# Thread: linear approximation and the derivative...

1. ## linear approximation and the derivative...

Writing g for the acceleration due to gravity, the period, T, of a pendulum of length is given by the following equation.

(a) Show that if the length of the pendulum changes by , the change in the period, T, is given by the following equation. (Do this on paper. Your teacher may ask you to turn in this work.)
(b) If the length of the pendulum increases by 4%, by what percent does the period change?

ok i just need to be directed in the right path....need some help on how and where to start for this problem...any help is greatly appreciated...

mathaction

2. Originally Posted by mathaction
Writing g for the acceleration due to gravity, the period, T, of a pendulum of length is given by the following equation.

(a) Show that if the length of the pendulum changes by , the change in the period, T, is given by the following equation. (Do this on paper. Your teacher may ask you to turn in this work.)
(b) If the length of the pendulum increases by 4%, by what percent does the period change?

ok i just need to be directed in the right path....need some help on how and where to start for this problem...any help is greatly appreciated...

mathaction
Taylor's theorem to a first order approximation says that
$T(l + \Delta l) \approx T(l) + T^{\prime}(0) \Delta l$
where the derivative is taken with respect to l. This only works if $\Delta l$ is small. (Typically that means that $\Delta l$ is only a few percent of the size of l.)

Work this out and then define $\Delta T = T(l + \Delta l) - T(l)$.

-Dan