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Calculus of Parametric Equations

Parametric equations can be quite handy, and we don't want to unravel them just to do Calculus. Some tricks can bend traditional derivative and integral methods to apply to 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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