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Calculus

Calculus of Parametric Equations

Parametric Equations - Derivative

         

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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