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Understanding Slope and Tangent Lines in Polar Coordinates

This section explores how to determine the slope of tangent lines to a polar graph defined by the function r = f(θ). To find the slope in polar coordinates, we utilize the parametric equations x = r cos(θ) = f(θ) cos(θ) and y = r sin(θ) = f(θ) sin(θ). The discussion includes identifying horizontal and vertical tangents, as well as locating cusp points at the origin. Finally, it provides exercises to find tangents and equations for lines tangent to specific polar curves.

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Understanding Slope and Tangent Lines in Polar Coordinates

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  1. Polar Derivatives Section 10-4 continued

  2. Slope and Tangent Lines To find the slope of a tangent line to a polar graph, consider a differentiable function given by r = f(Ɵ). To find the slope in polar form, use the parametric equations x = r cosƟ= f(Ɵ) cosƟ and y= r sin Ɵ= f(Ɵ) sin Ɵ.

  3. Slope and Tangent Lines Using the parametric form of dy/dx we have

  4. Horizontal and Vertical Tangent Lines • Horizontal • Vertical Cusp at (0, 0)

  5. Tangent Lines at the Pole If then Then the line Is tangent to the pole to the graph of

  6. 10) Find the tangents for

  7. 11) Find the tangents for

  8. 12) Find the equation of the line tangent to the polar curve

  9. Home Work Page 739 # 63,64,65-83 odd

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