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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.

Tangent

Consider the polar curve \( r\theta = 6 \). What is the slope of the tangent line of the polar curve at \( \theta = \pi \)?

Given the curve \[x=1-2t, y=5-4t+2t^2,\] what is the equation of the tangent line to the curve at \(x=5 ?\)

For the curve given by \[x=\frac{9t}{1+t^2}, y=\frac{1-t^2}{1+t^2},\] what is the equation of the tangent line to the curve for \(t=2 ?\)

Consider the polar curve \( r = 22 \sin \theta \). What is the slope of the tangent line at \( \theta = \frac{\pi}{6} \)?

A curve is defined by \((x,y) =\left(\cos^3 \theta, \sin^3 \theta \right)\). If the equation for the tangent line at \(\theta=\frac{2}{3}\pi\) is \(y=ax+b\), what is the value of \( a \times b \)?

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