Understanding Horizontal Tangents in Polar Coordinates

The video tutorial explains how to find the smallest positive value of theta for which the tangent line to a polar curve is horizontal. It covers the use of polar coordinates, the application of the product rule to find derivatives, and solving equations to determine theta. The tutorial concludes with calculating the correct angle in the second quadrant using a calculator.

What is the main objective of the problem discussed in the video?

• To find the largest angle for a vertical tangent line

• To determine the smallest positive theta for a horizontal tangent line

• To calculate the area under the polar curve

• To find the intersection points of two polar curves

In polar coordinates, what is the expression for x in terms of r and theta?

• x = r sin(theta)

• x = r

• x = r cos(theta)

• x = r tan(theta)

Which mathematical rule is applied to find dy/dtheta in polar coordinates?

• Power rule

• Chain rule

• Quotient rule

• Product rule

What is the derivative of the function r = 2e^(0.3theta) with respect to theta?

• e^(0.3theta)

• 0.3e^(0.3theta)

• 0.6e^(0.3theta)

• 2e^(0.3theta)

What condition must be met for a fraction to be zero?

• Numerator is zero

• Both numerator and denominator are zero

• Fraction is undefined

• Denominator is zero

What trigonometric identity is used to simplify the equation for horizontal tangents?

• sin(theta) = cos(theta)

• tan(theta) = cos(theta)/sin(theta)

• tan(theta) = sin(theta)/cos(theta)

• cos(theta) = sin(theta)/tan(theta)

What is the result of dividing -2 by 0.6 in the context of the problem?

• -2.6667

• -0.3333

• -1.6667

• -3.3333

Which function is used to find the angle theta from the tangent value?

• Exponential function

• Cosine function

• Inverse tangent function

• Sine function

Why is it necessary to add pi to the angle found using the inverse tangent function?

• To find the angle in the second quadrant

• To convert the angle to degrees

• To ensure the angle is positive

• To adjust for a vertical tangent

What is the smallest positive value for theta where the tangent line is horizontal?

• 3.1416 radians

• 1.8623 radians

• 1.2793 radians

• 0.8623 radians