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Calculus

Calculus of Parametric Equations

Parametric Equations - Velocity and Acceleration

         

Suppose the position of point \(P=(x(t), y(t))\) at time \(t\) is given by \[ \left ( 5{t}^2+4, -5{t}^3+4t \right ) .\] What is the magnitude of acceleration of \(P\) at time \( t=5 ?\)

Suppose the position of point \(P=(x, y)\) at time \(t\) is given by \( \left ( 2t, -2{t}^2+4t \right ).\) What is the magnitude of the velocity of \(P\) at time \( t=8 ?\)

Suppose the position of a particle \(P\) at time \(t\) is given by \[\left ( -8{e}^t\cos t + 2 , 8{e}^t\sin t + 6 \right ).\] What is the angle \( \alpha \) (\(0 < \alpha < \pi\)) between the \(x\)-axis and the velocity vector \(\vec{v}\) of \(P\) at time \(t= \frac{\pi}{2}?\)

Leaving from the origin at the same time, point \(P\) moves at a rate of \(5\) cm per second in the positive direction of the \(x\)-axis, while point \(Q\) moves at a rate of \(10\) cm per second in the positive direction of the \(y\)-axis. What is the velocity vector \( \vec{v}=\left ( \frac{dx}{dt} , \frac{dy}{dt} \right )\) of the intersection point between the line \( \overline {PQ} \) and the line \(y=3x \)?

Suppose the position of point \(P=(x(t), y(t))\) at time \(t\) is given by \[ \left ( 4t-10\sin t, 10\cos t + 10 \right ) .\] What is the maximum speed attained by point \(P?\)

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