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Calculus in Polar Coordinates quiz

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  • What is the general form for expressing a point in polar coordinates?
    A point in polar coordinates is given as (r, θ), where r is the radial distance from the origin and θ is the angle from the positive x-axis.
  • How do you convert a polar curve r = f(θ) into parametric equations for x and y?
    Set x = f(θ) cos(θ) and y = f(θ) sin(θ).
  • What formula is used to find the slope of the tangent line to a polar curve?
    The slope is given by dy/dx = (dy/dθ) / (dx/dθ).
  • When differentiating y = f(θ) sin(θ), what rule do you use?
    Use the product rule: derivative of the first times the second plus the first times the derivative of the second.
  • What is the derivative of y = 2 sin²(θ) with respect to θ?
    The derivative is 4 sin(θ) cos(θ), which can also be written as 2 sin(2θ).
  • How do you find dx/dθ if x = 2 sin(θ) cos(θ)?
    dx/dθ = 2 cos(2θ), using the double angle identity for sine.
  • What trigonometric identity allows you to simplify 2 sin(θ) cos(θ)?
    2 sin(θ) cos(θ) = sin(2θ).
  • If dy/dθ = 2 sin(2θ) and dx/dθ = 2 cos(2θ), what is dy/dx?
    dy/dx = tan(2θ).
  • What is the slope of the tangent line to r = 2 sin(θ) at θ = π/6?
    The slope is tan(π/3), which equals √3.
  • What is the formula for the area of a region bounded by a polar curve r = f(θ) from θ = α to θ = β?
    A = (1/2) ∫[α to β] r² dθ.
  • How do you set up the integral to find the area of one petal of the rose r = 4 sin(2θ)?
    Set up the integral as (1/2) ∫[0 to π/2] (4 sin(2θ))² dθ.
  • What is the value of (4 sin(2θ))²?
    It is 16 sin²(2θ).
  • What trigonometric identity is used to integrate sin²(2θ)?
    sin²(2θ) = (1 - cos(4θ))/2.
  • After applying the identity and simplifying, what does the area integral for one petal of r = 4 sin(2θ) become?
    It becomes 4 ∫[0 to π/2] (1 - cos(4θ)) dθ.
  • What is the area of one petal of the rose curve r = 4 sin(2θ)?
    The area is 2π.