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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)=?

  • Too many screen names -

    Hannah, Anastasia, Carmen,Yuka, Lena, Catalina, Jamima, ...
    I appreciate your inventiveness, but please, stop inventing screen names.

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