9.7 Defining Polar Coordinates and Differentiating in Polar Form

• The radius in polar coordinates represents the distance from the origin
• What formula is used to convert from polar to Cartesian coordinates for the x-coordinate?

  x = r \cos(\theta)</latex>
• How do you plot the polar coordinates (2, π/3)?

  Move 2 units at π/3
• Match the coordinate system with its components:

  Polar coordinates ↔️ (r, θ)

  Rectangular coordinates ↔️ (x, y)
• What formula is used to convert from rectangular to polar coordinates for the angle?

  $\theta =$$\arctan(\frac{y}{x})$
• In polar coordinates, the components are (r, θ)
• Polar coordinates use a distance and an angle, while rectangular coordinates use horizontal and vertical distances.


  True
• The angle in polar coordinates is measured from the positive x-axis.
• The term r'(θ) in the derivative formulas represents the derivative of r with respect to θ.


  True
• What is the formula for dy/dθ in polar coordinates?

  \frac{dy}{d\theta} = r'(\theta) \sin(\theta) + r(\theta) \cos(\theta)</latex>
• Polar coordinates use a distance called the radius and an angle to locate points on a plane.
• The formula to convert the x-coordinate from polar to Cartesian coordinates is x = r cos(θ).


  True
• The components of polar coordinates are (r, θ).
• What is the formula to calculate the radius when converting from rectangular to polar coordinates?

  r = √(x² + y²)
• Steps to convert the rectangular coordinates (3, 4) to polar coordinates:

  1️⃣ Calculate the radius: r = √(3² + 4²) = 5

  2️⃣ Calculate the angle: θ = arctan(4/3) ≈ 0.927

  3️⃣ Write the polar coordinates: (5, 0.927)
• The formula to convert from rectangular to polar coordinates is r = √(x² + y²) and θ = arctan(y/x).
• What is the formula for dx/dθ in polar coordinates?

  r'(θ) cos(θ) - r(θ) sin(θ)
• What is dx/dθ for the function r(θ) = 2θ?

  2 cos(θ) - 2θ sin(θ)
• The derivative formula for x in polar coordinates is dx/dθ = r'(θ) cos(θ) - r(θ) sin(θ)
• r'(θ) represents the derivative of r with respect to θ

  True
• Steps to find dy/dθ for r(θ) = 2θ

  1️⃣ Calculate r'(θ) = 2

  2️⃣ Substitute r'(θ) and r(θ) into the formula for dy/dθ

  3️⃣ Simplify the expression to dy/dθ = 2 sin(θ) + 2θ cos(θ)
• The slope of a curve in polar coordinates is given by dy/dx = (dy/dθ) / (dx/dθ), which involves the ratio of their derivatives
• For r(θ) = 2θ, the value of dx/dθ at θ = π/3 is 1 - √3
• Steps to find the slope of a polar curve

  1️⃣ Find r'(θ) of the polar equation r(θ)

  2️⃣ Substitute r(θ) and r'(θ) into the derivative formulas for dx/dθ and dy/dθ

  3️⃣ Compute the slope using dy/dx = (dy/dθ) / (dx/dθ)
• What two components are used in polar coordinates to locate points on a plane?

  Radius and angle
• The angle in polar coordinates is measured from the positive x-axis.


  True
• The formula to convert from polar to Cartesian coordinates for the y-coordinate is $y =$$r \sin(\theta)$
• Polar and rectangular coordinates are two ways to specify points on a plane.

  True
• The formula to convert from rectangular to polar coordinates for the radius is r = \sqrt{x^2 + y^2}</latex>
• Convert the rectangular coordinates (3, 4) to polar coordinates.

  1️⃣ Calculate the radius: r = √(3² + 4²) = 5

  2️⃣ Calculate the angle: θ = arctan(4/3) ≈ 0.927 radians

  3️⃣ The polar coordinates are approximately (5, 0.927)
• What does the radius (r) represent in polar coordinates?

  Distance from the origin
• Match the coordinate system with its conversion formula:

  Rectangular to Polar ↔️ $r =$ \sqrt{x^{2} + y^{2}} and $\theta =$$\arctan(\frac{y}{x})$

  Polar to Rectangular ↔️ $x =$$r \cos(\theta)$ and $y =$$r \sin(\theta)$
• What are the polar coordinates of the rectangular point (3, 4)?

  (5, 0.927)
• The formula for dx/dθ in polar coordinates is $\frac{dx}{d\theta} =$$r'(\theta) \cos(\theta) - r(\theta) \sin(\theta)$
• Polar coordinates use a distance and an angle to locate points on a plane.


  True
• What does the radius (r) represent in polar coordinates?

  Distance from the origin
• The angle in polar coordinates is measured from the positive x-axis.
• What is the angle in radians when plotting the polar coordinate (2, π/3)?

  π/3
• Match the coordinate system with its components:

  Polar ↔️ (r, θ)

  Rectangular ↔️ (x, y)
• The Great Wall of China is visible from the Moon with the naked eye.

  False