Calculus II - Parametric Equations and Curves

Introduction to Parametric Equations

• Traditional forms: y = f(x) or x = h(y)
• Not all curves fit these forms, e.g., circles
• Parametric equations: define both x and y in terms of a third variable (parameter), often denoted t
  • x = f(t)
  • y = g(t)
• Parametric curve: Graph of parametric equations as t varies

Visualizing Parametric Curves

• Example: Sketching parametric curve for x = t² + t, y = 2t - 1
  • Pick values of t, calculate x and y, plot points
  • Issues arise in picking effective t values
  • Direction of motion indicated by increasing t

Parametric Curves with Limits

• Parametric curves can be affected by limits on the parameter t
• Without limits: curve continues indefinitely
• With limits: sketch only a portion

Eliminating the Parameter

• Convert parametric equations to a single algebraic equation involving x and y
• Example: Eliminate parameter for x = t² + t, y = 2t - 1
  • Solve for t in terms of y
  • Substitute into x equation
• Issues:
  • Does not show direction of motion
  • Curve may not be fully sketched

Example: Parametric Equations with Trigonometric Functions

• x = 5cos(t), y = 2sin(t), 0 ≤ t ≤ 2π
• Eliminating parameter results in an ellipse
• Curve covers full ellipse as x and y range are complete
  • Direction of motion determined by derivatives

Parametric Equations with Different Speeds

• x = 5cos(3t), y = 2sin(3t), 0 ≤ t ≤ 2π
• Same ellipse, but curve traced out multiple times
• Speed change due to coefficient in trigonometric argument
• Importance of determining direction of motion accurately

Partial Parametric Curves

• Parametric equations may not trace full algebraic curve
• Example: x = sin²(t), y = 2cos(t)
  • Eliminates to a parabola
  • Parametric equations impose limits, resulting in partial curve

Practical Uses and Parameterization

• Parametric equations often describe paths of particles/objects
• Important for converting functions to parametric form (parameterization)
• Ellipses and circles can be parameterized
• Parameterization useful in advanced calculus (e.g., Calculus III)

Conclusion

• Parametric equations are crucial for dealing with complex curves
• Sketching requires careful selection of t values and consideration of direction
• Eliminating parameters provides useful algebraic form but comes with limitations
• Parameterization allows flexibility in representing curves