Understanding Polar Coordinates and Derivatives

The video tutorial explains a polar function r = 3θsinθ and its graph, which consists of two loops. It discusses why the graph has two loops and how the sine function affects the graph's shape. The tutorial then focuses on finding the rate of change of the x-coordinate with respect to θ using the product rule for derivatives. Finally, it evaluates the derivative at a specific point, confirming the result with an intuitive check.

What is the function r(θ) given in the video?

• r(θ) = 3θsin(θ)

• r(θ) = θsin(θ)

• r(θ) = 3θcos(θ)

• r(θ) = θcos(θ)

How many loops does the graph of r(θ) have in polar coordinates?

• One loop

• Two loops

• Three loops

• Four loops

Why does the second loop of the graph appear larger?

• Because r decreases

• Because θ increases

• Because θ decreases

• Because r increases

What happens to r when θ is between π and 2π?

• r becomes zero

• r remains constant

• r becomes negative

• r becomes positive

What is the task given at point P on the graph?

• Find the y-coordinate

• Find the x-coordinate

• Find the rate of change of the x-coordinate

• Find the rate of change of the y-coordinate

How is the x-coordinate expressed in terms of r and θ?

• x = r + cos(θ)

• x = r - sin(θ)

• x = r * sin(θ)

• x = r * cos(θ)

What is the expression for x(θ) derived in the video?

• x(θ) = 3θsin(θ)cos(θ)

• x(θ) = θsin(θ)cos(θ)

• x(θ) = θcos(θ)

• x(θ) = 3θcos(θ)

Which rule is applied to find the derivative of x with respect to θ?

• Quotient rule

• Chain rule

• Product rule

• Power rule

What is the value of x'(θ) at θ = π/2?

• π

• -3π/2

• 3π/2

• 0

Why does the rate of change of x with respect to θ make sense intuitively?

• Because x increases as θ increases

• Because x decreases as θ increases

• Because x becomes zero

• Because x remains constant