# BMCC MTH 253 - Analytic Geometry in Calculus (7 pages)

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## Analytic Geometry in Calculus

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## Analytic Geometry in Calculus

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Lecture Notes

Pages:
7
School:
Blue Mountain Community College
Course:
Mth 253 - CALCULUS
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MTH 253 Calculus Other Topics Chapter 11 Analytic Geometry in Calculus Section 11 2 Tangent Lines Arc Length for Parametric and Polar Curves Copyright 2006 by Ron Wallace all rights reserved Reminder Tangent Lines to a Cartesian Function at a Point y f x Find the point x0 f x0 Determine y f x f x0 x0 Evaluate f x0 m y f x0 f x0 x x0 Example x f t Slope at a Point of a Parametric Curve y g t Differentiate each equation giving dx f t dt dy g t dt Divide assuming dx dt 0 g t dy dt dy f t dx dt dx Therefore the slope can be determined without eliminating the parameter Slope at a Point of a Parametric Curve x f t y g t dy g t dx f t What is happening when dx dt i e f t is 0 Vertical tangent What is happening when dy dt i e g t is 0 Horizontal tangent What is happening when both are zero Singular Point Analyzed Case by Case Tangent Line to a Parametric Curve x f t Example Find the equation of the line tangent to the curve when t 2 y g t dy g t dx f t x 2sin t 2 y t 5 Pt 2sin t t 2 5 1 8 1 dy 2t m 4 8 dx 2 cos t y 1 4 8 x 1 8 4 8 x 7 64 When is the tangent line horizontal vertical Slope at a Point of a Polar Curve r f q From section 11 1 x r cos q y r sin q Essentially parametric equations parameter is so differentiating implicitly dr r cos q sin q dy dy dq dq dx dx dq r sin q dr cos q dq Tangent Lines to a Polar Curve Example Find the points where the tangent lines are horizontal or vertical dy 4 cos 2q cos q 8sin 2q sin q m dx 4 cos 2q sin q 8sin 2q cos q 2 4 cos q 6 cos q 5 2 4sin q 6sin q 5 horizontal tangents at 90 24 156 vertical tangents at 0 180 66 114 dr r cos q sin q dy dy dq dq dx dx dq r sin q dr cos q dq r f q r 4 cos 2q dr 8sin 2q dq

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