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Parametric Equations t-20123 x0-3-4-305 y-.50.511.5
Eliminating the Parameter 1) 2)
11.2 Slope and Concavity For the curve given by Find the slope and concavity at the point (2,3) At (2, 3) t = 4 and the slope is 8. The second derivative is positive so graph is concave up
Horizontal and Vertical tangents A horizontal tangent occurs when dy/dt = 0 but dx/dt 0. A vertical tangent occurs when dx/dt = 0 but dy/dt 0. Vertical tangents Horizontal tangent
Polar Coordinate Plane
Figure 9.37. Pole Polar axis Polar Coordinates
Polar/Rectangular Equivalences x 2 + y 2 = r 2 tan θ = y/x x = r cos θ y = r sin θ θ)
Figure 9.40(a-c). Symmetries
Figure 9.42(a-b). Graph r 2 = 4 cos θ
Figure 9.45. Finding points of intersection Third point does not show up. On r = 1-2 cos θ, point is (-1, 0) On r = 1, point is (1, π)
Slope of a polar curve Where x = r cos θ = f(θ) cos θ And y = r sin θ = f(θ) sin θ Horizontal tangent where dy/dθ = 0 and dx/dθ≠0 Vertical tangent where dx/dθ = 0 and dy/dθ≠0
Finding slopes and horizontal and vertical tangent lines For r = 1 – cos θ (a) Find the slope at θ = π/6 (b) Find horizontal tangents (c) Find vertical tangents
r = 1 – cos θ
Find Horizontal Tangents
Find Vertical Tangents Horizontal tangents at: Vertical tangents at:
Figure 9.47. Finding Tangent Lines at the pole r = 2 sin 3θ r = 2 sin 3θ = 0 3θ = 0, π, 2 π, 3 π θ = 0, π/3, 2 π/3, π
Figure 9.48. Area in the Plane
Figure 9.49. Area of region
Figure 9.51. Find Area of region inside smaller loop
Figure 9.52. Area between curves
Length of a Curve in Polar Coordinates Find the length of the arc for r = 2 – 2cosθ sin 2 A =(1-cos2A)/2 2 sin 2 A =1-cos2A 2 sin 2 (1/2θ) =1-cosθ
Polar Differentiation. Let r = f( θ ) and ( x,y) is the rectangular representation of the point having the polar representation ( r, θ ) Then x = f( θ.
Chapter 8 Plane Curves and Parametric Equations. Copyright © Houghton Mifflin Company. All rights reserved.8 | 2 Definition of a Plane Curve.
Warm Up Calculator Active The curve given can be described by the equation r = θ + sin(2θ) for 0 < θ < π, where r is measured in meters and θ is measured.
9.3: Calculus with Parametric Equations When a curve is defined parametrically, it is still necessary to find slopes of tangents, concavity, area, and.
Polar Equations and Graphs. 1. Transform each polar equation to an equation in rectangular coordinates. Then identify and graph the equation (Similar.
10 Conics, Parametric Equations, and Polar Coordinates
10.3 Polar Functions Quick Review 5.Find dy / dx. 6.Find the slope of the curve at t = 2. 7.Find the points on the curve where the slope is zero. 8.Find.
The Chain Rule Section 3.6c.
Derivatives of Parametric Equations
Copyright © Cengage Learning. All rights reserved.
10.2 – 10.3 Parametric Equations. There are times when we need to describe motion (or a curve) that is not a function. We can do this by writing equations.
Calculus with Polar Coordinates Ex. Find all points of intersection of r = 1 – 2cos θ and r = 1.
Conics, Parametric Equations, and Polar Coordinates Copyright © Cengage Learning. All rights reserved.
Chapter 10 – Parametric Equations & Polar Coordinates 10.2 Calculus with Parametric Curves 1Erickson.
Polar Coordinates Rectangular (Cartesian) coordinates plot a point by moving left/right and up/down (making a rectangle) Polar coordinates find the.
CHAPTER Continuity Arc Length Arc Length Formula: If a smooth curve with parametric equations x = f (t), y = g(t), a t b, is traversed exactly.
Chapter 9 Notes Honors Pre-Calculus.
Integration in polar coordinates involves finding not the area underneath a curve but, rather, the area of a sector bounded by a curve. Consider the region.
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