# Calculus

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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 -

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)=?

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