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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.

Derivative - Basic

         

If \(x = t^3\) and \(y = 3t^5\), where \(t\) is any real number, what is the derivative of \(y\) with respect to \(x\) at \(x = 216\)?

What is the equation of the tangent line to the curve \[x = 14 \sin t, y = 5\ \sin(t + \sin t) \] at \((0,0)\)?

If \(x = \ln t\) and \(y = 3 t^3 + 9 \ln t\) what is \(\displaystyle{\frac{d y}{d x}}?\)

If \(x=t+\frac{1}{t}\) and \(y=t^2+\frac{1}{t^2}\), what is the derivative of \(y\) with respect to \(x\) at \(x=21\)?

At time \(t\), the coordinates of a point \(P\) are given by \( (4 \cos t, 2 \sin t) \). At time \(t=\frac{\pi}{4}\), the magnitudes of velocity and acceleration are \(m\) and \(n\), respectively. What is \(m^2+n^2\)?

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