11.4 Graphing in Polar Coordinates Polar Symmetries

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Δορκάς Λιακόπουλος
6 χρόνια πριν
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1 .4 Graphing in Polar Coordinates Polar Symmetries x axis symmetry y axis symmetry origin symmetry r, θ = r, θ r, θ = r, θ r, θ = r, + θ

2 .4 Graphing in Polar Coordinates Polar Symmetries x axis symmetry y axis symmetry origin symmetry r, θ = r, θ r, θ = r, θ r, θ = r, + θ Find the symmetries r = cosθ 4 r 3 = 3. 5 r 3 = 3. 5 r 3 = 3. 5 r 3 = 3. 5 r 3 = 4. 5 r + 3 = 4. 5 x axis sym. no y axis sym. no origin sym.

3 .4 Graphing in Polar Coordinates Polar Symmetries x axis symmetry y axis symmetry origin symmetry r, θ = r, θ r, θ = r, θ r, θ = r, + θ Find the symmetries r = 6 sinθ r 6 = 2 r 6 = 2 r 6 = 2 r 6 = 2 r 6 = 2 r + 6 = 2 no x axis sym. y axis sym. no origin sym.

4 .4 Graphing in Polar Coordinates Slope of a Tangent for r = f θ m = dx dx = dθ dx dθ x = rcosθ x = f θ cosθ y = rsinθ y = f θ sinθ dθ = f θ sinθ + f θ cosθ dx dθ = f θ cosθ f θ sinθ dx = dθ dx dθ = f θ sinθ + f θ cosθ f θ cosθ f θ sinθ dx = f θ sinθ + f θ cosθ f θ cosθ f θ sinθ dx = r sinθ + rcosθ r cosθ rsinθ

5 .4 Graphing in Polar Coordinates Slope of a Tangent for r = f θ dx = f θ sinθ + f θ cosθ f θ cosθ f θ sinθ dx = r sinθ + rcosθ r cosθ rsinθ When r = f θ passes through the pole,r = and the slope of the tangent is: dx = f θ sinθ + f θ cosθ f θ cosθ f θ sinθ = f θ sinθ f θ cosθ = sinθ cosθ = tanθ

6 .4 Graphing in Polar Coordinates Slope of a Tangent for r = f θ dx = f θ sinθ + f θ cosθ f θ cosθ f θ sinθ dx = r sinθ + rcosθ r cosθ rsinθ Given the polar equation, find the polar coordinates and the slope of the tangent at θ = 4. r = sin(2θ) dx = ()sin 4 + ()cos r = 2cos(2θ) 4 ()cos 4 ()sin 4 r = sin r =, r = 2cos r = 2 4 dx = At, 4 = m =

7 .4 Graphing in Polar Coordinates Polar Equations Basic Equations: r = a θ = θ A circle with its center at the pole. A line through the pole at the angle θ. Examples: r = 3 θ = 6

8 Plotting Polar Curves.4 Graphing in Polar Coordinates Polar Equations (r, θ) Example: r = 4sinθ θ = r = θ = 6 r = 2 θ = 3 r = θ = 2 r = 4

9 .4 Graphing in Polar Coordinates Plotting Polar Curves Polar Equations (r, θ) Example: r = 2 cosθ θ r = und. θ = 6 r = θ = 3 r = 4 θ = 2 r = 2 θ = r =

10 .4 Graphing in Polar Coordinates Special Polar Curves r = a + bcosθ r = a + bcosθ r = a + bcosθ r = bcosθ a > b a = b a < b Limacon w/ a dimple Cardioid Limacon w/ a loop Circle

11 r = aθ r = b aθ.4 Graphing in Polar Coordinates Special Polar Curves r = acos(nθ) Spiral out Spiral in Roses r 2 = acos(2θ) r = 2 a acosθ Lemniscate Parabola

12 .5 Area and Lengths in Polar Coordinates Area of a Polar Curve circle area = r 2 r area of wedge = r 2 θ 2 area of wedge = 2 r2 θ area of a polar curve = α β 2 r2 dθ area of a polar curve = α β 2 f(θ)2 dθ

13 .5 Area and Lengths in Polar Coordinates Area of a Polar Curve r = 2 + 2cosθ area of a polar curve = Cardioid θ (2 + 2cosθ)2 dθ 2 (2( + cosθ))2 dθ 2( + cosθ) 2 dθ α β 2 r2 dθ 2 (2 + 2cosθ)2 dθ

14 α β 2 r2 dθ r = 5cos (3θ).5 Area and Lengths in Polar Coordinates Area of a Polar Curve Area of entire region 2 (5cos (3θ))2 dθ Area of a one petal (5cos (3θ))2 dθ Rose w/3 petals θ

15 α β 2 r2 dθ r = 5cos (3θ).5 Area and Lengths in Polar Coordinates Area of a Polar Curve Area of a one petal r = 5cos (3θ) = 5cos (3θ) 2 2 (5cos (3θ))2 dθ 6 3θ = 2, 3 2, 5 2, 7 2, θ = 6, 2, 5 6, 7 6, (5cos (3θ))2 dθ 2 Rose w/3 petals θ (5cos(3θ))2 dθ 6.545

16 α β 2 r2 dθ r = 3 4sinθ Limacon w/ a loop θ 2.5 Area and Lengths in Polar Coordinates Area of a Polar Curve Area of the inner loop r = 3 4sin θ = 3 4sinθ θ =.848 θ =.848 θ = (3 4sinθ)2 dθ (3 4sinθ)2 dθ.848

17 .5 Area and Lengths in Polar Coordinates

18 .5 Area and Lengths in Polar Coordinates Area of a Polar Curve Area of the region outside r = 2 and inside r = 4sin r = 2 r = 4sinθ Need Pts. Of Intersection 4sinθ = 2 sinθ = 2 θ = 6, 5 6 θ =.524 θ = 2.68 α β r 2 2 r 2 dθ sinθ dθ.524

19 .5 Area and Lengths in Polar Coordinates Calculate the area of the region outside r = 2 + 2sinθ, inside r = 2 + 2cosθ, and in the first quadrant. Points Of Intersection 2 + 2sinθ = 2 + 2cosθ 2sinθ = 2cosθ tanθ = θ = 4 =.785 α β 2 r 2 2 r 2 dθ cosθ sinθ 2 dθ 2.657

20 .5 Area and Lengths in Polar Coordinates Arc Length of a Polar Curve

21 .5 Area and Lengths in Polar Coordinates Find the arc length of the polar curve between the given angle interval. r = 5sinθ θ β L = r 2 + dr dθ α 2 dθ L = 25(sin 2 θ + cos 2 θ) dθ L = dr dθ = 5cosθ 5sinθ 2 + 5cosθ 2 dθ L = 5 dθ L = 5 θ L = 25sin 2 θ + 25cos 2 θ dθ L = 5 = 5.78

22 .5 Area and Lengths in Polar Coordinates Find the arc length of the polar curve between the given angle interval. r = e θ θ β L = r 2 + dr dθ α dr dθ = eθ 2 dθ L = e θ 2 + e θ 2 dθ L = e 2θ + e 2θ dθ L = 2e 2θ dθ L = 2 e θ dθ L = 2e θ L = 2e 2e L = 3.32

23 r = 2cosθ +.5 Area and Lengths in Polar Coordinates Find the arc length of half of the inside loop θ = r = 2cos + = 3 θ = r = 2cos + = = 2cosθ + θ = 2 3, 4 3 β L = r 2 + dr dθ α 2 dθ L = 4cos 2 θ + 4cosθ + + 4sin 2 θ dθ 2 3 L = 4cosθ + 4cos 2 θ + 4sin 2 θ + dθ 2 3 L = 4cosθ dθ 2 3 L = 4cosθ + 5 dθ 2 3 dr dθ = 2sinθ L = 2cosθ sinθ 2 dθ L =

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