Understanding Polar Functions and Tangent Lines

The video tutorial explains how to find the slope of the tangent line to a polar function r = 5 - 3sin(θ) at θ = 5π/6. It begins with visualizing the angle and tangent line on the polar graph. The instructor provides hints for converting polar to rectangular coordinates and calculating derivatives using the product rule. Finally, the slope is determined by evaluating the derivatives at the given angle, resulting in a slope of √3/4.

What is the polar function discussed in the video?

• r = 5 + 3 sin(θ)

• r = 5 sin(θ) - 3

• r = 5 - 3 sin(θ)

• r = 3 - 5 sin(θ)

At what angle θ is the slope of the tangent line being calculated?

• 2π/3

• 5π/6

• π/4

• π/3

Which formula represents the conversion from polar to rectangular coordinates for x?

• x = r sin(θ)

• x = r cos(θ)

• x = θ cos(r)

• x = θ sin(r)

What is the expression for dy/dx in terms of derivatives with respect to θ?

• dy/dx = r / θ

• dy/dx = dy/dθ / dx/dθ

• dy/dx = dx/dθ / dy/dθ

• dy/dx = θ / r

What rule is used to derive dy/dθ?

• Quotient Rule

• Product Rule

• Chain Rule

• Sum Rule

What is the expression for x in terms of r and θ?

• x = θ cos(r)

• x = r sin(θ)

• x = θ sin(r)

• x = r cos(θ)

What is the value of sin(5π/6)?

• -√3/2

• √3/2

• 1/2

• -1/2

What is the value of cos(5π/6)?

• √3/2

• 1/2

• -√3/2

• -1/2

What is the final slope of the tangent line at θ = 5π/6?

• √3/4

• 1/4

• -√3/4

• -1/4

What is the significance of the slope of the tangent line in this context?

• It represents the angle of the polar function.

• It determines the maximum value of r.

• It shows the rate of change of r with respect to θ.

• It indicates the rate of change of y with respect to x.