MathIsimple

Parametric & Polar Curves

Master parametric equations and polar coordinates. Learn to find derivatives, calculate arc lengths, convert between coordinate systems, and analyze beautiful curves like cardioids and rose curves.

12th Grade
Calculus
~90 min
🎮 Interactive Activity: Parametric Derivative Calculator

Find dy/dx for this parametric curve!

Parametric equations:

x(t) = t²
y(t) = 2t

At t = 1

🎮 Interactive Activity: Polar Coordinate Converter

Convert from rectangular to polar coordinates!

Rectangular coordinates:

(x, y) = (3, 4)

Find polar coordinates (r, θ):

1. Parametric Equations

Defining Curves with Parameters

Parametric equations define a curve using a parameter t, giving separate equations for x and y. This allows us to describe curves that cannot be expressed as y = f(x).

General Form

Parametric equations: x = f(t), y = g(t)

Parameter: t (often representing time or angle)

Example: x = t², y = t³ describes a curve in the plane

Advantage: Can represent curves with vertical tangents or loops

Example: Circle

Parametric form: x = r cos t, y = r sin t, 0 ≤ t ≤ 2π

Eliminating parameter: x² + y² = r² (standard circle equation)

Application: Motion along a circular path

2. Derivatives of Parametric Curves
3. Polar Coordinates
4. Coordinate Conversions
5. Arc Length
6. Area in Polar Coordinates
7. Applications
Frequently Asked Questions

Practice Time!

Practice Quiz
10
Questions
0
Correct
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Score
1
For parametric equations x = f(t), y = g(t), what is dy/dx?
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2
What are the polar coordinates of the point (3, 4)?
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3
What is the polar equation r = a(1 + cos θ)?
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4
How do you convert from polar to rectangular coordinates?
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5
What is the arc length formula for a parametric curve x = f(t), y = g(t)?
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6
What is the polar equation r = a cos(nθ)?
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7
For the parametric curve x = t², y = t³, what is dy/dx at t = 2?
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8
What is the relationship between rectangular and polar coordinates?
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9
What is the area enclosed by a polar curve r = f(θ)?
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10
What is a parametric curve?
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