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Differentiation in Polar Coordinates

This lesson reviews the relationship between polar and rectangular coordinate systems, introducing the formulas x = r cos(θ) and y = r sin(θ). It explains how to find the derivative dy/dx in polar coordinates, using the fact that r = f(θ). By applying these concepts to specific examples, such as finding the slope at a given point on the curve r = cos(3θ), students will learn how to evaluate derivatives in polar form. The lesson includes practice exercises on page 443 to reinforce understanding.

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Differentiation in Polar Coordinates

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  1. Differentiation in Polar Coordinates Lesson 10.7

  2. Review • Relationship of polar and rectangular systems • x = r cos θ y = r sin θ • Given r = f(θ), simple to find dr/dθ • However, we seek dy/dx

  3. Finding dy/dx • We know • r = f(θ) and y = r sin θ and x = r cos θ • Then • And

  4. Finding dy/dx • Since • Then

  5. Example • Given r = cos 3θ • Find the slope of the line tangent at (1/2, π/9) • dy/dx = ? • Evaluate •

  6. Define for Calculator • It is possible to define this derivative as a function on your calculator

  7. Try This! • Find where the tangent line is horizontal for r = 2 cos θ • Find dy/dx • Set equal to 0, solve for θ

  8. Assignment • Lesson 10.7 • Page 443 • Exercises 1 – 21 odd

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