Is the given statement is true or false? Justify your answer.

Question

If f is periodic, then f' is periodic.

Answer 1

The statement is True

if f is periodic, then f' is periodic or cyclical. they repeat. Also the periods are equal.

take a periodic example such as f = sinx = sin(x+npi) = sin(x+180n) where n = any integer

f' = cosx = sin(x + pi/2) = sin(x+npi/2) = sin(x + 90n)

both are periodic, just a phase shift of pi/2 or 90 degrees

y= sinx = cos(x +3pi/2)

any trigonometric function is periodic. sinx, cosx, tanx, cotx, secx, cscx. All their derivatives are also cyclical.

all can be rewritten as functions of sinx and cosx. tanx = sinx/cosx, cotx = cosx/sinx, secx =1/cosx, cscx =1/sinx

Their 1st, 2nd, 3rd and nth derivatives are also cyclical, as they can also be rewritten as functions of sinx or cosx which are cyclical

graph a unit circle, the function values repeat every 2pi or 360 degrees. "cyclical" "cycle" and "circle" are from the same root words. pi has a definition related to a circle, pi = the ratio of the circumference to the diameter, C=pid. pi=C/d

f(x) = e^ix is also cyclical with a period of 2pi. e^ix = cosx + isinx = sum of 2 cyclical trigonometric functions

e^ipi = -1 = e^3ipi = e^5ipi = e^7ipi = e^nip where n= any odd integer

f'(x) = ie^ix = -sinx + icosx which again is the sum of 2 cyclical trigonometric functions

f"(x)= -e^ix =-cosx -isinx

3rd derivative of f(x) = -ie^ix = sinx-icosx

4th derivative of f(x) = e^ix = cosx +isinx = f(x) which is cyclical

a piece wise linear function can be cyclical

f(x) = x/n if n-1<x<n which is a series of discontinuous 45 degree lines of distance sqr2, each slope = 1

f'(x) = 1 but discontinuous when x=an integer,

Answer 2

Hi Muhammad A.,

Think of this problem graphically. If f(x) is periodic, it means that you can shift a segment of it laterally by any multiple of the period length, and it will superpose (coincide). So since f'(x) is totally dependent on f(x), it can also be shifted, therefore it is also periodic.

However, please note that the converse statement is NOT always true, which in logic means it is NOT TRUE as a statement (a true statement must be true in every case!). Example: the f'(x) function = 1+sin(x) . This is a periodic function; however the associated f(x) looks like a cosine wave climbing along the line y=x , so it does NOT repeat when shifted laterally (shifting vertically is not permitted as an allowed operation within the definition of "periodic"). The essential reason for this is that the differentiation process has no arbitrary constants generated, but the integration process DOES have (in general) arbitrary constant(s) generated. I say "in general", because integrating y=0 doesn't result in any area being obtained: however, there is still a hidden constant involved, it is just multiplied by zero, hence it appears to disappear!

-- Cheers, --Mr. d.