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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.
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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.
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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.
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Polar Coordinates Rectangular (Cartesian) coordinates plot a point by moving left/right and up/down (making a rectangle) Polar coordinates find the.
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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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Graphing Polar Graphs Calc AB- Section10.6A. Symmetry Tests for Polar Graphs 1.Symmetry about the x -axis: If the point lies on the graph, the point ________.
Paige McNaney, Luke Glaser, Freeman Judd. Vocabulary Polar Curves: 1. Cardioids 2. Limacons 3. Rose Curves Parametric Equations: 1. Parameter 2. Orientation.
Sec 5 Symmetry in polar coordinate. Definitions Symmetry about the Polar Axis The curve is symmetric about the polar axis ( the x-axis)if replacing the.
Clicker Question 1 If x = e 2t + 1 and y = 2t 2 + t, then what is y as a function of x ? – A. y = (1/2)(ln 2 (x – 1) + ln(x – 1)) – B. y = ln 2 (x – 1)
Parametric Equations. In a rectangular coordinate system, you will recall, a point in the plane is represented by an ordered pair of number (x,y), where.
Find the slope of the tangent line to the graph of f at the point ( - 1, 10 ). f ( x ) = 6 - 4x
AREA BOUNDED BY A POLAR CURVE. The area of the region bounded by a polar curve, r = f (θ ) and the lines θ = α and θ = β is given by: β α A = 1 2 r 2.
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Trigonometry Review Find sin( /4) = cos( /4) = tan( /4) = Find sin( /4) = cos( /4) = tan( /4) = csc( /4) = sec( /4) = cot( /4) = csc(
Derivatives - Equation of the Tangent Line Now that we can find the slope of the tangent line of a function at a given point, we need to find the equation.
Conics, Parametric Equations, and Polar Coordinates 10 Copyright © Cengage Learning. All rights reserved.
Chapter 8 – Polar Coordinates and Parametric Equations Graphs of Polar Equations1.
10.3 Polar Coordinates. Converting Polar to Rectangular Use the polar-rectangular conversion formulas to show that the polar graph of r = 4 sin.
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P OLAR E QUATIONS Section Polar Coordinates Given: r: Directed distance from the Polar axis (pole) to point P Ɵ: Directed angle from the Polar axis.
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