Dashawn Robbins

2022-05-03

Antiderivative of periodic function

I practise before the real analysis exam and I got stuck on few excercises.

"if F is an antiderivative of a continuous periodic function and F is bounded, then F is periodic"

$\text{}\to \text{}$ one should give a proof or counterexample.

(I'm convinced it's true, but how to prove it?)

I practise before the real analysis exam and I got stuck on few excercises.

"if F is an antiderivative of a continuous periodic function and F is bounded, then F is periodic"

$\text{}\to \text{}$ one should give a proof or counterexample.

(I'm convinced it's true, but how to prove it?)

Kendrick Fritz

Beginner2022-05-04Added 12 answers

Step 1

What is special about $1+\mathrm{sin}(x)$ that gives it the following property?

$\underset{R\to \mathrm{\infty}}{lim}{\int}_{0}^{R}1+\mathrm{sin}(x)\text{}dx=\mathrm{\infty}$

Step 2

If this integral was to have a maximum, what points would you look for first?

Further, $\frac{d}{dR}{\int}_{0}^{R}f(x)dx=f(R)$

What is special about $1+\mathrm{sin}(x)$ that gives it the following property?

$\underset{R\to \mathrm{\infty}}{lim}{\int}_{0}^{R}1+\mathrm{sin}(x)\text{}dx=\mathrm{\infty}$

Step 2

If this integral was to have a maximum, what points would you look for first?

Further, $\frac{d}{dR}{\int}_{0}^{R}f(x)dx=f(R)$

Klanglinkmgk

Beginner2022-05-05Added 13 answers

Step 1

Start with the definition of the antiderivative(s) of f:

$F(x)={\int}_{0}^{x}f(t)dt+C$

where C is a constant. Since f is periodic, with period $T>0$, then

$F(2T)={\int}_{0}^{2T}f(t)dt+C={\int}_{0}^{T}f(t)dt+{\int}_{T}^{2T}f(t)dt+C=2{\int}_{0}^{T}f(t)dt+C$

$F(3T)={\int}_{0}^{3T}f(t)dt+C=F(2T)+{\int}_{T}^{2T}f(t)dt=3{\int}_{0}^{T}f(t)dt+C$

Step 2

Can you generalize this to find an expression for F(kT) for any integer k (for example by induction)? What does this tell you about ${\int}_{0}^{T}f(t)dt$ given the conditions F have to satisfy?

Finally what is the value of $F(x+T)-F(x)={\int}_{x}^{x+T}f(t)dt={\int}_{x}^{T}f(t)dt+{\int}_{T}^{x+T}f(t)dt$ given the result found above?

Start with the definition of the antiderivative(s) of f:

$F(x)={\int}_{0}^{x}f(t)dt+C$

where C is a constant. Since f is periodic, with period $T>0$, then

$F(2T)={\int}_{0}^{2T}f(t)dt+C={\int}_{0}^{T}f(t)dt+{\int}_{T}^{2T}f(t)dt+C=2{\int}_{0}^{T}f(t)dt+C$

$F(3T)={\int}_{0}^{3T}f(t)dt+C=F(2T)+{\int}_{T}^{2T}f(t)dt=3{\int}_{0}^{T}f(t)dt+C$

Step 2

Can you generalize this to find an expression for F(kT) for any integer k (for example by induction)? What does this tell you about ${\int}_{0}^{T}f(t)dt$ given the conditions F have to satisfy?

Finally what is the value of $F(x+T)-F(x)={\int}_{x}^{x+T}f(t)dt={\int}_{x}^{T}f(t)dt+{\int}_{T}^{x+T}f(t)dt$ given the result found above?

Find the local maximum and minimum values and saddle points of the function. If you have three-dimensional graphing software, graph the function with a domain and viewpoint that reveal all the important aspects of the function

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