# What is the instantaneous velocity of an object moving in accordance to  f(t)= (t^2sin(t-pi),tcost)  at  t=pi/3 ?

Jan 2, 2016

$0.172261$

#### Explanation:

The instantaneous velocity is equal to $f ' \left(\frac{\pi}{3}\right)$.

$x \left(t\right) = {t}^{2} \sin \left(t - \pi\right)$

To find $x ' \left(t\right)$, use the product rule.

$x ' \left(t\right) = 2 t \sin \left(t - \pi\right) + {t}^{2} \cos \left(t - \pi\right)$

We also know that

$y \left(t\right) = t \cos t$

Again, differentiate with the product rule.

$y ' \left(t\right) = \cos t - t \sin t$

The derivative of the entire parametric equation is found as follows:

$f ' \left(t\right) = \frac{y ' \left(t\right)}{x ' \left(t\right)} = \frac{\cos t - t \sin t}{2 t \sin \left(t - \pi\right) + {t}^{2} \cos \left(t - \pi\right)}$

Find $f ' \left(\frac{\pi}{3}\right)$.

$f ' \left(\frac{\pi}{3}\right) = \frac{\cos \left(\frac{\pi}{3}\right) - \frac{\pi}{3} \sin \left(\frac{\pi}{3}\right)}{2 \left(\frac{\pi}{3}\right) \sin \left(\frac{\pi}{3} - \pi\right) + {\left(\frac{\pi}{3}\right)}^{2} \cos \left(\frac{\pi}{3} - \pi\right)}$

$\approx 0.172261$