Posted by **Hannah** on Saturday, February 26, 2011 at 10:25pm.

An ant moves along the x-axis with velocity given by v(t)=tsin(t^2), t is greater than or equal to zero.

Given that x(t) is the position of the particle at time t and that x(0)=3, find x(2)

Find the total distance traveled by the particle from t=0 to t=2

- Calculus-IVP -
**MathMate**, Saturday, February 26, 2011 at 10:39pm
This is an initial value problem, where you would solve a differential equation (or an integral), and with a given set of initial values, determine the integration constant.

Let x(t)=position at time t

v(t)=tsin(t^2)

∫v(t)dt = ∫tsin(t²)dt

Integral of velocity gives distance

For the right-hand side, substitute u=t^2,

du=2tdt

x(t)=∫(1/2)sin(u)du

=-cos(u)/2+C'

=-cos(t^2)+C

Given x(0)=3

=>

x(0)=-cos(0)+C = 3

C=3+cos(0)=4

=>

x(t)=-cos(t^2)+4

x(2)=4-cos(2^2)=4-cos(4)

Distance travaelled = x(2)-x(0)=?

- Too many screen names -
**MathMate**, Saturday, February 26, 2011 at 10:52pm
Hannah, Anastasia, Carmen,Yuka, Lena, Catalina, Jamima, ...

I appreciate your inventiveness, but please, stop inventing screen names.

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