That's odd - Solution

Graphs of Odd Functions with their Derivatives

y= t

y=t and its derivative. t is an odd function and its derivative is even.

From the image it appears that the derivative is an even function because if the derivative graph is reflected in the y axis it results in a copy of itself.

y=sin(t)

From the image it appears that the derivative is an even function.

y=t^3

t^3 and its derivative. t cubed is an odd function and its derivative is even.

y=t^5

t^5 and its derivative. t^5 is an odd function and its derivative is even.

y=sinh(t) using sinh(t)=0.5(e^t-e^(-1t))

sinh(t), using sinh(t)=0.5*(e^t-e^(-1*t)), and its derivative. sinh(t) is an odd function and its derivative is even

The Maclaurin series for a general function, f(t), defined around t=0 and with all its derivatives defined at t=0 is given by:

The Maclaurin series for a general function, defined around t=0 and with all its derivatives defined at t=0

If a function is odd then it will only have odd powers of t in its Maclaurin series, i.e. f(0) and f''(0) and f⁽ⁿ⁾(0) for n even, must all be 0.

Differentiating above we get:

The power series for the derivative of an odd function. We see that the power series for the derivative consists of only a constant term and even powers of t and therefore is even

We see that the power series for the derivative consists of only a constant term and even powers of t and therefore is even.

plotXpose app is available on Google Play

Google Play and the Google Play logo are trademarks of Google LLC.
Versions will shortly be available for iOS and Windows.

plotXpose app is a companion to the book Mathematics for Electrical Engineering and Computing by Mary Attenborough, published by Newnes, 2003.