# differentiation

• Jun 10th 2010, 12:57 PM
hunty
differentiation
A vibration is identified as having the sinusoidal wave form equation of: -

s=0.55sin(3t/10-0.12)

i) Using calculus determine, and prove, the (first) times at which the displacement is a MAXIMUM and also a MINIMUM to 6 dec places.

ii) Determine to 6 decimal places the value of the maximum and minimum displacements at these times.

if you can just tell me what the method is it would be great
• Jun 10th 2010, 01:15 PM
e^(i*pi)
Quote:

Originally Posted by hunty
A vibration is identified as having the sinusoidal wave form equation of: -

s=0.55sin(3t/10-0.12)

i) Using calculus determine, and prove, the (first) times at which the displacement is a MAXIMUM and also a MINIMUM to 6 dec places.

ii) Determine to 6 decimal places the value of the maximum and minimum displacements at these times.

if you can just tell me what the method is it would be great

Use the chain rule to find $\displaystyle s'(t)$

To find turning points set $\displaystyle s'(t)=0$.

To establish whether it is a maximum or a minimum find $\displaystyle s''(t)$ using the values given from $\displaystyle s'(t) =0$.

If $\displaystyle s''(t) < 0$ you have a maximum and if $\displaystyle s''(t) >0$ it is a minimym
• Jun 10th 2010, 01:30 PM
hunty
sorry what is the chain rule?
• Jun 10th 2010, 01:34 PM
e^(i*pi)
Quote:

Originally Posted by hunty
sorry what is the chain rule?

Given that y = f[g(x)] you must differentiate according to the chain rule which says that $\displaystyle y'= f'[g(x)] \cdot g'(x)$

In practical terms it means differentiating the function inside the brackets and multiplying by the derivative of the function as if there was just x in the brackets.

For example $\displaystyle \frac{d}{dx} \cos(ax) = -a\sin(ax)$ and $\displaystyle \frac{d^2y}{dx^2} = -a^2\cos(ax)$