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Understanding Tangent Lines and Polar Curves without a Calculator

This guide explores the mathematical concepts of tangent lines and polar curves. It begins with parametric equations, describing how to derive the equation of a tangent line to a curve at a specific point. It then shifts focus to graphing polar curves without the use of a calculator, outlining generalizations for various scalar values. You'll learn about different curve types such as cardioids, limaçons, and rose curves, and how to identify their graphs and intersections. Enhance your understanding of these key topics in calculus!

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Understanding Tangent Lines and Polar Curves without a Calculator

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Presentation Transcript

  1. Warm Up No Calculator • A curve is described by the parametric equations • x = t2 + 2t, y = t3 + t2. An equation of the line tangent to the curve at the point where t = 1 is 3) A particle moves along the x-axis so that at any time t > 0 the acceleration of the particle is a(t) = e-2t . If at t = 0 the velocity of the particle is 5/2 and its position is 17/4, then its position at any time t > 0 is x(t) =

  2. Graphing Polar curves without a calculator

  3. Use your calculator to make generalizations…Graph various values of each scalar (a,b and n), then generalize. 1. a) r = a b) r = acos θ c) r = asin θ Generalizations that will help you graph each without a calculator:

  4. 2. a) r = a  b cos θ b) r = a  b sin θIf a = b, generalizations that will help you graph without a calculator… These are called “cardioids”

  5. 2. a) r = a  bcos θ b) r = a  bsin θ If a < b, generalizations that will help you graph without a calculator… These are called “limaçons”

  6. 2. a) r = a  bcos θ b) r = a  bsin θ If a > b, generalizations that will help you graph without a calculator… These are called “limaçons with an inner loop”

  7. 3. a) r = acos(nθ) b) r = asin(nθ) If n is an odd number, generalizations that will help you graph without a calculator… If n is an even number, generalizations that will help you graph without a calculator… These are called “rose curves”

  8. Where do the graphs intersect?

  9. Where do the graphs intersect?

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