Calculus

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A particle moves along a line so that at any time t its position is given by x(t)=2(pi)t+cos(2(pi)t). Determine the particle's maximum velocity.

  • Calculus -

    max velocity occurs when the second derivative is zero

    so ...
    x(t) = 2πt + cos(2πt)
    x '(t) = 2π - 2π(sin 2πt)
    x ''(t) = 0 - 4π^2(cos 2πt)
    = 0 for a max/min of velocity
    cos 2πt = 0
    2πt = π/2 or 2πt = 3π/2
    t = 1/4 or t = 3/4

    if t= 1/4
    v(1/4) = x '(1/4) = 2π - 2π(sin π/2) = 2π - 2π = 0
    if t=3/4
    v(3/4) = 2π - 2π(sin 3π/2) = 2π + 2π = 4π

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