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Parametric Derivatives

Parametric Derivatives. Section 10-3-a. The Parametric form of a Derivative : If a smooth curve C is given by the equations Then the slope of C at (x, y) is . Derivatives of Parametric Equations. Horizontal: Vertical: . Tangents.

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Parametric Derivatives

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  1. Parametric Derivatives Section 10-3-a

  2. The Parametric form of a Derivative: If a smooth curve C is given by the equations Then the slope of C at (x, y) is Derivatives of Parametric Equations

  3. Horizontal: Vertical: Tangents

  4. Tangent lines: Because parametric equations are seldom functions, they may loop around and have more than one tangent line at a given point Tangent Lines

  5. 1) Find the derivative at t = 1, then find both the vertical and horizontal tangents to the curve

  6. 2nd Derivative: 3rd Derivative: Higher Order Derivatives

  7. 2) Find the 1st and 2nd derivative at t = 1, then find the equation of the tangent line(s)

  8. A particle’s position in the xy plane at time t is given by: Find a) the x-component of the particles velocity at t = 5 b) the derivative in terms of x c) the times at which the x and y components of the velocity are the same

  9. Properties of Derivatives apply to Graphs of Parametric Equations • Derivative gives us slope at any point • Increasing and Decreasing • Concavity • Extrema • Equations for tangent lines

  10. 4) Determine the time intervals on which the curve is concave down or concave up

  11. Home Work Page 727 # 1-4, 5-15 odd, 30-44 even

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