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Calculus 2 Curve Cheat Sheet

Calculus 2 Curve Cheat Sheet











Curve Cheat Sheet

Parametric & Polar Forms








Shape

Cartesian Equation

Parametric / Polar Equation

Key Characteristics







Line / Segment

$$ y = mx+b $$


Parametric: $$ x = a+bt, y = c+dt $$
Polar: $$ \theta = c $$


The parametric form defines a line segment if \(t\) is restricted.





Circle

$$ x^2 + y^2 = R^2 $$


Parametric: $$ x=R\cos t, y=R\sin t $$
Polar: $$ r = R $$
Polar: $$ r = a\cos\theta $$


Standard parametric form traces counter-clockwise. In polar, \(r=a\cos\theta\) is on the x-axis.





Ellipse

$$ \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 $$


Parametric: $$ x=a\cos t, y=b\sin t $$


A stretched circle where horizontal radius is \(a\) and vertical radius is \(b\).





Parabola

$$ y = Ax^2+Bx+C $$


Parametric: $$ x = t, y = At^2+Bt+C $$
Parametric (Trig): $$ x=\tan t, y=\sec^2 t $$


Often formed by one linear and one quadratic parametric equation.





Cardioid

(Complex)


Polar: $$ r = a \pm a\sin\theta $$
Polar: $$ r = a \pm a\cos\theta $$


Heart-shaped curve. Key is that the coefficients are equal (e.g., \(r=2+2\cos\theta\)).





Rose Curve

(Complex)


Polar: $$ r=a\cos(n\theta) $$
Polar: $$ r=a\sin(n\theta) $$


If \(n\) is odd, there are \(n\) petals. If \(n\) is even, there are \(2n\) petals.