgroup$
add a comment |
$begingroup$
I have this one problem in my Calculus textbook which consists of finding all of the local maxima of the Fresnel function : $S(x)= int_0^xsin(pi x^2/2) dx$
I solved for the critical points of the derivative $S'(x)=sin(pi x^2/2)$ and found out that there are two local maximums at $x = -2$ and $x = sqrt2$.
My question is : Since I'm working with a sinusoidal function (which means that it's constantly repeating itself on a period) how should I denote my answer?
For example :
Should my answer resemble something of this sort?
$$x = {-2+4n,sqrt2+4n}$$ where $nin Bbb Z$
Moreover, when you graph the derivative of the Fresnel function I see that the period is gradually shrinking so I wonder how this will affect the answer.
P.S. I recently started solving problems that consist of the Fundamental Theorem of Calculus so I know nothing more than that.
calculus definite-integrals
edited Jan 5 at 4:41
Randall
9,71111230
asked Jan 5 at 4:30
DBlykDBlyk
204
$endgroup$
$begingroup$
you are using the same name for the variable of $S$ and the integrand, but they are different
$endgroup$
– Masacroso
Jan 5 at 5:50
add a comment |
$begingroup$
I have this one problem in my Calculus textbook which consists of finding all of the local maxima of the Fresnel function : $S(x)= int_0^xsin(pi x^2/2) dx$
I solved for the critical points of the derivative $S'(x)=sin(pi x^2/2)$ and found out that there are two local maximums at $x = -2$ and $x = sqrt2$.
My question is : Since I'm working with a sinusoidal function (which means that it's constantly repeating itself on a period) how should I denote my answer?
For example :
Should my answer resemble something of this sort?
$$x = {-2+4n,sqrt2+4n}$$ where $nin Bbb Z$
Moreover, when you graph the derivative of the Fresnel function I see that the period is gradually shrinking so I wonder how this will affect the answer.
P.S. I recently started solving problems that consist of the Fundamental Theorem of Calculus so I know nothing more than that.
calculus definite-integrals
edited Jan 5 at 4:41
Randall
9,71111230
asked Jan 5 at 4:30
DBlykDBlyk
204
$endgroup$
I have this one problem in my Calculus textbook which consists of finding all of the local maxima of the Fresnel function : $S(x)= int_0^xsin(pi x^2/2) dx$
I solved for the critical points of the derivative $S'(x)=sin(pi x^2/2)$ and found out that there are two local maximums at $x = -2$ and $x = sqrt2$.
My question is : Since I'm working with a sinusoidal function (which means that it's constantly repeating itself on a period) how should I denote my answer?
For example :
Should my answer resemble something of this sort?
$$x = {-2+4n,sqrt2+4n}$$ where $nin Bbb Z$
Moreover, when you graph the derivative of the Fresnel function I see that the period is gradually shrinking so I wonder how this will affect the answer.
P.S. I recently started solving problems that consist of the Fundamental Theorem of Calculus so I know nothing more than that.
calculus definite-integrals
calculus definite-integrals
edited Jan 5 at 4:41
Randall
9,71111230
asked Jan 5 at 4:30
DBlykDBlyk
204
edited Jan 5 at 4:41
Randall
9,71111230
asked Jan 5 at 4:30
DBlykDBlyk
204
edited Jan 5 at 4:41
Randall
9,71111230
edited Jan 5 at 4:41
Randall
9,71111230
edited Jan 5 at 4:41
Randall
9,71111230
9,71111230
asked Jan 5 at 4:30
DBlykDBlyk
204
asked Jan 5 at 4:30
DBlykDBlyk
204
asked Jan 5 at 4:30
DBlykDBlyk
204
204
$begingroup$
you are using the same name for the variable of $S$ and the integrand, but they are different
$endgroup$
– Masacroso
Jan 5 at 5:50
add a comment |
$begingroup$
you are using the same name for the variable of $S$ and the integrand, but they are different
$endgroup$
– Masacroso
Jan 5 at 5:50
$begingroup$
you are using the same name for the variable of $S$ and the integrand, but they are different
$endgroup$
– Masacroso
Jan 5 at 5:50
$begingroup$
you are using the same name for the variable of $S$ and the integrand, but they are different
$endgroup$
– Masacroso
Jan 5 at 5:50
add a comment |
1 Answer
1
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$begingroup$
No, you're not going to be adding something regular to $x$ to find the other maxima. You said it yourself; a "gradually shrinking period" means that it's not actually periodic.
Instead, we're going to have maxima whenever $frac{pi x^2}{2}$ reaches certain values; one pattern for positive $x$ and one for negative $x$ (Since $S$ is an odd function, switching between positive and negative $x$ interchanges local maxima and local minima).
answered Jan 5 at 4:53
jmerryjmerry
7,327920
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add a comment |
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1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
No, you're not going to be adding something regular to $x$ to find the other maxima. You said it yourself; a "gradually shrinking period" means that it's not actually periodic.
Instead, we're going to have maxima whenever $frac{pi x^2}{2}$ reaches certain values; one pattern for positive $x$ and one for negative $x$ (Since $S$ is an odd function, switching between positive and negative $x$ interchanges local maxima and local minima).
answered Jan 5 at 4:53
jmerryjmerry
7,327920
$endgroup$
add a comment |
$begingroup$
No, you're not going to be adding something regular to $x$ to find the other maxima. You said it yourself; a "gradually shrinking period" means that it's not actually periodic.
Instead, we're going to have maxima whenever $frac{pi x^2}{2}$ reaches certain values; one pattern for positive $x$ and one for negative $x$ (Since $S$ is an odd function, switching between positive and negative $x$ interchanges local maxima and local minima).
answered Jan 5 at 4:53
jmerryjmerry
7,327920
$endgroup$
add a comment |
$begingroup$
No, you're not going to be adding something regular to $x$ to find the other maxima. You said it yourself; a "gradually shrinking period" means that it's not actually periodic.
Instead, we're going to have maxima whenever $frac{pi x^2}{2}$ reaches certain values; one pattern for positive $x$ and one for negative $x$ (Since $S$ is an odd function, switching between positive and negative $x$ interchanges local maxima and local minima).
answered Jan 5 at 4:53
jmerryjmerry
7,327920
$endgroup$
No, you're not going to be adding something regular to $x$ to find the other maxima. You said it yourself; a "gradually shrinking period" means that it's not actually periodic.
Instead, we're going to have maxima whenever $frac{pi x^2}{2}$ reaches certain values; one pattern for positive $x$ and one for negative $x$ (Since $S$ is an odd function, switching between positive and negative $x$ interchanges local maxima and local minima).
answered Jan 5 at 4:53
jmerryjmerry
7,327920
answered Jan 5 at 4:53
jmerryjmerry
7,327920
answered Jan 5 at 4:53
jmerryjmerry
7,327920
answered Jan 5 at 4:53
jmerryjmerry
7,327920
7,327920
add a comment |
add a comment |
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$begingroup$
you are using the same name for the variable of $S$ and the integrand, but they are different
$endgroup$
– Masacroso
Jan 5 at 5:50