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AP Calculus BC: Polar Coordinates

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Section 1

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Slope Formula

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Calculus Math

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Mar 1, 2020

Cards (27)

Section 1

(27 cards)

Slope Formula

Front

dy/dx=r'sin(θ)+rcos(θ)/r'cos(θ)-rsin(θ)

Back

Graph: Limacon

Front

r=a+bsin(θ) a<b

Back

Area of One Curve

Front

Back

What is the meaning of dr/dθ (r')?

Front

Curve is moving towards the origin (if negative) Curve is moving away from origin (if possible)

Back

dr/dt =

Front

dr/dθ (r') × dθ/dt

Back

Area with One Curve Inside and One Curve Outside

Front

- 2 integrals - Same limits - 1st Integral is the one with the area inside - 2nd Integral is the one with the area outside

Back

What is the meaning of dx/dθ (bottom of slope equation)?

Front

Moving right (if positive) Moving left (if negative)

Back

Graph: Rose Curve with Sine (Even)

Front

r=sin(nθ) 2n = # of petals y-axis symmetry

Back

Find θ given an x or y coordinate

Front

1. Remember x = rcosθ and y = rsinθ (use respectively depending on which coordinate is given) 2. Plug in r (should be given) 3. Set expression equal to x/y coordinate 4. Put expression into y₁ in function mode 5. Put x/y coordinate into y₂ in function mode 6. Find the point of intersection

Back

Graph: Circle

Front

r= (a constant, that constant is the radius of the circle)

Back

Graph: Circle Shifted right/left

Front

r=ccos(θ) c = constant + = right - = left

Back

Graph: Lemniscate

Front

r²=a²cos(2θ)

Back

Graph: Rose Curve with Cosine (Even)

Front

r=cos(nθ) 2n = # of petals x-axis symmetry

Back

How do you find max distance?

Front

(*Refer to 2005 AP Calculus BC Free Response Question 2 Part D) 1. Solve for r' (aka dr/dθ) 2. Set r' equal to 0 and solve 3. Make a t-chart with limits (interval) and value found from step 2 4. Plug θ values into r equation Highest value is the maximum

Back

Graph: Cardioid with Cosine

Front

r=a+bcos(θ) a=b

Back

y =

Front

rsin(θ)

Back

Graph: Rose Curve with Cosine (Odd)

Front

r=cos(nθ) n = # of petals x-axis symmetry

Back

Graph: Rose Curve with Sine (Odd)

Front

r=sin(nθ) n = # of petals y-axis symmetry

Back

How do you find dr/dθ or dy/dx in the calculator?

Front

dy/dx: 2nd → trace → 2 → type θ value (bar shows up at the bottom as you type it) → enter dr/dθ: 2nd → trace → 3 → type θ value (bar shows up at the bottom as you type it) → enter

Back

x =

Front

rcos(θ)

Back

Graph: Line

Front

θ= # (in radians)

Back

Graph: Circle Shifted up/down

Front

r=csin(θ) c = constant + = up - = down

Back

What is the meaning of dy/dθ (top of slope equation)?

Front

Moving up (if positive) Moving down (if negative)

Back

Graph: Cardioid with Sine

Front

r=a+bsin(θ) a=b

Back

Vertical Tangent

Front

dx/dθ = 0 dy/dθ ≠ 0

Back

Horizontal Tangent

Front

dy/dθ = 0 dx/dθ ≠0

Back

Area Inside Both Curves

Front

- Think Perimeter! c (½∫r² dθ + ½∫r² dθ) c = constant (you may have to multiply accordingly) *Limits aren't shown, but it would be from a to b in the first integral, and from b to c in the second integral

Back