# Chapter 10 parametric and polar curves 101 tangent

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Chapter 10: Parametric and Polar Curves 10.1 Tangent Lines and Arc Length for Parametric Curves 10.2 Polar Coordinates 10.3 Tangent Lines, Arc Length and Area for Polar Curves Example 10: r = sin θ + cos θ Figure: Graph of the function r = sin θ + cos θ Figure: Circle with polar equation: r = sin θ + cos θ Chapter 10 Lecture Notes MAT187H1F Lec0101 Burbulla Chapter 10: Parametric and Polar Curves 10.1 Tangent Lines and Arc Length for Parametric Curves 10.2 Polar Coordinates 10.3 Tangent Lines, Arc Length and Area for Polar Curves Example 10, Continued We can confirm that the polar curve of r = sin θ + cos θ is a circle, by finding its Cartesian equation. r = sin θ + cos θ r 2 = r sin θ + r cos θ x 2 + y 2 = y + x x 2 - x + 1 4 + y 2 - y + 1 4 = 1 4 + 1 4 x - 1 2 2 + y - 1 2 2 = 1 2 So the circle has centre 1 2 , 1 2 and radius 1 / 2 . Chapter 10 Lecture Notes MAT187H1F Lec0101 Burbulla
Chapter 10: Parametric and Polar Curves 10.1 Tangent Lines and Arc Length for Parametric Curves 10.2 Polar Coordinates 10.3 Tangent Lines, Arc Length and Area for Polar Curves Example 11: r = θ/ 25 Figure: Graph of the function r = θ/ 25 , θ 0 Figure: Archimedean spiral with polar equation: r = θ/ 25 , θ 0 Chapter 10 Lecture Notes MAT187H1F Lec0101 Burbulla Chapter 10: Parametric and Polar Curves 10.1 Tangent Lines and Arc Length for Parametric Curves 10.2 Polar Coordinates 10.3 Tangent Lines, Arc Length and Area for Polar Curves Example 12: r = e θ/ 25 Figure: Graph of the function r = e θ/ 25 Figure: Logarithmic spiral with polar equation: r = e θ/ 25 Chapter 10 Lecture Notes MAT187H1F Lec0101 Burbulla
Chapter 10: Parametric and Polar Curves 10.1 Tangent Lines and Arc Length for Parametric Curves 10.2 Polar Coordinates 10.3 Tangent Lines, Arc Length and Area for Polar Curves Derivatives for Polar Curves Since the parametric equations for a polar curve are x = r cos θ, y = r sin θ, the first derivative of a polar curve is dy dx = dy d θ dx d θ = sin θ dr d θ + r cos θ cos θ dr d θ - r sin θ . The formula for the second derivative is so messy we won’t even try to write it down! Chapter 10 Lecture Notes MAT187H1F Lec0101 Burbulla Chapter 10: Parametric and Polar Curves 10.1 Tangent Lines and Arc Length for Parametric Curves 10.2 Polar Coordinates 10.3 Tangent Lines, Arc Length and Area for Polar Curves Example 1 Find the slope of the tangent line to the circle with polar equation r = 4 cos θ at the point with θ = π/ 6 . dy dx = sin θ dr d θ + r cos θ cos θ dr d θ - r sin θ = - 4 sin 2 θ + 4 cos 2 θ - 4 cos θ sin θ - 4 cos θ sin θ = sin 2 θ - cos 2 θ 2 sin θ cos θ = - cot(2 θ ) So at θ = π/ 6 the slope is - cot( π/ 3) = - 1 / 3 . Chapter 10 Lecture Notes MAT187H1F Lec0101 Burbulla
Chapter 10: Parametric and Polar Curves 10.1 Tangent Lines and Arc Length for Parametric Curves 10.2 Polar Coordinates 10.3 Tangent Lines, Arc Length and Area for Polar Curves Example 2: Tangent Lines to Polar Curves at the Origin If r = 0 for θ = α, and dr / d θ = 0 at θ = α, then dy dx θ = α = sin α dr d θ θ = α + 0 cos α dr d θ θ = α - 0 = tan α. That is, the line θ = α is tangent to the curve at the origin. The figure to the right illustrates the example r = sin 3 θ for which the tangent lines at the origin have equations θ = 0 , π/ 3 or 2 π/ 3 . Chapter 10 Lecture Notes MAT187H1F Lec0101 Burbulla Chapter 10: Parametric and Polar Curves