Polar Curves and Tangent Lines

This video tutorial explains how to find horizontal and vertical tangent lines to polar curves. It begins with a review of the formula for dy/dx in polar form and explains how to determine when tangent lines are horizontal or vertical. The video then demonstrates the process of finding horizontal tangent lines by setting dy/dtheta to zero and solving for theta. It continues with finding vertical tangent lines by setting dx/dtheta to zero. The tutorial emphasizes the importance of verifying results with a graph, especially at points where both dy/dtheta and dx/dtheta are zero, as no conclusion can be made about tangent lines at these points.

What condition must be met for a tangent line to be horizontal in a polar curve?

• dy/dx equals infinity

• dy/dx equals zero

• dx/d theta equals zero

• dy/d theta equals zero

In the context of polar curves, what does dx/d theta measure?

• The change in y with respect to theta

• The change in x with respect to theta

• The change in radius with respect to theta

• The change in angle with respect to radius

What is the derivative of the function r = 1 - sin(theta) with respect to theta?

• -sin(theta)

• cos(theta)

• -cos(theta)

• sin(theta)

At which angle does cosine(theta) equal zero?

• pi/2

• pi/6

• 3pi/2

• pi

Which of the following points does NOT have a horizontal tangent line?

• (1/2, pi/6)

• (1/2, 5pi/6)

• (0, pi/2)

• (2, 3pi/2)

What condition must be met for a tangent line to be vertical in a polar curve?

• dy/d theta equals zero

• dx/d theta equals zero

• dy/dx equals zero

• dy/dx equals infinity

What identity is used to replace cosine squared in terms of sine squared?

• 1 + cot^2(theta) = csc^2(theta)

• sin^2(theta) + cos^2(theta) = 1

• sin(theta) = cos(theta)

• tan^2(theta) + 1 = sec^2(theta)

At which angle is sine(theta) equal to negative 1/2?

• pi/6

• 11pi/6

• 5pi/6

• 7pi/6

Which point is expected to have a vertical tangent line?

• (0, pi/2)

• (2, 3pi/2)

• (3/2, 7pi/6)

• (1/2, pi/6)

Why is it important to verify points on a graph when finding tangent lines?

• To verify the accuracy of calculus techniques

• To confirm the points are on the curve

• To ensure the graph is drawn correctly

• To check for sharp points where no tangent exists