Understanding Tangent Lines and Derivatives

The video tutorial explains how to find points on a curve where the tangent line is vertical. It begins by discussing the concept of tangent lines and their slopes, particularly focusing on vertical lines where the slope is undefined. The tutorial then presents the equation of the curve and its derivative, and demonstrates how to solve for points where the tangent is vertical by setting the derivative of x with respect to y to zero. Finally, it finds the specific point on the curve where the tangent line is vertical, using substitution and solving the equations.

What is the slope of a horizontal line?

• One

• Zero

• Undefined

• Infinity

How is the slope of a vertical line often described?

• Undefined

• Negative

• One

• Zero

What happens to the derivative of y with respect to x when the tangent line is vertical?

• It becomes zero

• It becomes undefined

• It becomes negative

• It becomes positive

What is the equation of the curve given in the video?

• y^2 - x = 2

• y^3 - xy = 2

• x^2 + y = 2

• x^3 - y = 2

What is the derivative of y with respect to x for the given curve?

• x over y

• y over 3y^2 - x

• 3y^2 - x over y

• x over 3y^2

What condition must be met for the derivative of x with respect to y to be zero?

• The numerator must be zero

• The denominator must be zero

• Both numerator and denominator must be zero

• Neither numerator nor denominator can be zero

What is the relationship between x and y when the tangent line is vertical?

• x = y^2

• x = 3y^2

• x = y^3

• x = 2y^2

What is the value of y when the tangent line is vertical?

• 1

• -1

• 2

• 0

What is the value of x when y is -1?

• 1

• 3

• 2

• 4

What is the point on the curve where the tangent line is vertical?

• (-3, -1)

• (3, 1)

• (-3, 1)

• (3, -1)